An Analytical Formula for the Transition Density of a Conic Combination of Independent Squared Bessel Processes with Time-Dependent Dimensions and Financial Applications

Abstract

The squared Bessel process plays a central role in stochastic analysis, with broad applications in mathematical finance, physics, and probability theory. While explicit expressions for its transition probability density function (TPDF) under constant parameters are well known, analytical results in the case of time-dependent dimensions remain scarce. In this paper, we address a significantly challenging problem by deriving an analytical formula for the TPDF of a conic combination of independent squared Bessel processes with time-dependent dimensions. The result is expressed in terms of a Laguerre series expansion. Furthermore, we obtain closed-form expressions for the conditional moments of such conic combinations, represented via generalized hypergeometric functions. These results also yield new analytical formulas for the TPDF and conditional moments of both squared Bessel processes and Bessel processes with time-dependent dimensions. The proposed formulas provide a unified analytical framework for modeling and computation involving a broad class of time-inhomogeneous diffusion processes. The accuracy and computational efficiency of our formulas are verified through Monte Carlo simulations. As a practical application, we provide an analytical valuation of an interest rate swap, where the underlying short rate follows a conic combination of independent squared Bessel processes with time-dependent dimensions, thereby illustrating the theoretical and practical significance of our results in mathematical finance.

1. Introduction

The squared Bessel process is a fundamental continuous-time stochastic process with extensive applications in mathematical finance, physics, and probability theory. It is particularly valued for modeling non-negative quantities, such as variances in stochastic volatility models, short-term interest rates in term structure models, and radial diffusion in multidimensional spaces (see, e.g., [1,2,3,4,5]). Analytical results for its transition probability density function (TPDF) and conditional moments under constant parameters are well-established; see, for example, the classical references Revuz and Yor [6] and Karatzas and Shreve [1].

However, the analytical study of squared Bessel processes under time-dependent parameters, particularly time-varying dimensions, remains relatively underdeveloped (see, e.g., [7]), despite its relevance in modeling non-stationary systems in real-world applications. In recent years, time-inhomogeneous stochastic processes have gained increasing attention due to their flexibility in capturing dynamic structural changes in underlying systems.

For instance, in finance, the behavior of volatility and interest rates is influenced by market regimes, macroeconomic shocks, and policy changes—contexts where time-homogeneous models often fail (see, e.g., [8,9,10,11]). In physics and engineering, non-uniform or anisotropic media often give rise to stochastic dynamics with time-varying coefficients (see, e.g., [5,12]). In light of this, a surge of recent works has aimed to understand time-inhomogeneous stochastic processes, including deep learning approaches for solving forward and backward Kolmogorov equations associated with TPDFs; see, for example, Su, Tretyakov, and Newton [13], Al-Aradi et al. [14], and Haji-Ali and Sprungk [15]. Other efforts rely on exact simulations and multilevel Monte Carlo (MC) schemes for path-dependent stochastic differential equations (see [16,17,18]).

Despite these advances, analytical formulas for the TPDF and conditional moments of squared Bessel processes with time-dependent dimensions remain scarce. A key challenge lies in understanding the distribution of conic combinations of independent noncentral chi-square random variables—a classical yet unsolved problem with implications across statistics, wireless communications, and physics. Recent progress in this direction has been made by Rujivan et al. [19], who utilized a Laguerre series expansion for the probability density function (PDF) of such combinations to derive the conditional moments.

This paper contributes to the literature by addressing the analytical gap through the following key innovations:

(1)

We derive closed-form formulas for the TPDF and conditional moments of a conic combination of independent squared Bessel processes with time-dependent dimensions.

(2)

The resulting expressions are formulated using Laguerre polynomials and generalized hypergeometric functions, providing a unified analytical framework that accommodates both squared Bessel and Bessel processes in non-stationary settings.

(3)

The proposed formulas recover classical results as special cases when the dimension is constant, thereby extending the existing theory to encompass time-inhomogeneous systems.

(4)

We illustrate the practical utility of the framework by analytically valuing interest rate swaps under the assumption that the underlying short rate follows a conic combination of squared Bessel processes with time-dependent dimensions.

(5)

We perform extensive MC simulations to validate the accuracy, robustness, and computational efficiency of the derived formulas under various specifications of time-dependent dimensions.

To the best of our knowledge, this is the first work to provide fully tractable analytical expressions for both the TPDF and conditional moments of such a general class of processes while also establishing their convergence and computational advantages through theoretical and empirical evaluations.

The remainder of this paper is organized as follows: Section 2 presents the essential background and preliminaries, including the definition of the squared Bessel process and the review of its properties under both constant and time-dependent dimensions. The section also introduces the PDF of a conic combination of independent noncentral chi-square random variables as a key analytical tool. Section 3 derives the TPDF of a conic combination of independent squared Bessel processes with time-dependent dimensions and further specializes the results to obtain new analytical formulas for the TPDF and conditional moments of both squared Bessel and Bessel processes, along with rigorous truncation error analyses. Section 4 demonstrates the practical relevance of the proposed framework through an application for the analytical valuation of an interest rate swap. Section 5 presents comprehensive numerical experiments using MC simulations to validate the accuracy and computational efficiency of the derived formulas. Finally, Section 6 concludes the paper.

2. Preliminaries and Background

The Bessel process and its squared counterpart, the squared Bessel process, are essential stochastic processes with extensive applications in mathematical finance, physics, and probability theory. These processes are defined on a probability space and are characterized by the time-dependent dimension parameter $\delta \left(t\right)$, which introduces non-stationary dynamics. This section provides rigorous definitions of these processes, highlights their key properties, and derives explicit TPDFs in the special case where $\delta \left(t\right)$ is constant. Additionally, this section reviews a theorem that establishes an explicit formula for the PDF of a conic combination of independent noncentral chi-square random variables. This result is foundational and will be applied in subsequent sections to derive key findings in this paper.

2.1. Definition and Dynamics

Let $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$ be a complete probability space with the filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$ satisfying the usual conditions. The stochastic processes considered in this work are adapted to this filtration, and the expectation $\mathbb{E}[·]$ is computed with respect to the probability measure $\mathbb{P}$, unless otherwise stated.

The Bessel process and the squared Bessel process are closely related and generalize the radial part of Brownian motion in Euclidean space. Both processes are characterized by the time-dependent dimension parameter $\delta \left(t\right)$, which governs their drift terms and drives their evolution.

2.1.1. Bessel Processes

The Bessel process ${\left(R_t\right)}_{t\ge 0}$ with the time-dependent dimension $\delta \left(t\right)$ is defined as the unique strong solution to the stochastic differential equation (SDE):

$$d{R}_t={\displaystyle \frac{\delta \left(t\right)-1}{2R_t}}dt+d{W}_t,R_0=r_0>0.$$

(1)

In this equation, $\delta \left(t\right)>0$ is a continuous, time-dependent parameter referred to as the dimension of the process. The term ${\displaystyle \frac{\delta \left(t\right)-1}{2R_t}}dt$ represents the drift, which depends inversely on $R_t$ and is scaled by $\delta \left(t\right)-1$. The term $d{W}_t$ is driven by ${\left(W_t\right)}_{t\ge 0}$, a standard one-dimensional Brownian motion adapted to the filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$.

The Bessel process ${\left(R_t\right)}_{t\ge 0}$ describes the radial distance of a Brownian motion in a dynamic dimension space characterized by the time-varying parameter $\delta \left(t\right)$. This process is widely studied in the literature (e.g., [1,6]) and provides a flexible framework for modeling systems where the spatial dimension evolves over time. It has significant applications in finance, physics, and probability theory, particularly for modeling non-stationary systems.

2.1.2. Squared Bessel Processes

The squared Bessel process ${\left(X_t\right)}_{t\ge 0}$ is defined as $X_t=R_t^2$, where ${\left(R_t\right)}_{t\ge 0}$ is a Bessel process. It satisfies the SDE:

$$d{X}_t=\delta \left(t\right)dt+2\sqrt{X_t}d{W}_t,X_0=x_0>0,$$

(2)

where the time-dependent parameter $\delta \left(t\right)$ governs the drift term. The dynamics of ${\left(X_t\right)}_{t\ge 0}$ ensure that the process remains non-negative for all $t\ge 0$, making it particularly suitable for modeling quantities that must remain non-negative.

The squared Bessel process has extensive applications across various disciplines. In mathematical finance, it is fundamental to stochastic volatility models, such as the Cox–Ingersoll–Ross (CIR) model, where it describes the evolution of variances. The process ensures the realistic modeling of non-negative quantities like variance and interest rates [20,21]. In physics, the squared Bessel process models radial diffusion in $\delta \left(t\right)$-dimensional spaces, applicable in studying heat diffusion in cylindrical geometries and constrained Brownian motion [1,3,4,7]. Similarly, in risk management, it models aggregate claims or liabilities that evolve over time but cannot become negative [22]. The flexibility of the squared Bessel process to adapt to non-stationary behaviors through the time-dependent dimension parameter $\delta \left(t\right)$ makes it invaluable for modeling dynamic systems across finance, physics, biology, and operation research.

2.2. Key Properties

The Bessel process and the squared Bessel process exhibit several fundamental properties critical to their analysis and applications.

2.2.1. Non-Negativity

The squared Bessel process ${\left(X_t\right)}_{t\ge 0}$ remains non-negative for all $t\ge 0$ if $X_0\ge 0$. This follows from the positive drift term $\delta \left(t\right)dt$ and the vanishing volatility term $2\sqrt{X_t}d{W}_t$ as $X_t\to 0$. Similarly, the Bessel process ${\left(R_t\right)}_{t\ge 0}$, as the square root of ${\left(X_t\right)}_{t\ge 0}$, is strictly positive for $R_0>0$ [20,21].

2.2.2. Boundary Behavior

The behavior of the Bessel process at the boundary $R_t=0$ depends on $\delta \left(t\right)$. If $\delta \left(t\right)\ge 2$ for all $t\ge 0$, the process almost never hits zero. However, if $0<\delta \left(t\right)<2$ for some t, the process hits zero in finite time with a positive probability [20,21]. The squared Bessel process reflects this behavior since $X_t=R_t^2$.

2.3. The TPDF of the Squared Bessel Process with a Constant Dimension

When the dimension parameter is constant, $\delta \left(t\right)=\delta >0$; the TPDFs of the Bessel process and the squared Bessel process admit explicit closed-form formulas. These results are foundational and serve as a benchmark for generalizations involving time-dependent $\delta \left(t\right)$.

For the squared Bessel process ${\left(X_t\right)}_{t\ge 0}$ with a constant dimension of $\delta >0$, the TPDF is given by

$$f_{X_t}(x,t|x_0,t_0)={\displaystyle \frac{e^{-\frac{x+x_0}{2(t-t_0)}}}{2(t-t_0)}}{\left({\displaystyle \frac{x}{x_0}}\right)}^{{\displaystyle \frac{\delta }{4}}-{\displaystyle \frac{1}{2}}}{\mathit{I}}_{{\displaystyle \frac{\delta }{2}}-1}\left({\displaystyle \frac{\sqrt{x{x}_0}}{t-t_0}}\right),x,x_0>0,0\le t_0\le t.$$

(3)

where $I_{\nu }\left(z\right)$ denotes the modified Bessel function of the first kind of order $\nu$. This formula describes the probability density of $X_t$ at time t, given that $X_{t_0}=x_0$. The term $I_{{\displaystyle \frac{\delta }{2}}-1}$ reflects the dependency on the dimension parameter $\delta$, while the exponential and power terms ensure proper decay and scaling of the density.

The explicit TPDFs for the Bessel and squared Bessel processes with constant dimensions are fundamental for understanding their behavior, providing closed-form solutions that facilitate the computation of probabilities and expectations and serving as benchmarks for numerical approximations [1,6].

However, when the dimension parameter $\delta \left(t\right)$ becomes time-dependent, deriving explicit TPDFs is significantly more challenging due to the non-stationarity introduced by the time-varying drift term $\delta \left(t\right)dt$. This disrupts the mathematical structure underlying classical methods, such as eigenfunction expansions and the separation of variables, and leads to complex, often non-separable partial differential equations.

Additionally, the functional dependence of $\delta \left(t\right)$ introduces extra stochastic dependencies, requiring novel approaches to extend solutions that typically rely on special functions like modified Bessel functions. Addressing these challenges is a key focus of this work, which develops new frameworks for deriving TPDFs under time-dependent dimensions.

2.4. The PDF of a Conic Combination of Independent Noncentral Chi-Square Random Variables

Let $Y_n$ be a random variable defined as

$$Y_n:=\sum _{i=1}^n{\alpha }_i{X}_i,$$

(4)

where $n\ge 2$ is an integer, ${\alpha }_i>0$, and $X_i\sim {\chi }_{{\nu }_i}^2\left({\delta }_i\right)$ are independent random variables. Each $X_i$ follows a noncentral chi-square distribution with ${\nu }_i>0$ degrees of freedom and noncentrality parameter ${\delta }_i\ge 0$ for $i=1,\dots ,n$.

The PDF of a conic combination of independent noncentral chi-square random variables has been widely studied due to its relevance in statistics, physics, and engineering (see, e.g., [23,24,25,26,27]). These studies have led to various representations of the PDF over the past several decades. In this work, we utilize the recent results established by Rujivan et al. [19], summarized in the following theorem, as a key tool for deriving explicit formulas referenced in Section 1.

Proposition 1. 

Let $\nu :={\sum }_{i=1}^n{\nu }_i$. The PDF of $Y_n$ can be expressed as

$$f_{Y_n}\left(y\right)=f_{Y_n}^{\left(\beta \right)}\left(y\right):={\displaystyle \frac{e^{-\frac{y}{2\beta }}{y}^{\frac{\nu }{2}-1}}{{\left(2\beta \right)}^{\frac{\nu }{2}}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\nu }{2}+k\right)}}{c}_k{\mathit{L}}_k^{\left({\displaystyle \frac{\nu }{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right),$$

(5)

for $y>0$, and the parameter $\beta >0$ is chosen arbitrarily. In this expression, $\mathsf{\Gamma }\left(x\right)$ is the gamma function, defined as $\mathsf{\Gamma }\left(x\right):={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt$. The function ${\mathit{L}}_k^{\left(\eta \right)}\left(x\right)$ is the generalized Laguerre polynomial of parameter η and fractional order k, defined as ${\mathit{L}}_k^{\left(\eta \right)}\left(x\right):={\sum }_{r=0}^k{\displaystyle \frac{{(-1)}^r(\eta +k)!x^r}{(k-r)!(\eta +r)!r!}}.$

Furthermore, the coefficients $c_k$ for $k=0,1,2,\dots$ are determined by the recurrence relations:

$$c_0=1,$$

(6)

and

$$c_k={\displaystyle \frac{1}{k}}\sum _{j=0}^{k-1}{c}_j{d}_{k-j},$$

(7)

for $k\ge 1$. The auxiliary terms $d_j$ are defined as

$$d_1=-{\displaystyle \frac{1}{2\beta }}\sum _{i=1}^n{\delta }_i{\alpha }_i+{\displaystyle \frac{1}{2}}\sum _{i=1}^n{\nu }_i\left(1-{\displaystyle \frac{{\alpha }_i}{\beta }}\right),$$

(8)

and for $j\ge 2$,

$$d_j=-{\displaystyle \frac{j}{2}}{\left({\displaystyle \frac{1}{\beta }}\right)}^j\sum _{i=1}^n{\delta }_i{\alpha }_i{\left(\beta -{\alpha }_i\right)}^{j-1}+{\displaystyle \frac{1}{2}}\sum _{i=1}^n{\nu }_i{\left(1-{\displaystyle \frac{{\alpha }_i}{\beta }}\right)}^j.$$

(9)

Proof. 

See [19,23,28].    □

The parameter $\beta$ in Equation (5) plays a crucial role in determining both the PDF and the coefficients $c_k$. Although $\beta$ can be chosen arbitrarily, its value strongly impacts the convergence rate of the infinite series in Equation (5). In particular, if the coefficients $c_k$ grow rapidly, the series may converge slowly, thereby compromising the reliability of numerical approximations. Based on the error analysis provided by Rujivan et al. [19], this paper recommends selecting $\beta$ such that $\beta >{\displaystyle \frac{1}{2}}{max}_{i\in \{1,\dots ,n\}}{\alpha }_i$ to guarantee the convergence of the infinite series in Equation (5).

3. A Conic Combination of Independent Squared Bessel Processes with Time-Dependent Dimensions

Let $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$ be a probability space equipped with the filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$ as introduced in Section 2.1. Fix a positive integer $m\ge 1$, and consider the random variable

$$Y_t^{\left(m\right)}:=\sum _{j=1}^m{\omega }_j{X}_t^{\left(j\right)},t\ge 0,$$

(10)

where each process ${\left(X_t^{\left(j\right)}\right)}_{t\ge 0}$, for $j=1,\dots ,m$, satisfies the SDE:

$$d{X}_t^{\left(j\right)}={\delta }_j\left(t\right)dt+2\sqrt{X_t^{\left(j\right)}}d{W}_t^{\left(j\right)},t>0,$$

(11)

with time-dependent dimension ${\delta }_j\left(t\right)$ and initial condition $X_{t_0}^{\left(j\right)}=x_0^{\left(j\right)}>0$ for $t_0\ge 0$. The processes ${\left(W_t^{\left(j\right)}\right)}_{t\ge 0}$, $j=1,\dots ,m$ are independent standard Brownian motions, and the weights ${\omega }_j>0$ are distinct real numbers.

We refer to $Y_t^{\left(m\right)}$ as a conic combination of independent squared Bessel processes with time-dependent dimensions. This formulation generalizes classical constructions and captures a wide class of non-stationary and heterogeneous stochastic dynamics.

In this section, we build upon the analytical frameworks developed by Peng and Schellhorn [29] and Rujivan et al. [19] to derive an analytical expression for the TPDF of $Y_t^{\left(m\right)}$, as defined in Equation (10), using Laguerre series expansions. In addition, we employ this result to obtain analytical formulas for the conditional moments of $Y_t^{\left(m\right)}$ in terms of generalized hypergeometric functions. The detailed derivations are provided in the subsequent subsections.

3.1. An Analytical Formula for the TPDF of a Conic Combination of Independent Squared Bessel Processes with Time-Dependent Dimensions

3.1.1. Derivation of the Analytical Formula

To derive the TPDF of the conic combination $Y_t^{\left(m\right)}$ defined in Equation (10), we begin by imposing regularity assumptions on each individual squared Bessel process $X_t$ governed by the SDE (2).

Assumption 1. 

The time-dependent dimension parameter $\delta \left(t\right)$ satisfies $\delta \left(t\right)\ge 2$ for all $t\ge 0$.

Assumption 2. 

The first derivative of $\delta \left(t\right)$ with respect to time, denoted by ${\delta }^{\left(1\right)}\left(t\right)$, exists and satisfies $0\le {\delta }^{\left(1\right)}\left(t\right)<\infty$ for all $t\ge 0$.

Under these assumptions and following the analytical approach developed by Peng and Schellhorn [29], we demonstrate that the process $X_t$ admits a distributional representation as a convergent sum of weighted independent (noncentral) chi-square random variables, as formalized in the following proposition.

Proposition 2. 

Let Assumptions 1 and 2 be satisfied, and define $\tau (t,t_0):=t-t_0$ for $0\le t_0\le t<\infty$. Suppose that $X_t$ is the solution to SDE (2) with initial condition $X_{t_0}=x_0>0$. Then, for $t>t_0$, the random variable $X_t$ admits the distributional representation

$$X_t\stackrel{\mathit{law}}{=}\underset{n\to \infty }{lim}\sum _{i=1}^n{\widehat{\alpha }}_i{\widehat{X}}_i,$$

(12)

where ${\left\{{\widehat{X}}_i\right\}}_{i=1}^n$ are independent random variables such that

$${\widehat{X}}_i\sim {\chi }_{{\widehat{\nu }}_i}^2\left({\widehat{\delta }}_i\right);$$

(13)

i.e., each ${\widehat{X}}_i$ follows a (central or noncentral) chi-square distribution with degrees of freedom ${\widehat{\nu }}_i$ and noncentrality parameter ${\widehat{\delta }}_i$.

The coefficients ${\widehat{\alpha }}_i$ and the distributional parameters ${\widehat{\nu }}_i$ and ${\widehat{\delta }}_i$ depend on t, $t_0$, and $x_0$, and the dimension $\delta \left(t\right)$ is explicitly given by

$${\widehat{\alpha }}_i=\tau \left(t,t_0+(i-1){\displaystyle \frac{t-t_0}{n}}\right),i=1,\dots ,n,$$

(14)

$${\widehat{\nu }}_1=\delta \left(t_0\right),{\widehat{\delta }}_1={\displaystyle \frac{x_0}{\tau (t,t_0)}},$$

(15)

and for $i=2,\dots ,n$,

$${\widehat{\nu }}_i={\delta }^{\left(1\right)}\left(t_0+(i-1){\displaystyle \frac{t-t_0}{n}}\right)·{\displaystyle \frac{t-t_0}{n}},{\widehat{\delta }}_i=0.$$

(16)

Proof. 

The result follows directly from Theorem 3.1 in Peng and Schellhorn [29], which establishes the distributional equivalence in Equation (12) for a squared Bessel process with a time-dependent dimension. Hence, the proof is omitted here for brevity.    □

Rujivan et al. [19] built upon the foundational results by Peng and Schellhorn [29] to derive the exact TPDF of the extended Cox–Ingersoll–Ross (ECIR) process. In contrast, one of the principal contributions of the present paper lies in significantly extending this framework to a more general setting. Specifically, we go beyond the ECIR process by considering a conic combination of independent squared Bessel processes with time-dependent dimensions. By leveraging the analytical tools developed in Propositions 1 and 2, we derive a closed-form expression for the TPDF of the random variable $Y_t^{\left(m\right)}$, which represents a broader and more complex class of stochastic models. This generalization not only advances the theoretical landscape but also enhances practical applicability, particularly in modeling time-inhomogeneous dynamics in interest rate derivatives.

Theorem 1. 

Let $Y_t^{\left(m\right)}$ be the random variable defined in Equation (10). Suppose that for each $j=1,\dots ,m$, the squared Bessel process ${\left(X_t^{\left(j\right)}\right)}_{t\ge 0}$ governed by the SDE (11) satisfies Assumptions 1 and 2. Then, the TPDF of $Y_t^{\left(m\right)}$ is given by

$$f_{Y_t^{\left(m\right)}}^{\left(\beta \right)}(y,t\mid {\mathit{x}}_{t_0},t_0)={\displaystyle \frac{e^{-\frac{y}{2\beta }}{y}^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2\beta \right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right),$$

(17)

for $y>0$ and $t>t_0$, where ${\mathit{x}}_{t_0}=(x_0^{\left(1\right)},\dots ,x_0^{\left(m\right)})$, $\delta \left(t\right):={\sum }_{j=1}^m{\delta }_j\left(t\right)$, $\beta >0$ is an arbitrary positive constant, and $L_k^{\left(\eta \right)}$ is the generalized Laguerre polynomial as defined in Proposition 1. The coefficients ${\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})$ are defined recursively by

$$\begin{array}{cc}\hfill {\widehat{b}}_0^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})& =1,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})& ={\displaystyle \frac{1}{k}}\sum _{l=0}^{k-1}{\widehat{b}}_l^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){\widehat{d}}_{k-l}^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}),k\ge 1.\hfill \end{array}$$

(19)

The auxiliary terms ${\widehat{d}}_l^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})$ are given explicitly by

$$\begin{array}{cc}\hfill {\widehat{d}}_1^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})& =-{\displaystyle \frac{1}{2\beta }}\sum _{j=1}^m{\omega }_j{x}_0^{\left(j\right)}+{\displaystyle \frac{1}{2}}\sum _{j=1}^m{\delta }_j\left(t_0\right)\left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int }_{t_0}^t\sum _{j=1}^m{\delta }_j^{\left(1\right)}\left(u\right)\left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)du,\hfill \end{array}$$

(20)

and for $l\ge 2$,

$$\begin{array}{cc}\hfill {\widehat{d}}_l^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})& =-{\displaystyle \frac{l}{2}}{\left({\displaystyle \frac{1}{\beta }}\right)}^l\sum _{j=1}^m{\omega }_j{x}_0^{\left(j\right)}{\left(\beta -{\omega }_j(t-t_0)\right)}^{l-1}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{j=1}^m{\delta }_j\left(t_0\right){\left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)}^l\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int }_{t_0}^t\sum _{j=1}^m{\delta }_j^{\left(1\right)}\left(u\right){\left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)}^{l}du.\hfill \end{array}$$

(21)

Proof. 

To prove this theorem, we employ a two-step approach. First, we represent each squared Bessel process $X_t^{\left(j\right)}$ as a limit of a weighted sum of independent noncentral chi-square random variables using Proposition 2. Then, by applying Proposition 1, we obtain the analytical form of the TPDF of the conic combination $Y_t^{\left(m\right)}$.

By Proposition 2, we obtain

$${\omega }_j{X}_t^{\left(j\right)}\stackrel{\mathrm{law}}{=}\underset{n\to \infty }{lim}\sum _{i=1}^n{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}{\widehat{X}}_i^{\left(j\right)},$$

(22)

where ${\widehat{X}}_i^{\left(j\right)}\sim {\chi }_{{\widehat{\nu }}_i^{\left(j\right)}}^2\left({\widehat{\delta }}_i^{\left(j\right)}\right)$. This leads to

$$Y_t^{\left(m\right)}=\sum _{j=1}^m{\omega }_j{X}_t^{\left(j\right)}\stackrel{\mathrm{law}}{=}\underset{n\to \infty }{lim}\sum _{j=1}^m\sum _{i=1}^n{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}{\widehat{X}}_i^{\left(j\right)},$$

(23)

where the parameters and coefficients are defined as follows:

$$\begin{array}{cc}\hfill {\widehat{\alpha }}_i^{\left(j\right)}& =\tau \left(t,t_0+(i-1){\displaystyle \frac{t-t_0}{n}}\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\widehat{\nu }}_1^{\left(j\right)}& ={\delta }_j\left(t_0\right),{\widehat{\delta }}_1^{\left(j\right)}={\displaystyle \frac{x_0^{\left(j\right)}}{\tau (t,t_0)}},\hfill \end{array}$$

(25)

for all $j=1,\dots ,m$ and $i=1,\dots ,n$. For $i=2,\dots ,n$, we define

$${\widehat{\nu }}_i^{\left(j\right)}={\delta }_j^{\left(1\right)}\left(t_0+(i-1){\displaystyle \frac{t-t_0}{n}}\right){\displaystyle \frac{t-t_0}{n}},{\widehat{\delta }}_i^{\left(j\right)}=0.$$

(26)

We define the approximating random variable as

$${\widehat{Y}}_n^{\left(m\right)}:=\sum _{j=1}^m\sum _{i=1}^n{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}{\widehat{X}}_i^{\left(j\right)},$$

(27)

and consider the limit ${lim}_{n\to \infty }{f}_{{\widehat{Y}}_n^{\left(m\right)}}\left({\widehat{y}}_n^{\left(m\right)}\right)$.

By Proposition 1, the PDF of ${\widehat{Y}}_n^{\left(m\right)}$ is given by

$$f_{{\widehat{Y}}_n^{\left(m\right)}}^{\left(\beta \right)}\left({\widehat{y}}_n^{\left(m\right)}\right)={\displaystyle \frac{e^{-\frac{{\widehat{y}}_n^{\left(m\right)}}{2\beta }}{\left({\widehat{y}}_n^{\left(m\right)}\right)}^{\frac{{\widehat{\nu }}_n^{\left(m\right)}}{2}-1}}{{\left(2\beta \right)}^{\frac{{\widehat{\nu }}_n^{\left(m\right)}}{2}}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{{\widehat{\nu }}_n^{\left(m\right)}}{2}+k\right)}}{\widehat{b}}_{k,n}^{\left(m\right)}{\mathbf{L}}_k^{\left({\displaystyle \frac{{\widehat{\nu }}_n^{\left(m\right)}}{2}}-1\right)}\left({\displaystyle \frac{{\widehat{y}}_n^{\left(m\right)}}{2\beta }}\right),$$

(28)

valid for ${\widehat{y}}_n^{\left(m\right)}>0$, where the parameter ${\widehat{\nu }}_n^{\left(m\right)}$ is

$$\begin{array}{cc}\hfill {\widehat{\nu }}_n^{\left(m\right)}& =\sum _{j=1}^m\sum _{i=1}^n{\widehat{\nu }}_i^{\left(j\right)}\hfill \\ \hfill & =\sum _{j=1}^m\left({\delta }_j\left(t_0\right)+\sum _{i=2}^n{\delta }_j^{\left(1\right)}\left(t_0+(i-1){\displaystyle \frac{t-t_0}{n}}\right){\displaystyle \frac{t-t_0}{n}}\right).\hfill \end{array}$$

Thus,

$$\underset{n\to \infty }{lim}{\widehat{\nu }}_n^{\left(m\right)}=\sum _{j=1}^m{\delta }_j\left(t_0\right)+{\int }_{t_0}^t\sum _{j=1}^m{\delta }_j^{\left(1\right)}\left(u\right)du=\sum _{j=1}^m{\delta }_j\left(t\right).$$

(29)

The coefficients ${\widehat{b}}_{k,n}^{\left(m\right)}$ are defined recursively by

$$\begin{array}{cc}\hfill {\widehat{b}}_{0,n}^{\left(m\right)}& =1,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\widehat{b}}_{k,n}^{\left(m\right)}& ={\displaystyle \frac{1}{k}}\sum _{l=0}^{k-1}{\widehat{b}}_l^{\left(m\right)}{\widehat{d}}_{k-l}^{\left(m\right)},\hfill \end{array}$$

(31)

with

$$\begin{array}{cc}\hfill {\widehat{d}}_{1,n}^{\left(m\right)}& =-{\displaystyle \frac{1}{2\beta }}\sum _{j=1}^m\sum _{i=1}^n{\widehat{\delta }}_i^{\left(j\right)}{\widehat{\alpha }}_i^{\left(j\right)}{\omega }_j+{\displaystyle \frac{1}{2}}\sum _{j=1}^m\sum _{i=1}^n{\widehat{\nu }}_i^{\left(j\right)}\left(1-{\displaystyle \frac{{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}}{\beta }}\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2\beta }}\sum _{j=1}^m{\displaystyle \frac{x_0^{\left(j\right)}}{\tau (t,t_0)}}{\omega }_j\tau (t,t_0)+{\displaystyle \frac{1}{2}}\sum _{j=1}^m{\delta }_j\left(t_0\right)\left(1-{\displaystyle \frac{{\omega }_j\tau (t,t_0)}{\beta }}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{j=1}^m\sum _{i=2}^n{\delta }_j^{\left(1\right)}\left(t_0+(i-1){\displaystyle \frac{t-t_0}{n}}\right){\displaystyle \frac{t-t_0}{n}}\left(1-{\displaystyle \frac{{\omega }_j\tau (t,t_0+(i-1)\frac{t-t_0}{n})}{\beta }}\right),\hfill \end{array}$$

which leads to

$$\underset{n\to \infty }{lim}{\widehat{d}}_{1,n}^{\left(m\right)}={\widehat{d}}_1^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}).$$

(32)

For $l\ge 2$, we similarly obtain

$$\begin{array}{cc}\hfill {\widehat{d}}_{l,n}^{\left(m\right)}& =-{\displaystyle \frac{l}{2}}{\left({\displaystyle \frac{1}{\beta }}\right)}^l\sum _{j=1}^m{\omega }_j{x}_0^{\left(j\right)}{(\beta -{\omega }_j\tau (t,t_0))}^{l-1}+{\displaystyle \frac{1}{2}}\sum _{j=1}^m{\delta }_j\left(t_0\right){\left(1-{\displaystyle \frac{{\omega }_j\tau (t,t_0)}{\beta }}\right)}^l\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{j=1}^m\sum _{i=2}^n{\delta }_j^{\left(1\right)}\left(t_0+(i-1){\displaystyle \frac{t-t_0}{n}}\right){\displaystyle \frac{t-t_0}{n}}{\left(1-{\displaystyle \frac{{\omega }_j\tau (t,t_0+(i-1)\frac{t-t_0}{n})}{\beta }}\right)}^l,\hfill \end{array}$$

and hence

$$\underset{n\to \infty }{lim}{\widehat{d}}_{l,n}^{\left(m\right)}={\widehat{d}}_l^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}),\mathrm{for}\mathrm{all}l\ge 2.$$

(33)

Using the convergence results from Equations (29)–(33), as well as from Equation (23), we obtain

$$y=\underset{n\to \infty }{lim}{\widehat{y}}_n^{\left(m\right)},$$

and thus,

$$\underset{n\to \infty }{lim}{f}_{{\widehat{Y}}_n^{\left(m\right)}}^{\left(\beta \right)}\left({\widehat{y}}_n^{\left(m\right)}\right)={\displaystyle \frac{e^{-\frac{y}{2\beta }}{y}^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2\beta \right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right),$$

which is the desired TPDF $f_{Y_t^{\left(m\right)}}^{\left(\beta \right)}(y,t\mid {\mathit{x}}_{t_0},t_0)$ as in Equation (17). This completes the proof.    □

As previously established, the analytical expression in (17) is represented as a Laguerre series expansion. Consequently, the convergence properties of this series are of fundamental importance and must be carefully examined. Moreover, for practical computational purposes, it is necessary to truncate the infinite series to a finite sum. This inevitably introduces truncation errors, which must be rigorously analyzed to ensure the reliability and accuracy of the resulting approximation. The following subsection is therefore devoted to a detailed discussion of the truncation error analysis associated with our analytical formula.

3.1.2. Truncation Error Analysis

Before analyzing the convergence of the series expansion in the TPDF of Formula (17) and the associated truncation error, it is essential to establish a boundary on the coefficients ${\widehat{b}}_k^{\left(m\right)}$. This boundedness result plays a crucial role in ensuring the exponential decay of the Laguerre series terms, thereby justifying truncation. The following proposition presents the required estimate.

Proposition 3. 

The coefficients ${\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})$, for $k\in \mathbb{N}$, defined recursively in Equations (18)–(21), satisfy the inequality

$$\begin{array}{cc}\hfill |{\widehat{b}}_k^{\left(m\right)}|& \le {\rho }^{-k}exp\left(\sum _{j=1}^m\right({\displaystyle \frac{x_0^{\left(j\right)}{\omega }_j\rho }{2\beta }}{\left(1+\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)}^{-1}\hfill \\ \hfill & -\left({\displaystyle \frac{{\delta }_j\left(t_0\right)}{2}}ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }_j^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)\right)du\right)\left)\right),\hfill \end{array}$$

(34)

where $1<\rho <{\displaystyle \frac{1}{{max}_{i,j}\left|{\widehat{\gamma }}_i^{\left(j\right)}\right|}}$ and

$${\widehat{\gamma }}_i^{\left(j\right)}:=1-{\displaystyle \frac{{\omega }_j\left(t-t_0-(i-1)\frac{t-t_0}{n}\right)}{\beta }}$$

for any positive constant $\beta >{\displaystyle \frac{1}{2}}{max}_j{\omega }_j(t-t_0)$. Moreover, the sequence satisfies ${\widehat{b}}_k^{\left(m\right)}\to 0$ as $k\to \infty$.

Proof. 

We begin by considering the Laplace transform of the random variable ${\widehat{Y}}_n^{\left(m\right)}$ defined in Equation (27). This Laplace transform can be expressed as a power series, with coefficients ${\widehat{b}}_{k,n}^{\left(m\right)}$. We then apply Cauchy’s inequality, following the approach in Kotz et al. [30], to derive an upper bound for these coefficients.

From Equation (27), we obtain

$${\widehat{Y}}_n^{\left(m\right)}:=\sum _{j=1}^m\sum _{i=1}^n{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}{\widehat{X}}_i^{\left(j\right)}.$$

Using the method proposed by Castaño-Martinez et al. [23], the Laplace transform of the PDF of the conic combination of noncentral chi-square random variables ${\widehat{Y}}_n^{\left(m\right)}$ is given by

$$\mathcal{L}\left\{f_{{\widehat{Y}}_n^{\left(m\right)}}\right\}\left(\lambda \right)=exp\left(-\sum _{j=1}^m\sum _{i=1}^n{\displaystyle \frac{{\widehat{\delta }}_i^{\left(j\right)}{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\lambda }{1+2{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\lambda }}\right)\prod _{j=1}^m\prod _{i=1}^n{\left(1+2{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\lambda \right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}},$$

(35)

making it valid for any complex number $\lambda$.

Using the technique from Chapter 3 of Mathai and Provost  [31], we observe that

$$\prod _{j=1}^m\prod _{i=1}^n{\left(1+2{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\lambda \right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}=\prod _{j=1}^m\prod _{i=1}^n{\left(1+2\beta \lambda -2\beta \lambda \left(1-{\displaystyle \frac{{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}}{\beta }}\right)\right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}.$$

Let

$${\widehat{\gamma }}_i^{\left(j\right)}:=1-{\displaystyle \frac{{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}}{\beta }},\mathrm{and}\theta :={\displaystyle \frac{2\beta \lambda }{1+2\beta \lambda }},$$

for any positive constant $\beta$. Then,

$$\prod _{j=1}^m\prod _{i=1}^n{\left(1+2{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\lambda \right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}={(1+2\beta \lambda )}^{-{\displaystyle \frac{{\widehat{\nu }}_n^{\left(m\right)}}{2}}}\prod _{j=1}^m\prod _{i=1}^n{(1-{\widehat{\gamma }}_i^{\left(j\right)}\theta )}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}},$$

(36)

and

$${(1+2{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\lambda )}^{-1}=(1-\theta ){(1-{\widehat{\gamma }}_i^{\left(j\right)}\theta )}^{-1}.$$

(37)

Substituting Equations (36) and (37) into Equation (35) gives

$$\begin{array}{cc}\hfill \mathcal{L}\left\{f_{{\widehat{Y}}_n^{\left(m\right)}}\right\}\left(\lambda \right)& =exp\left(-\sum _{j=1}^m\sum _{i=1}^n{\displaystyle \frac{{\widehat{\delta }}_i^{\left(j\right)}{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\theta }{2\beta }}{(1-{\widehat{\gamma }}_i^{\left(j\right)}\theta )}^{-1}\right){(1+2\beta \lambda )}^{-{\displaystyle \frac{{\widehat{\nu }}_n^{\left(m\right)}}{2}}}\hfill \\ \hfill & \times \prod _{j=1}^m\prod _{i=1}^n{(1-{\widehat{\gamma }}_i^{\left(j\right)}\theta )}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}.\hfill \end{array}$$

Define

$$\mathcal{M}\left(\theta \right):=exp\left(-\sum _{j=1}^m\sum _{i=1}^n{\displaystyle \frac{{\widehat{\delta }}_i^{\left(j\right)}{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\theta }{2\beta }}{(1-{\widehat{\gamma }}_i^{\left(j\right)}\theta )}^{-1}\right)\prod _{j=1}^m\prod _{i=1}^n{(1-{\widehat{\gamma }}_i^{\left(j\right)}\theta )}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}.$$

(38)

Under the condition of $|{\widehat{\gamma }}_i^{\left(j\right)}\theta |<1$ for all $i=1,\dots ,n$ and $j=1,\dots ,m$, the function $\mathcal{M}\left(\theta \right)$ admits a power series representation:

$$\mathcal{M}\left(\theta \right)=\sum _{k=0}^{\infty }{\widehat{b}}_{k,n}^{\left(m\right)}{\theta }^k.$$

By applying Cauchy’s inequality, we obtain

$$\left|{\widehat{b}}_{k,n}^{\left(m\right)}\right|\le {\rho }^{-k}\underset{\left|\theta \right|=\rho }{max}\left|\mathcal{M}\left(\theta \right)\right|,$$

(39)

for any $0<\rho <{\displaystyle \frac{1}{{max}_{i,j}\left|{\widehat{\gamma }}_i^{\left(j\right)}\right|}}$.

Using Lemma 2 in Kotz et al. [30] and the expression in Equation (38), Inequality (39) becomes

$$\left|{\widehat{b}}_{k,n}^{\left(m\right)}\right|\le {\rho }^{-k}exp\left(\sum _{j=1}^m\sum _{i=1}^n{\displaystyle \frac{{\widehat{\delta }}_i^{\left(j\right)}{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}\rho }{2\beta }}{(1+{\widehat{\gamma }}_i^{\left(j\right)}\rho )}^{-1}\right)\prod _{j=1}^m\prod _{i=1}^n{\left(1-{\widehat{\gamma }}_i^{\left(j\right)}\rho \right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}.$$

(40)

From the setup in Theorem 1, we simplify Equation (40) as

$$\left|{\widehat{b}}_{k,n}^{\left(m\right)}\right|\le {\rho }^{-k}exp\left(\sum _{j=1}^m{\displaystyle \frac{x_0^{\left(j\right)}{\omega }_j\rho }{2\beta (1+{\widehat{\gamma }}_1^{\left(j\right)})}}\right)\prod _{j=1}^m\prod _{i=1}^n{\left(1-{\widehat{\gamma }}_i^{\left(j\right)}\rho \right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}.$$

(41)

To ensure the decay of ${\widehat{b}}_{k,n}^{\left(m\right)}$ as $k\to \infty$, we restrict $\rho >1$ and enforce

$$1<\rho <{\displaystyle \frac{1}{{max}_{i,j}\left|{\widehat{\gamma }}_i^{\left(j\right)}\right|}},$$

(42)

which holds if

$$0<\underset{i,j}{max}\left|1-{\displaystyle \frac{{\omega }_j{\widehat{\alpha }}_i^{\left(j\right)}}{\beta }}\right|<1,$$

leading to

$$\beta >{\displaystyle \frac{1}{2}}\underset{j}{max}\left({\omega }_j(t-t_0)\right).$$

Returning to Equation (41), we take limits on both sides:

$$\underset{n\to \infty }{lim}\left|{\widehat{b}}_{k,n}^{\left(m\right)}\right|\le {\rho }^{-k}exp\left(\sum _{j=1}^m{\displaystyle \frac{x_0^{\left(j\right)}{\omega }_j\rho }{2\beta (1+{\widehat{\gamma }}_1^{\left(j\right)})}}\right)\prod _{j=1}^m\underset{n\to \infty }{lim}\prod _{i=1}^n{\left(1-{\widehat{\gamma }}_i^{\left(j\right)}\rho \right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}.$$

(43)

Let

$$g_n:=\prod _{i=1}^n{\left(1-{\widehat{\gamma }}_i^{\left(j\right)}\rho \right)}^{-{\displaystyle \frac{{\widehat{\nu }}_i^{\left(j\right)}}{2}}}.$$

Taking logarithms and passing the limit results in

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}ln{g}_n& =-{\displaystyle \frac{{\delta }_j\left(t_0\right)}{2}}ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }_j^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)\right)du,\hfill \end{array}$$

which implies

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{g}_n& =exp(-{\displaystyle \frac{{\delta }_j\left(t_0\right)}{2}}ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }_j^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)\right)du).\hfill \end{array}$$

(44)

Substituting Equation (44) into Equation (43) yields

$$\begin{array}{cc}\hfill |{\widehat{b}}_k^{\left(m\right)}|& \le {\rho }^{-k}exp\left(\sum _{j=1}^m{\displaystyle \frac{x_0^{\left(j\right)}{\omega }_j\rho }{2\beta (1+{\widehat{\gamma }}_1^{\left(j\right)})}}\right)\hfill \\ \hfill & \times \prod _{j=1}^{m}exp\left(-{\displaystyle \frac{{\delta }_j\left(t_0\right)}{2}}ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }_j^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)\right)du\right).\hfill \end{array}$$

(45)

A straightforward algebraic rearrangement of Equation (45) yields Equation (34). Since $1<\rho <{\displaystyle \frac{1}{{max}_{i,j}\left|{\widehat{\gamma }}_i^{\left(j\right)}\right|}}$, it follows that $0<{\rho }^{-1}<1$, and hence $\left|{\widehat{b}}_k^{\left(m\right)}\right|\to 0$ as $k\to \infty$.

This completes the proof.    □

After establishing the boundedness of the coefficients ${\widehat{b}}_k^{\left(m\right)}$, the next step is to analyze the truncation error associated with the TPDF formula.

Let ${\mathit{G}}_{k_1,k_2}$ denote the truncation error of the TPDF of a conic combination of independent squared Bessel processes, defined by

$${\mathit{G}}_{k_1,k_2}(y,t;{\mathit{x}}_{t_0},t_0):={\displaystyle \frac{e^{-\frac{y}{2\beta }}{y}^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2\beta \right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=k_1+1}^{k_2}{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right),$$

(46)

where $k_1,k_2\in \mathbb{N}\cup \{0,\infty \}$, and $k_1+1\le k_2$. The following proposition investigates the behavior of the truncation error ${\mathit{G}}_{K,\infty }$ as the number of retained terms K tends to infinity. Since ${\mathit{G}}_{K,\infty }$ is expressed in terms of generalized Laguerre polynomials, we begin with a classical bound, as shown in Szegő  [32].

Lemma 1. 

A classical global uniform estimate (with respect to k, y, and α) according to Szegő  [32] is given by

$$\begin{array}{cc}\hfill \left|{\mathit{L}}_k^{\left(\alpha \right)}\left(y\right)\right|& \le {\displaystyle \frac{{(\alpha +1)}_k}{k!}}exp\left({\displaystyle \frac{y}{2}}\right),\alpha \ge 0,\hfill \\ \hfill \left|{\mathit{L}}_k^{\left(\alpha \right)}\left(y\right)\right|& \le \left(2-{\displaystyle \frac{{(\alpha +1)}_k}{k!}}\right)exp\left({\displaystyle \frac{y}{2}}\right),-1<\alpha <0.\hfill \end{array}$$

Proposition 4. 

The truncation error of the order K in Equation (46) associated with the TPDF defined in Equation (17) satisfies the inequality:

$$\left|{\mathit{G}}_{K,\infty }(y,t;{\mathit{x}}_{t_0},t_0)\right|\le D(y,t;{\mathit{x}}_{t_0},t_0)\sum _{k=K+1}^{\infty }{\rho }^{-k},$$

(47)

where

$$\begin{array}{cc}\hfill D(y,t;{\mathit{x}}_{t_0},t_0)& ={\displaystyle \frac{y^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2\beta \right)}^{\frac{\delta \left(t\right)}{2}}}}exp\left(\sum _{j=1}^m\right({\displaystyle \frac{x_0^{\left(j\right)}{\omega }_j\rho }{2\beta }}{\left(1+\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)}^{-1}\hfill \\ \hfill & -\left({\displaystyle \frac{{\delta }_j\left(t_0\right)}{2}}ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }_j^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)\right)du\right)\left)\right).\hfill \end{array}$$

(48)

Furthermore, the truncation error vanishes in the limit:

$$\underset{K\to \infty }{lim}{\mathit{G}}_{K,\infty }(y,t;{\mathit{x}}_{t_0},t_0)=0.$$

(49)

Proof. 

The truncation error of order K for the TPDF of the conic combination of independent squared Bessel processes is given by

$${\mathit{G}}_{K,\infty }(y,t;{\mathit{x}}_{t_0},t_0):={\displaystyle \frac{e^{-\frac{y}{2\beta }}{y}^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2\beta \right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=K+1}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right),$$

for $K\in \mathbb{N}$. Taking the absolute value yields the bound:

$$\left|{\mathit{G}}_{K,\infty }\right|\le {\displaystyle \frac{e^{-\frac{y}{2\beta }}{y}^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2\beta \right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=K+1}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}\left|{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})\right|\left|{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right)\right|.$$

(50)

By substituting the upper bound for $|{\widehat{b}}_k^{\left(m\right)}|$ (from Proposition 3) and applying the Laguerre polynomial bound in Lemma 1, we obtain

$$\left|{\mathit{G}}_{K,\infty }\right|\le D(y,t;{\mathit{x}}_{t_0},t_0)\sum _{k=K+1}^{\infty }{\rho }^{-k},$$

where the constant $D(y,t;{\mathit{x}}_{t_0},t_0)$ is defined in Equation (48).

From Proposition 3, we know that $0<{\rho }^{-1}<1$, so the sum ${\sum }_{k=0}^{\infty }{\rho }^{-k}$ is a convergent geometric series. Therefore,

$$\underset{K\to \infty }{lim}\sum _{k=K+1}^{\infty }{\rho }^{-k}=0.$$

Finally, by the squeeze theorem, we conclude that

$$\underset{K\to \infty }{lim}{\mathit{G}}_{K,\infty }(y,t;{\mathit{x}}_{t_0},t_0)=0.$$

   □

3.2. An Analytical Formula for the Conditional Moments of the Conic Combination of Independent Squared Bessel Processes with Time-Dependent Dimensions

After successfully deriving an analytical formula for the TPDF of the conic combination of independent squared Bessel processes $Y_t^{\left(m\right)}$, we now proceed to utilize this formula to obtain an analytical expression for the conditional moments of the random variable $Y_t^{\left(m\right)}$.

3.2.1. Derivation of an Analytical Formula for the Conditional Moments

Theorem 2. 

Let $Y_t^{\left(m\right)}$ be the conic combination of independent squared Bessel processes defined in Equation (10). Suppose that the time-dependent dimension functions ${\delta }_j\left(t\right)$ satisfy Assumptions 1 and 2 for all $j=1,\dots ,m$, and let $\gamma >0$ be a real number. Then, the γ-th conditional moment of $Y_t^{\left(m\right)}$, given the initial values ${\mathit{x}}_{t_0}=(x_0^{\left(1\right)},\dots ,x_0^{\left(m\right)})$, is given by

$$\mathbb{E}\left[{\left(Y_t^{\left(m\right)}\right)}^{\gamma }|{\mathit{x}}_{t_0}\right]={\left(2\beta \right)}^{\gamma }\sum _{k=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right),$$

(51)

for $t>t_0$, where the coefficients ${\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})$ are recursively defined in Theorem 1.

The function ${}_2\mathbf{F}_1(a_1,a_2;b_1;z)$, known as the generalized hypergeometric function, is defined by the series:

$${}_2\mathbf{F}_1(a_1,a_2;b_1;z)=\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(a_1\right)}_r{\left(a_2\right)}_r}{{\left(b_1\right)}_r}}{\displaystyle \frac{z^r}{r!}},$$

(52)

where ${\left(a\right)}_r=a(a+1)(a+2)\cdots (a+r-1)$ denotes the Pochhammer symbol.

Proof. 

To prove this result, we apply the definition of the conditional moment of a random variable in conjunction with the analytical formula for the TPDF of $Y_t^{\left(m\right)}$ given in Theorem 1. In addition, we employ a change in variable and utilize a classical integral identity involving Laguerre polynomials.

We begin with the expression for the $\gamma$-th conditional moment:

$$\mathbb{E}\left[{\left(Y_t^{\left(m\right)}\right)}^{\gamma }|{\mathit{x}}_{t_0}\right]={\int }_0^{\infty }{y}^{\gamma }{f}_{Y_t^{\left(m\right)}}(y,t\mid {\mathit{x}}_{t_0},t_0)dy.$$

By substituting the TPDF formula from Equation (17), we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\left(Y_t^{\left(m\right)}\right)}^{\gamma }|{\mathit{x}}_{t_0}\right]& =\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{{\left(2\beta \right)}^{\frac{\delta \left(t\right)}{2}}\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})\hfill \\ \hfill & \times {\int }_0^{\infty }{y}^{{\displaystyle \frac{\delta \left(t\right)}{2}}-1+\gamma }{e}^{-{\displaystyle \frac{y}{2\beta }}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right)dy.\hfill \end{array}$$

(53)

To evaluate the integral, we apply the change of variable $z={\displaystyle \frac{y}{2\beta }}$ so that $y=2\beta z$ and $dy=2\beta dz$. The integral becomes

$${\int }_0^{\infty }{y}^{{\displaystyle \frac{\delta \left(t\right)}{2}}-1+\gamma }{e}^{-{\displaystyle \frac{y}{2\beta }}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{y}{2\beta }}\right)dy={\left(2\beta \right)}^{{\displaystyle \frac{\delta \left(t\right)}{2}}+\gamma }{\int }_0^{\infty }{z}^{{\displaystyle \frac{\delta \left(t\right)}{2}}-1+\gamma }{e}^{-z}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left(z\right)dz.$$

Using the known integral formula for Laguerre polynomials (see, e.g., Erdélyi et al. [33] or Gradshteyn and Ryzhik [34]), we obtain

$${\int }_0^{\infty }{z}^{{\displaystyle \frac{\delta \left(t\right)}{2}}-1+\gamma }{e}^{-z}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left(z\right)dz={\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}{k!\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}{}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right).$$

Substituting this result into Equation (53) and simplifying it completes the proof and yields the expression in Equation (51).    □

The formula in Theorem 2 holds for all $\gamma \in {\mathbb{R}}_{+}$. Notably, when $\gamma =N\in {\mathbb{N}}_0$, the hypergeometric function ${}_2\mathbf{F}_1(-k,N+{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1)$ reduces to a finite polynomial in k, which truncates the infinite series in Equation (51) to a finite sum. The following corollary formalizes this simplification.

Corollary 1. 

Let $Y_t^{\left(m\right)}$ be the conic combination of independent squared Bessel processes defined in Equation (10). Suppose ${\delta }_j\left(t\right)$ satisfies Assumptions 1 and 2 for all $j=1,\dots ,m$, and let $N\in {\mathbb{N}}_0$ be a non-negative integer. Then, the N-th conditional moment of $Y_t^{\left(m\right)}$, given the initial values ${\mathit{x}}_{t_0}=(x_0^{\left(1\right)},\dots ,x_0^{\left(m\right)})$, is given by

$$\mathbb{E}\left[{\left(Y_t^{\left(m\right)}\right)}^N|{\mathit{x}}_{t_0}\right]={\left(2\beta \right)}^N\sum _{k=0}^N{\displaystyle \frac{\mathsf{\Gamma }\left(N+\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){}_2\mathbf{F}_1\left(-k,N+{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right),$$

(54)

for $t>t_0$, where the coefficients ${\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})$ are defined recursively in Theorem 1.

Proof. 

This result follows directly from Theorem 2 by setting $\gamma =N\in {\mathbb{N}}_0$. When $\gamma$ is an integer, the hypergeometric function simplifies due to the identity

$${}_2\mathbf{F}_1\left(-k,N+{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right)={\displaystyle \frac{{(-N)}_k}{{\left(\frac{\delta \left(t\right)}{2}\right)}_k}},$$

and the Pochhammer symbol in the numerator vanishes for $k>N$ since ${(-N)}_k=0$ for $k>N$. Thus, the series truncates at $k=N$, and the infinite sum in Equation (51) becomes a finite sum as expressed in (54), completing the proof.    □

3.2.2. Truncation Error Analysis

As previously mentioned in the preceding subsection, the analytical formula for the conditional moments (51) is expressed as an infinite series expansion. In practice, however, this series must be truncated to a finite number of terms for implementation in computer programs or any other computational purposes. Therefore, truncation error analysis is essential to ensure that the contribution of the remaining terms vanishes as the number of retained terms increases. This guarantees both the accuracy and the reliability of the computational results.

Let ${\mathit{V}}^{(\gamma ,K)}$ denote the truncation error of order K for the $\gamma$-th conditional moment of $Y_t^{\left(m\right)}$, defined by

$${\mathit{V}}^{(\gamma ,K)}(t,t_0;{\mathit{x}}_{t_0}):={\left(2\beta \right)}^{\gamma }\sum _{k=K+1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0}){}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right).$$

(55)

We further define the following constant:

$$\begin{array}{cc}\hfill H^{(\rho ,\beta )}(t,t_0;{\mathit{x}}_{t_0})& :={\left(2\beta \right)}^{\gamma }{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}exp\left(\sum _{j=1}^m\right({\displaystyle \frac{x_0^{\left(j\right)}{\omega }_j\rho }{2\beta }}{\left(1+\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)}^{-1}\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_j\left(t_0\right)}{2}}ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }_j^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)\right)du\left)\right),\hfill \end{array}$$

(56)

where $\rho$ is defined as in Proposition 3 and $\beta$ is a positive constant. The following proposition establishes the absolute boundedness of the truncation error ${\mathit{V}}^{(\gamma ,K)}$ and its limiting behavior as $K\to \infty$.

Proposition 5. 

The truncation error ${\mathit{V}}^{(\gamma ,K)}$ defined in Equation (55) satisfies the bound:

$$\left|{\mathit{V}}^{(\gamma ,K)}(t,t_0;{\mathit{x}}_{t_0})\right|\le H^{(\rho ,\beta )}(t,t_0;{\mathit{x}}_{t_0})\sum _{k=K+1}^{\infty }{\rho }^{-k}\left|{}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right)\right|,$$

(57)

where $H^{(\rho ,\beta )}(t,t_0;{\mathit{x}}_{t_0})$ is defined in Equation (56). Furthermore, if $\beta >{\displaystyle \frac{1}{2}}{max}_j{\omega }_j(t-t_0)$, then

$$\underset{K\to \infty }{lim}{\mathit{V}}^{(\gamma ,K)}(t,t_0;{\mathit{x}}_{t_0})=0.$$

(58)

Proof. 

The absolute boundedness of ${\mathit{V}}^{(\gamma ,K)}$ follows directly from its definition:

$$\left|{\mathit{V}}^{(\gamma ,K)}\right|\le {\left(2\beta \right)}^{\gamma }\sum _{k=K+1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}\left|{\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})\right|\left|{}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right)\right|.$$

Using the upper bound for $\left|{\widehat{b}}_k^{\left(m\right)}\right|$ from Proposition 3 and recalling that the generalized Laguerre polynomial does not appear in this context, we obtain

$$\left|{\mathit{V}}^{(\gamma ,K)}\right|\le H^{(\rho ,\beta )}(t,t_0;{\mathit{x}}_{t_0})\sum _{k=K+1}^{\infty }{\rho }^{-k}\left|{}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right)\right|,$$

where $H^{(\rho ,\beta )}$ is defined in Equation (56).

Now, define

$$B_k:={\rho }^{-k}\left|{}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right)\right|={\rho }^{-k}\left|{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)\mathsf{\Gamma }(k-\gamma )}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)\mathsf{\Gamma }(-\gamma )}}\right|,$$

which is valid for $k>\gamma$, where we have used the closed-form evaluation of the generalized hypergeometric function at 1 when $a=-k\in {\mathbb{Z}}_{\le 0}$.

We observe that

$$\underset{k\to \infty }{lim}\left|{\displaystyle \frac{B_{k+1}}{B_k}}\right|={\rho }^{-1}<1.$$

Hence, by the ratio test, the series ${\sum }_{k=0}^{\infty }{B}_k$ converges absolutely. It follows that

$$\underset{K\to \infty }{lim}\sum _{k=K+1}^{\infty }{B}_k=0.$$

Using this, we conclude that

$$\begin{array}{cc}\hfill \underset{K\to \infty }{lim}\left|{\mathit{V}}^{(\gamma ,K)}\right|& \le {\left(2\beta \right)}^{\gamma }{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}exp\left(\sum _{j=1}^m\right({\displaystyle \frac{x_0^{\left(j\right)}{\omega }_j\rho }{2\beta }}{\left(1+\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)}^{-1}\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_j\left(t_0\right)}{2}}ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-t_0)}{\beta }}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }_j^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{{\omega }_j(t-u)}{\beta }}\right)\right)du\left)\right)\underset{K\to \infty }{lim}\sum _{k=K+1}^{\infty }{B}_k=0.\hfill \end{array}$$

Therefore,

$$\underset{K\to \infty }{lim}{\mathit{V}}^{(\gamma ,K)}(t,t_0;{\mathit{x}}_{t_0})=0.$$

   □

3.3. An Analytical Formula for the TPDF of the Squared Bessel Process with the Time-Dependent Dimension

In this section, we derive analytical expressions for the TPDF and conditional moments of the squared Bessel process with the time-dependent dimension. These results are obtained as special cases of the general formulas derived earlier for the conic combination of independent squared Bessel processes.

3.3.1. Derivation of an Analytical Formula for the TPDF

Theorem 3. 

Let $X_t$ be the squared Bessel process governed by the SDE (2), with initial condition $X_{t_0}=x_0>0$. Suppose that Assumptions 1 and 2 hold. Then, the TPDF of $X_t$ is given by

$$f_{X_t}(x,t\mid x_0,t_0)={\displaystyle \frac{e^{-\frac{x}{2(t-t_0)}}{x}^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2(t-t_0)\right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{c}}_k(t,t_0;x_0){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{x}{2(t-t_0)}}\right),$$

(59)

for $x>0$ and $t>t_0$, where $L_k^{\left(\eta \right)}$ is the generalized Laguerre polynomial as defined in Proposition 1. The coefficients ${\widehat{c}}_k(t,t_0;x_0)$ are defined recursively as follows:

$${\widehat{c}}_0(t,t_0;x_0)=1,$$

(60)

$${\widehat{c}}_k(t,t_0;x_0)={\displaystyle \frac{1}{k}}\sum _{j=0}^{k-1}{\widehat{c}}_j(t,t_0;x_0){\widehat{d}}_{k-j}(t,t_0;x_0),k\ge 1,$$

(61)

where the auxiliary terms ${\widehat{d}}_j(t,t_0;x_0)$ are given by

$${\widehat{d}}_1(t,t_0;x_0)=-{\displaystyle \frac{x_0}{2(t-t_0)}}+{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }^{\left(1\right)}\left(u\right)\left(1-{\displaystyle \frac{t-u}{t-t_0}}\right)du,$$

(62)

and for all $j\ge 2$,

$${\widehat{d}}_j(t,t_0;x_0)={\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }^{\left(1\right)}\left(u\right){\left(1-{\displaystyle \frac{t-u}{t-t_0}}\right)}^{j}du.$$

(63)

Proof. 

The result follows directly by observing that the process $X_t$ is a special case of the conic combination of squared Bessel processes $Y_t^{\left(m\right)}$ defined in Equation (10). Specifically, by setting $m=1$, ${\omega }_1=1$, and $\beta =t-t_0$, the general TPDF formula presented in Theorem 1 reduces to the expression stated in Equation (59). Therefore, the proof is an immediate consequence of Theorem 1.    □

According to Theorem 1, if the positive constants ${\omega }_j=1$ for all $j=1,\dots ,m$, then the random variable $Y_t^{\left(m\right)}$ in Equation (10) simplifies to

$$Y_t^{\left(m\right)}:=\sum _{j=1}^m{X}_t^{\left(j\right)},$$

(64)

which implies that $Y_t^{\left(m\right)}$ is itself a squared Bessel process with time-dependent dimension $\delta \left(t\right):={\sum }_{j=1}^m{\delta }_j\left(t\right)$ and initial value $x_0:={\sum }_{j=1}^m{x}_0^{\left(j\right)}$. The following corollary is a direct consequence of Theorem 1 and Theorem 3.

Corollary 2. 

Let $Y_t^{\left(m\right)}$ be defined as in Equation (64). Then, the TPDF of $Y_t^{\left(m\right)}$ coincides with the TPDF of the squared Bessel process $X_t$ given in Theorem 3, with the dimension and initial value replaced by $\delta \left(t\right)$ and $x_0$, respectively.

Proof. 

As noted before the corollary, when ${\omega }_j=1$ for all $j=1,\dots ,m$, the random variable $Y_t^{\left(m\right)}$ defined in Equation (64) becomes a squared Bessel process with time-dependent dimension

$$\delta \left(t\right)=\sum _{j=1}^m{\delta }_j\left(t\right)$$

(65)

and initial value

$$x_0=\sum _{j=1}^m{x}_0^{\left(j\right)}.$$

(66)

By substituting ${\omega }_j=1$ for all $j=1,\cdots ,m$, $\beta =t-t_0$, $\delta \left(t\right)$ from (65), and $x_0$ from (66) into the general expressions in (17)–(21), we observe that the resulting coefficients ${\widehat{b}}_k^{\left(m\right)}(t,t_0;{\mathit{x}}_{t_0})$ coincide with the coefficients ${\widehat{c}}_k(t,t_0;x_0)$ defined in Equation (61). Therefore, the density $f_{Y_t^{\left(m\right)}}^{\left(\beta \right)}(y,t\mid {\mathit{x}}_{t_0},t_0)$ matches the density $f_{X_t}(x,t\mid x_0,t_0)$ given in Equation (59). This completes the proof.    □

According to Theorem 3, the TPDF of the squared Bessel process can be expressed using a Laguerre series expansion. When the dimension $\delta \left(t\right)$ is constant, i.e., $\delta \left(t\right)=\delta \ge 2$, the recursion for the coefficients and the Laguerre expansion simplify significantly. These simplifications lead to a closed-form expression for the TPDF in terms of the modified Bessel function of the first kind, $I_{\nu }\left(z\right)$. The following corollary presents this result in its canonical form, which is widely recognized for squared Bessel processes with constant dimensions.

Corollary 3. 

According to Theorem 3, when the dimension is constant, i.e., $\delta \left(t\right)=\delta \ge 2$, the TPDF of the squared Bessel process given in Equation (59) simplifies to the closed-form expression on the right-hand side of Equation (3).

Proof. 

From Equation (59), when $\delta \left(t\right)=\delta \ge 2$ is constant, we have ${\delta }^{\left(1\right)}\left(t\right)=0$. This implies that all integrals in the auxiliary terms vanish, yielding

$${\widehat{d}}_1(t,t_0;x_0)=-{\displaystyle \frac{x_0}{2(t-t_0)}},{\widehat{d}}_j(t,t_0;x_0)=0\mathrm{for}j\ge 2.$$

By substituting these into the recursion (61), we obtain

$${\widehat{c}}_k(t,t_0;x_0)={\displaystyle \frac{1}{k}}{\widehat{c}}_{k-1}(t,t_0;x_0)·\left(-{\displaystyle \frac{x_0}{2(t-t_0)}}\right)={\displaystyle \frac{{\left(-\frac{x_0}{2(t-t_0)}\right)}^k}{k!}}.$$

By substituting this into the Laguerre expansion (59), the TPDF becomes

$$f_{X_t}(x,t\mid x_0,t_0)=C\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-\frac{x_0}{2(t-t_0)}\right)}^k}{\mathsf{\Gamma }\left(\frac{\delta }{2}+k\right)}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta }{2}}-1\right)}\left({\displaystyle \frac{x}{2(t-t_0)}}\right),$$

(67)

where we denote

$$C:={\displaystyle \frac{e^{-\frac{x}{2(t-t_0)}}{x}^{\frac{\delta }{2}-1}}{{\left(2(t-t_0)\right)}^{\frac{\delta }{2}}}}.$$

Next, we express the modified Bessel function of the first kind on the right-hand side of Equation (3) as a Laguerre expansion:

$$I_{{\displaystyle \frac{\delta }{2}}-1}\left({\displaystyle \frac{\sqrt{x{x}_0}}{t-t_0}}\right)=C_1\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-\frac{x_0}{2(t-t_0)}\right)}^k}{\mathsf{\Gamma }\left(\frac{\delta }{2}+k\right)}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta }{2}}-1\right)}\left({\displaystyle \frac{x}{2(t-t_0)}}\right),$$

(68)

where $C_1$ is a prefactor to be determined.

From Equation (3), we write the prefactor as

$$C_2:={\displaystyle \frac{e^{-\frac{x+x_0}{2(t-t_0)}}}{2(t-t_0)}}{\left({\displaystyle \frac{x}{x_0}}\right)}^{{\displaystyle \frac{\delta }{4}}-{\displaystyle \frac{1}{2}}}.$$

Comparing Equations (3), (67), and (68), it suffices to verify that $C=C_1{C}_2$.

To establish this, we employ a known identity connecting the modified Bessel function and Laguerre polynomials (see, e.g., Gradshteyn and Ryzhik [34]):

$$I_{{\displaystyle \frac{\delta }{2}}-1}\left(\sqrt{\alpha y}\right)={\left({\displaystyle \frac{\sqrt{\alpha y}}{2}}\right)}^{{\displaystyle \frac{\delta }{2}}-1}{e}^{-\xi }\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-\frac{\alpha }{2}\right)}^k}{\mathsf{\Gamma }\left(k+\frac{\delta }{2}\right)}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta }{2}}-1\right)}\left(-{\displaystyle \frac{\alpha y}{4\xi }}\right),$$

(69)

for $y>0$, $\alpha >0$, and $\xi \ne 0$.

Letting

$$y={\displaystyle \frac{x}{t-t_0}},\alpha ={\displaystyle \frac{x_0}{t-t_0}},\mathrm{and}\xi =-{\displaystyle \frac{x_0}{2(t-t_0)}},$$

and substituting them into Equation (69), we recover the series expansion (68). Comparing terms yields

$$C_1={\left({\displaystyle \frac{1}{2(t-t_0)}}\right)}^{{\displaystyle \frac{\delta }{2}}-1}{\left(x{x}_0\right)}^{{\displaystyle \frac{\delta }{4}}-{\displaystyle \frac{1}{2}}}{e}^{{\displaystyle \frac{x_0}{2(t-t_0)}}}.$$

Multiplying $C_1$ and $C_2$, we obtain

$$C_1{C}_2={\displaystyle \frac{e^{-\frac{x}{2(t-t_0)}}{x}^{\frac{\delta }{2}-1}}{{\left(2(t-t_0)\right)}^{\frac{\delta }{2}}}}=C,$$

thus verifying that the Laguerre expansion in Equation (67) coincides with the classical closed-form expression involving the modified Bessel function. This completes the proof.    □

3.3.2. Truncation Error Analysis

In this part, we investigate the boundedness of the truncation error for the TPDF of the squared Bessel process, as defined in (59). Before establishing the truncation error bound, we first derive a bound for the coefficients ${\widehat{c}}_k$, which are defined recursively in Equations (60)–(63).

Proposition 6. 

The coefficients ${\widehat{c}}_k(t,t_0;x_0)$, for $k\in \mathbb{N}$, defined recursively in Equations (60)–(63), satisfy the inequality

$$\left|{\widehat{c}}_k\right|\le {\rho }^{-k}exp\left({\displaystyle \frac{x_0}{2(t-t_0)}}·{\displaystyle \frac{\rho }{1+{\widehat{\gamma }}_1\rho }}\right)exp\left(-{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{t-u}{t-t_0}}\right)\right)du\right),$$

(70)

where $1<\rho <{\displaystyle \frac{1}{{max}_i\left|{\widehat{\gamma }}_i\right|}}$ and ${\widehat{\gamma }}_i=1-{\displaystyle \frac{{\widehat{\alpha }}_i}{t-t_0}}$. Moreover, the coefficients satisfy ${\widehat{c}}_k(t,t_0;x_0)\to 0$ as $k\to \infty$.

Proof. 

The result follows directly from Proposition 3, which provides a general bound for the coefficients in the Laguerre expansion of a conic combination of squared Bessel processes. By setting $m=1$, $\omega =1$, and $\beta =t-t_0$, the general case reduces to the current setting, yielding the bound in Equation (70).    □

To derive the boundedness of the truncation error of the TPDF in Equation (59), we define

$${\mathit{E}}_{k_1,k_2}(x,t;x_0,t_0):={\displaystyle \frac{e^{-\frac{x}{2(t-t_0)}}{x}^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2(t-t_0)\right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=k_1+1}^{k_2}{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{c}}_k(t,t_0;x_0){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{x}{2(t-t_0)}}\right),$$

(71)

where $k_1,k_2\in \mathbb{N}\cup \{0,\infty \}$ with $k_1+1\le k_2$. Our goal is to derive a bound for the truncation error ${\mathit{E}}_{K,\infty }$.

Proposition 7. 

The truncation error of order K, denoted as ${\mathit{E}}_{K,\infty }$, is bounded as follows:

$$\left|{\mathit{E}}_{K,\infty }(x,t;x_0,t_0)\right|\le B(x,t;x_0,t_0)\sum _{k=K+1}^{\infty }{\rho }^{-k},$$

(72)

where

$$\begin{array}{cc}\hfill B(x,t;x_0,t_0)& :={\displaystyle \frac{x^{\frac{\delta \left(t\right)}{2}-1}}{{\left(2(t-t_0)\right)}^{\frac{\delta \left(t\right)}{2}}\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}exp\left({\displaystyle \frac{x_0}{2(t-t_0)}}·{\displaystyle \frac{\rho }{1+{\widehat{\gamma }}_1\rho }}\right)\hfill \\ \hfill & \times exp\left(-{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{t-u}{t-t_0}}\right)\right)du\right),\hfill \end{array}$$

(73)

and $\rho \in \left(1,{\displaystyle \frac{1}{{max}_i\left|{\widehat{\gamma }}_i\right|}}\right)$, with ${\widehat{\gamma }}_i:=1-{\displaystyle \frac{{\widehat{\alpha }}_i}{t-t_0}}$.

Furthermore,

$$\underset{K\to \infty }{lim}{\mathit{E}}_{K,\infty }(x,t;x_0,t_0)=0.$$

(74)

Proof. 

The result follows by applying the same argument as in the proof of Proposition 4, adapted to the single squared Bessel process case. Specifically, we use the bound on ${\widehat{c}}_k$ from Proposition 6 in combination with the uniform bound on generalized Laguerre polynomials and then apply the ratio test to show convergence of the tail series.    □

3.3.3. An Analytical Formula for the Conditional Moments of the Squared Bessel Process with the Time-Dependent Dimension

In this subsection, we derive an analytical formula for the $\gamma$-th conditional moment of the squared Bessel process ${\left(X_t\right)}_{t\ge 0}$ with time-dependent dimension $\delta \left(t\right)$ for any $\gamma \ge 0$. The result is obtained by utilizing the TPDF derived in Theorem 3 and is presented in the following theorem.

Theorem 4. 

Let $X_t$ be the squared Bessel process governed by the SDE (2), with initial condition $X_{t_0}=x_0>0$. Suppose the time-dependent dimension function $\delta \left(t\right)$ satisfies Assumptions 1 and 2, and let $\gamma \ge 0$. Then, the γ-th conditional moment of $X_t$ is given by

$$\begin{array}{cc}\hfill \mathbb{E}[X_t^{\gamma }\mid X_{t_0}=x_0]& ={\left(2(t-t_0)\right)}^{\gamma }\left({\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}\right)\hfill \\ \hfill & \times \sum _{k=0}^{\infty }{}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right){\widehat{c}}_k(t,t_0;x_0),\hfill \end{array}$$

(75)

for $t>t_0$, where the coefficients ${\widehat{c}}_k(t,t_0;x_0)$ are recursively defined in Theorem 3.

Proof. 

This result is an immediate consequence of Theorem 2. It follows by setting $m=1$, $\omega =1$, and $\beta =t-t_0$, which reduces the conic combination $Y_t^{\left(m\right)}$ to the single squared Bessel process $X_t$. Substituting these values into the moment formula in Theorem 2 yields the expression in Equation (75).    □

The following corollary provides a simplified expression for the $\gamma$-th conditional moment of the squared Bessel process in the special case where $\gamma =N\in {\mathbb{N}}_0$. In this setting, the generalized hypergeometric function appearing in Theorem 4 reduces to a finite polynomial because its first argument is a non-positive integer. This truncation transforms the infinite series into a finite sum, making the computation of integer moments more efficient. The result also clarifies the dependence of the moment on the time-dependent dimension $\delta \left(t\right)$, the moment order N, and the recursively defined coefficients ${\widehat{c}}_k(t,t_0;x_0)$, providing a practical formula for evaluating integer moments.

Corollary 4. 

Let $X_t$ be the squared Bessel process governed by the SDE (2), with initial condition $X_{t_0}=x_0>0$. Suppose that the dimension function $\delta \left(t\right)$ satisfies Assumptions 1 and 2. If $\gamma =N\in {\mathbb{N}}_0$, then the N-th conditional moment of $X_t$ is given by

$$\begin{array}{cc}\hfill \mathbb{E}[X_t^N\mid X_{t_0}=x_0]& ={\left(2(t-t_0)\right)}^N\left({\displaystyle \frac{\mathsf{\Gamma }\left(N+\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}\right)\hfill \\ \hfill & \times \sum _{k=0}^N{}_2\mathbf{F}_1\left(-k,N+{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right){\widehat{c}}_k(t,t_0;x_0),\hfill \end{array}$$

(76)

for $t>t_0$, where the coefficients ${\widehat{c}}_k(t,t_0;x_0)$ are recursively defined in Theorem 3.

Proof. 

The proof of this corollary follows directly from the proof of Theorem 4 and the proof of Corollary 1 by using the property of the hypergeometric function to truncate the infinite sum on the right-hand side of Equation (75) into the finite sum which is in the same situation as Corollary 1. This completes the proof.    □

The formula for the N-th conditional moment of $X_t$ presented in Corollary 4 is a generalized expression that reduces seamlessly to the classical result derived by Jeanblanc, Yor, and Chesney [21], when the dimension is constant, i.e., $\delta \left(t\right)=\delta \ge 2$. In this case, the coefficients ${\widehat{c}}_k(t,t_0;x_0)$ in the Laguerre expansion are time-invariant, and the generalized hypergeometric function ${}_2\mathbf{F}_1\left(-k,N+{\displaystyle \frac{\delta }{2}};{\displaystyle \frac{\delta }{2}};1\right)$ becomes a finite sum due to the non-positive integer parameter $-k$. Consequently, the moment formula simplifies to

$$\begin{array}{cc}\hfill \mathbb{E}[X_t^N\mid X_{t_0}=x_0]& ={\left(2(t-t_0)\right)}^N\left({\displaystyle \frac{\mathsf{\Gamma }\left(N+\frac{\delta }{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta }{2}\right)}}\right)\hfill \\ \hfill & \times \sum _{k=0}^N{\displaystyle \frac{{(-1)}^k}{k!}}{}_2\mathbf{F}_1\left(-k,N+{\displaystyle \frac{\delta }{2}};{\displaystyle \frac{\delta }{2}};1\right){\left({\displaystyle \frac{x_0}{2(t-t_0)}}\right)}^k,\hfill \end{array}$$

which exactly matches the closed-form result found in Jeanblanc, Yor, and Chesney [21] for constant-dimension squared Bessel processes. This confirms that Corollary 4 not only extends the classical formulation to the case of time-varying dimensions $\delta \left(t\right)$ but also retains consistency with known special cases.

For squared Bessel processes with the time-dependent dimension, an alternative approach developed by Revuz and Yor [6] and Chesney [21] computes the N-th conditional moment via a recursive method. This technique is based on the infinitesimal generator of the process and yields a Volterra-type integral relation:

$$\mathbb{E}[X_t^N\mid X_{t_0}=x_0]=x_0^N+N{\int }_{t_0}^t\left(\delta \left(s\right)+2(N-1)\right)\mathbb{E}[X_s^{N-1}\mid X_{t_0}=x_0]ds.$$

While this recursive method provides a theoretical framework for deriving moments, it requires computing all lower-order moments and performing numerical integration, which becomes computationally intensive for large N or complex forms of $\delta \left(t\right)$.

By contrast, the analytical formula presented in Corollary 4 expresses the N-th moment in closed form, involving only generalized hypergeometric functions, Laguerre expansion coefficients, and gamma functions. This eliminates the need for iterative numerical integration, significantly reducing computational complexity and increasing efficiency, especially for high-order moments. Moreover, the analytical formulation automatically incorporates the dynamics of the time-dependent dimension $\delta \left(t\right)$, making it particularly suitable for applications in stochastic modeling where time-inhomogeneous behavior plays a critical role. Overall, it offers a robust and computationally efficient framework for evaluating conditional moments of squared Bessel processes in both stationary and non-stationary settings.

3.3.4. Truncation Error Analysis

As shown in the previous subsection, the formula for the conditional moment of the squared Bessel process is expressed as a series expansion. In this part, we analyze the truncation error associated with this expansion.

Let

$$U^{(\gamma ,K)}(t,t_0;x_0):={\left(2(t-t_0)\right)}^{\gamma }\sum _{k=K+1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}{\widehat{c}}_k(t,t_0;x_0){}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right),$$

(77)

denote the truncation error of order K for the conditional moment $\mathbb{E}[X_t^{\gamma }\mid X_{t_0}=x_0]$. We define the following bound:

$$\begin{array}{cc}\hfill C^{\left(\rho \right)}(t,t_0;x_0)& ={\displaystyle \frac{{\left(2(t-t_0)\right)}^{\gamma }\mathsf{\Gamma }\left(\gamma +\frac{\delta \left(t\right)}{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}exp\left({\displaystyle \frac{x_0}{2(t-t_0)}}·{\displaystyle \frac{\rho }{1+{\widehat{\gamma }}_1\rho }}\right)\hfill \\ \hfill & \times exp\left(-{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\delta }^{\left(1\right)}\left(u\right)ln\left(1-\rho \left(1-{\displaystyle \frac{t-u}{t-t_0}}\right)\right)du\right)\hfill \\ \hfill & \times \sum _{k=K+1}^{\infty }{\rho }^{-k}\left|{}_2\mathbf{F}_1\left(-k,\gamma +{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right)\right|,\hfill \end{array}$$

(78)

where the constants $\rho >1$ and ${\widehat{\gamma }}_1$ are defined as in Proposition 6.

Proposition 8. 

The truncation error $U^{(\gamma ,K)}(t,t_0;x_0)$ defined in Equation (77) satisfies

$$\left|U^{(\gamma ,K)}(t,t_0;x_0)\right|\le C^{\left(\rho \right)}(t,t_0;x_0),\forall \gamma \in \mathbb{R},K\in \mathbb{N},$$

(79)

and

$$\underset{K\to \infty }{lim}{U}^{(\gamma ,K)}(t,t_0;x_0)=0.$$

Proof. 

The result follows directly by applying the same argument as in the proof of Proposition 5, using the bound on ${\widehat{c}}_k$ from Proposition 6 and the decay properties of the generalized hypergeometric function.    □

3.4. Analytical Formulas for the TPDF and Conditional Moments of the Bessel Process with the Time-Dependent Dimension

In this section, we derive analytical formulas for the TPDF and conditional moments of the Bessel process with the time-dependent dimension by leveraging the results obtained for the squared Bessel process and applying an appropriate transformation.

3.4.1. Derivation of an Analytical Formula for the TPDF

Proposition 9. 

Let ${\left(R_t\right)}_{t\ge 0}$ be the Bessel process governed by the SDE (1), with initial value $R_{t_0}=r_0>0$. Suppose Assumptions 1 and 2 hold. Then, the TPDF of $R_t$ is given by

$$f_{R_t}(r,t\mid r_0,t_0)={\displaystyle \frac{2e^{-\frac{r^2}{2(t-t_0)}}{r}^{\delta \left(t\right)-1}}{{\left(2(t-t_0)\right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{c}}_k(t,t_0;r_0){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{r^2}{2(t-t_0)}}\right),$$

(80)

for $r>0$ and $t>t_0$, where the coefficients ${\widehat{c}}_k(t,t_0;r_0)$ are defined recursively as in Theorem 3.

Proof. 

The Bessel and squared Bessel processes satisfy the relationship

$$X_t=R_t^2.$$

(81)

Under Assumption 1, where $\delta \left(t\right)\ge 2$, the Bessel process $R_t$ almost surely does not hit zero. Therefore, we also have

$$R_t=\sqrt{X_t}.$$

(82)

This implies that the density functions are related via the change in the variable formula:

$$f_{R_t}(r,t\mid x_0^2,t_0)=2r·f_{X_t}(r^2,t\mid x_0^2,t_0).$$

(83)

Substituting the analytical formula for $f_{X_t}$ from Theorem 3 into Equation (83) yields:

$$f_{R_t}(r,t\mid x_0^2,t_0)={\displaystyle \frac{2e^{-\frac{r^2}{2(t-t_0)}}{r}^{\delta \left(t\right)-1}}{{\left(2(t-t_0)\right)}^{\frac{\delta \left(t\right)}{2}}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}}{\widehat{c}}_k(t,t_0;x_0^2){\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{r^2}{2(t-t_0)}}\right).$$

(84)

Finally, by setting $r_0=x_0^2$, we obtain Formula (80).    □

3.4.2. Analytical Formulas for the Conditional Moments of the Bessel Process

We now derive the analytical formula for the conditional moments of the Bessel process $R_t$.

Proposition 10. 

Let ${\left(R_t\right)}_{t\ge 0}$ be the Bessel process governed by the SDE (1), with initial value $R_{t_0}=r_0>0$. Suppose $\delta \left(t\right)\ge 2$, and let $\gamma >0$ be a real number. Then, the γ-th conditional moment of $R_t$ is given by

$$\mathbb{E}[R_t^{\gamma }\mid R_{t_0}=r_0]={\left(2(t-t_0)\right)}^{{\displaystyle \frac{\gamma }{2}}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+\frac{\gamma }{2}\right)}{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}\sum _{k=0}^{\infty }{\widehat{c}}_k(t,t_0;r_0){}_2\mathbf{F}_1\left(-k,{\displaystyle \frac{\delta \left(t\right)}{2}}+{\displaystyle \frac{\gamma }{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right),$$

(85)

for $t>t_0$, where the coefficients ${\widehat{c}}_k(t,t_0;r_0)$ are defined recursively as in Theorem 3.

Proof. 

By starting from the definition of the $\gamma$-th conditional moment, where

$$\mathbb{E}[R_t^{\gamma }\mid R_{t_0}=r_0]={\int }_0^{\infty }{r}^{\gamma }{f}_{R_t}(r,t\mid r_0,t_0)dr;$$

(86)

substituting Equation (80) into Equation (86); interchanging the summation and integration; and applying the change in variables

$$z={\displaystyle \frac{r^2}{2(t-t_0)}},\mathrm{so}\mathrm{that}dr={\displaystyle \frac{(t-t_0)}{r}}dz,$$

the integral becomes

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{r}^{\delta \left(t\right)-1+\gamma }{e}^{-{\displaystyle \frac{r^2}{2(t-t_0)}}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left({\displaystyle \frac{r^2}{2(t-t_0)}}\right)dr& ={\displaystyle \frac{1}{2}}{\left(2(t-t_0)\right)}^{{\displaystyle \frac{\delta \left(t\right)}{2}}+{\displaystyle \frac{\gamma }{2}}}\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-z}{z}^{{\displaystyle \frac{\delta \left(t\right)}{2}}-1+{\displaystyle \frac{\gamma }{2}}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left(z\right)dz.\hfill \end{array}$$

(87)

Using the standard integral identity for Laguerre polynomials, where

$${\int }_0^{\infty }{e}^{-z}{z}^{{\displaystyle \frac{\delta \left(t\right)}{2}}-1+{\displaystyle \frac{\gamma }{2}}}{\mathbf{L}}_k^{\left({\displaystyle \frac{\delta \left(t\right)}{2}}-1\right)}\left(z\right)dz={\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+\frac{\gamma }{2}\right)\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}+k\right)}{k!\mathsf{\Gamma }\left(\frac{\delta \left(t\right)}{2}\right)}}{}_2\mathbf{F}_1\left(-k,{\displaystyle \frac{\delta \left(t\right)}{2}}+{\displaystyle \frac{\gamma }{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1\right),$$

(88)

and combining all steps leads to Formula (85).    □

Remark 1. 

The formula presented in Proposition 10 holds for all $\gamma \in {\mathbb{R}}_{+}$. However, in the special case where $\gamma =N$ with $N\in {\mathbb{N}}_0$ being an even integer, the series on the right-hand side of Equation (85) simplifies. Specifically, following the same reasoning as in the proof of Corollary 1, the hypergeometric function ${}_2\mathbf{F}_1(-k,N+{\displaystyle \frac{\delta \left(t\right)}{2}};{\displaystyle \frac{\delta \left(t\right)}{2}};1)$ vanishes for $k>N$, allowing the infinite series to be truncated to a finite sum over $k=0,1,\dots ,N$.

4. Financial Applications

In this section, we demonstrate the applicability of our analytical results to the pricing of interest rate swaps (IRSs). An IRS is a forward contract in which a floating interest rate is exchanged for a fixed rate over a predefined period.

There exist several types of IRSs depending on the contractual agreement between counterparties. In this work, we follow the formulation described by Thamrongrat and Rujivan [35], where the value of an IRS, denoted by $V_{\mathrm{swap}}$, is given by

$$V_{\mathrm{swap}}=P_{\mathrm{fl}}-P_{\mathrm{fix}},$$

(89)

where $P_{\mathrm{fl}}$ and $P_{\mathrm{fix}}$ represent the present values of the floating-rate and fixed-rate legs of the swap, respectively.

According to Thamrongrat and Rujivan [35], the fixed-rate leg is given by

$$P_{\mathrm{fix}}=\sum _{i=1}^N{k}_i{e}^{-(t_i-t_0)r_{t_i}}+L{e}^{-(t_N-t_0)r_{t_N}},$$

(90)

where $N>2$ is the number of payment dates, $t_0$ is the initiation time, $t_i$ denotes the i-th payment date, L is the notional principal, $k_i$ is the fixed payment at time $t_i$, and $r_{t_i}$ is the short rate with maturity $t_i$.

The floating-rate leg is expressed as

$$P_{\mathrm{fl}}=(L+k_0)e^{-(t_1-t_0)r_{t_1}},$$

(91)

where $k_0$ denotes the known floating-rate payment scheduled for the next payment date.

By substituting Equations (90) and (91) into Equation (89), the swap value becomes

$$V_{\mathrm{swap}}(r_{t_i},t_i;t_0,N)=\sum _{i=1}^N{K}_i{e}^{-(t_i-t_0)r_{t_i}},$$

(92)

where the coefficients $K_i$ are defined as

$$K_i=\left\{\begin{array}{cc}L+k_0-k_1,\hfill & \mathrm{if}i=1,\hfill \\ -k_i,\hfill & \mathrm{if}i=2,\dots ,N-1,\hfill \\ -(k_N+L),\hfill & \mathrm{if}i=N.\hfill \end{array}\right.$$

(93)

According to Thamrongrat and Rujivan [35], the short rate $r_{t_i}$ is modeled by a one-factor extended Cox–Ingersoll–Ross (ECIR) process. In this paper, we consider a more general and realistic framework by modeling the short rate as a conic combination of independent squared Bessel processes:

$$r_t=\sum _{j=1}^m{\omega }_j{x}_t^{\left(j\right)},$$

(94)

where ${\omega }_j>0$, $t\in (t_0,T]$, and each component $x_t^{\left(j\right)}$ evolves according to

$$d{x}_t^{\left(j\right)}={\delta }_j\left(t\right)dt+2\sqrt{x_t^{\left(j\right)}}d{W}_t^{\left(j\right)},x_{t_0}^{\left(j\right)}=x_0^{\left(j\right)}>0,$$

(95)

for $j=1,\dots ,m$. The functions ${\delta }_j\left(t\right)>0$ are time-dependent and satisfy Assumptions 1 and 2, while ${\left(W_t^{\left(j\right)}\right)}_{t\ge 0}$ are independent Brownian motions on a filtered probability space $(\mathsf{\Omega },\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbb{Q})$ under the risk-neutral measure $\mathbb{Q}$.

The motivation for using a multifactor model stems from empirical studies showing that interest rate dynamics are influenced by multiple economic factors. This modeling approach aligns with the literature on multifactor term structure models, including Chen and Scott [36], Longstaff and Schwartz [37], and Chen and Scott [38], which demonstrate improved flexibility and accuracy in the pricing of interest rate derivatives.

To determine the fair price of the IRS at time $t_0$, we compute the conditional expectation under the risk-neutral measure:

$$\mathrm{FIRS}(r_{t_0},t_0;N):={\mathbb{E}}^{\mathbb{Q}}\left[V_{\mathrm{swap}}(r_{t_i},t_i;t_0,N)|{\mathcal{F}}_{t_0}\right]=\sum _{i=1}^N{K}_i{\mathbb{E}}^{\mathbb{Q}}\left[e^{-(t_i-t_0)r_{t_i}}|r_{t_0}=\sum _{j=1}^m{\omega }_j{x}_0^{\left(j\right)}\right].$$

(96)

Computing a closed-form solution to Equation (96) is challenging without explicit knowledge of the distribution of $r_t$. However, since $r_t$ is modeled as a conic combination of independent squared Bessel processes, we can directly apply the analytical formulas derived in Section 3 to obtain a series representation for the fair IRS value. The result is formally stated in the following theorem.

Theorem 5. 

An IRS has predetermined dates $t_0<t_1<\cdots <t_N=T$. If the short rate is generated by the stochastic process of Equations (94) and (95), then the fair price of the IRS at time $t_0$ is given by

$${\mathrm{FIRS}}^{\left(\beta \right)}(r_{t_0},t_0;N)=\sum _{k=0}^{\infty }\sum _{i=1}^N{K}_i{\widehat{b}}_k^{\left(m\right)}(t_i,t_0;r_{t_0}){\displaystyle \frac{{\left(2{\beta }_i(t_i-t_0)\right)}^k}{{(2{\beta }_i(t_i-t_0)+1)}^{\frac{\delta \left(t_i\right)}{2}+k}}},$$

(97)

where the coefficients ${\widehat{b}}_k^{\left(m\right)}$ are defined recursively as in Theorem 1 and ${\beta }_i$ are any positive constants that satisfy ${\beta }_i>{\displaystyle \frac{1}{2}}{max}_j{\omega }_{i,j}(t_i-t_0)$ for $i=1,\cdots ,N$ and $j=1,\cdots ,m$.

Proof. 

In this proof, we will begin with the definition of expectation in Equation (96). After that we will use a formula of the integral in terms of Laguerre polynomial to simplify the expression.

From Equation (96), by using the formula of expectation, we observe that

$${\mathbb{E}}^{\mathbb{Q}}\left[e^{-(t_i-t_0)r_{t_i}}|r_{t_0}=\sum _{j=0}^m{\omega }_j{x}_0^{\left(j\right)}\right]={\int }_0^{\infty }exp(-(t_i-t_0)x)f_{r_{t_i}}(x,t_i|{\mathit{x}}_{t_0},t_0)dx,$$

(98)

for $x>0$ and ${\mathit{x}}_{t_0}=(x_0^{\left(1\right)},\cdots ,x_0^{\left(m\right)})$.

From Equation (94), we see that $r_{t_i}$ is distributed according to a conic combination of independent squared Bessel processes; then by using Equation (17), the right-hand side of Equation (98) can be rewritten as

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{e}^{-(t_i-t_0)x}{f}_{r_{t_i}}(x,t_i|{\mathit{x}}_{t_0},t_0)dx& =\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{{\left(2{\beta }_i\right)}^{\frac{\delta \left(t_i\right)}{2}}\mathsf{\Gamma }(\frac{\delta \left(t_i\right)}{2}+k)}}{\widehat{b}}_k^{\left(m\right)}(t_i;t_0,{\mathit{x}}_{t_0})\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-(t_i-t_0+{\displaystyle \frac{1}{2{\beta }_i}})x}{x}^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}{\mathrm{L}}_k^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}\left({\displaystyle \frac{x}{2{\beta }_i}}\right)dx.\hfill \end{array}$$

(99)

So $z={\displaystyle \frac{x}{2{\beta }_i}}$ implies that $x=2{\beta }_{i}z$ and $dx=2{\beta }_{i}dz$. Then the integral in Equation (99) becomes

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{e}^{-(t_i-t_0+{\displaystyle \frac{1}{2{\beta }_i}})x}{x}^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}{\mathrm{L}}_k^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}\left({\displaystyle \frac{x}{2{\beta }_i}}\right)dx& ={\left(2{\beta }_i\right)}^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}}\hfill \\ \hfill & \times {\int }_0^{\infty }{e}^{-(2{\beta }_i(t_i-t_0)+1)z}{z}^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}{\mathrm{L}}_k^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}\left(z\right)dz.\hfill \end{array}$$

(100)

According to Gradshteyn and Ryzhik [34], the integral on the right-hand side of Equation (100) can be written as

$${\int }_0^{\infty }{e}^{-(2{\beta }_i(t_i-t_0)+1)z}{z}^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}{\mathrm{L}}_k^{{\displaystyle \frac{\delta \left(t_i\right)}{2}}-1}\left(z\right)dz={\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\delta \left(t_i\right)}{2}+k\right){\left(2{\beta }_i(t_i-t_0)\right)}^k}{k!{(2{\beta }_i(t_i-t_0)+1)}^{\frac{\delta \left(t_i\right)}{2}+k}}}.$$

(101)

By substituting Equation (101) into Equation (100), followed by Equation (99) and then finally to Equation (98), we obtain

$${\mathbb{E}}^{\mathbb{Q}}\left[e^{-(t_i-t_0)r_{t_i}}|r_{t_0}=\sum _{j=0}^m{\omega }_j{x}_0^{\left(j\right)}\right]=\sum _{k=0}^{\infty }{\widehat{b}}_k^{\left(m\right)}(t_i,t_0;{\mathit{x}}_{t_0}){\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\delta \left(t_i\right)}{2}+k\right){\left(2{\beta }_i(t_i-t_0)\right)}^k}{{(2{\beta }_i(t_i-t_0)+1)}^{\frac{\delta \left(t_i\right)}{2}+k}}}.$$

(102)

Eventually, by substituting Equation (102) into Equation (96), we obtain Equation (97). This completes the proof.    □

This result highlights the analytical tractability of our framework and its utility in the valuation of fixed-income derivatives. The series representation in Equation (97) enables efficient computation and provides insights into the effects of time-dependent model parameters on interest rate dynamics.

5. Numerical Results and Discussions

This section is dedicated to demonstrating the accuracy and robustness of the analytical formulas derived in Section 3 for the TPDF and conditional moments of a conic combination of independent squared Bessel processes with time-dependent dimensions. In addition to the theoretical developments, we have also performed a rigorous truncation error analysis, establishing convergence properties and providing bounds for the approximation errors associated with the series representations.

To validate the correctness and computational efficiency of the proposed formulas, we present two numerical examples comparing the analytical results with those obtained via MC simulations. The first example evaluates the accuracy of the TPDF approximation, while the second focuses on the conditional moments of various orders. These experiments confirm the theoretical predictions and highlight the practical applicability of our framework in both probabilistic modeling and financial computation.

The MC simulations were conducted by generating $N_p$ sample paths of the conic combination of squared Bessel processes over prescribed time intervals using the Euler–Maruyama method. All numerical computations and graphical visualizations were implemented in Mathematica 11 and executed on a standard notebook with the following specifications: Intel(R) Core(TM) i5-8265U CPU @ 1.60GHz, 8 GB of RAM, and running Windows 11 Pro (64-bit operating system). Detailed numerical comparisons and graphical illustrations are presented in the following subsections.

5.1. The Accuracy of the Laguerre Series Expansion for the TPDF of the Conic Combination

In this subsection, we validate the accuracy of the analytical formula for the TPDF of a conic combination of independent squared Bessel processes with time-dependent dimensions, as established in Theorem 1. Our main objective is to compare the Laguerre series expansion in Formula (17) with MC simulation results, thereby illustrating the precision and applicability of the derived series representation in practical settings.

We consider the conic combination

$$Y_t^{\left(m\right)}:=\sum _{j=1}^m{\omega }_{j,m}{X}_t^{(j,m)},$$

(103)

where ${\left(X_t^{(j,m)}\right)}_{t\ge 0}$ are independent squared Bessel processes governed by the SDE (2). The time settings $t_0=0.5\mathrm{to}t=5$, weights, and initial values are given by

$${\omega }_{j,m}={\displaystyle \frac{j}{m}},\mathrm{and}{x}_0^{(j,m)}={\displaystyle \frac{3j}{2}},\mathrm{for}j=1,\dots ,m.$$

(104)

To examine the effect of dimensional complexity, we test two cases: $m=3$ and $m=10$. The time-dependent dimensions ${\delta }_j\left(t\right)$ are set as follows:

• Case 1: $m=3$,

  $${\delta }_{j,m}\left(t\right)=3\sqrt{j}+(j-1)\left(t+0.1(j-2)sin\left(t\right)\right),$$

  (105)
• Case 2: $m=10$,

$${\delta }_{j,m}\left(t\right)=\left\{\begin{array}{cc}3\sqrt{j},\hfill & \mathrm{for}j=1,2,\hfill \\ 3\sqrt{j}+t,\hfill & \mathrm{for}j=3,4,5,6,\hfill \\ 4+0.1sin\left(t\right),\hfill & \mathrm{for}j=7,8,9,10.\hfill \end{array}\right.$$

(106)

These specifications ensure that the model incorporates a variety of functional forms—constant, linear, and trigonometric—while satisfying Assumptions 1 and 2, thus justifying the use of our analytical framework.

Example 1. 

In this example, we consider the random variable $Y_t^{\left(m\right)}$ as defined in Equation (103) for two representative cases: $m=3$ and $m=10$. The objective of these two settings is to examine the performance and scalability of our analytical TPDF formula under different levels of dimensional complexity. The case $m=3$ represents a low-dimensional setting, which allows us to assess the baseline accuracy of our series expansion in a simpler regime, while the case $m=10$ presents a more computationally demanding and high-dimensional setting to evaluate the robustness and convergence behavior of the Laguerre series expansion in more realistic financial or physical applications. The parameters involved in Equation (103) are specified by Equations (104), (105), and (106), incorporating diverse forms of time-dependent dimensions including linear, trigonometric, and constant functions to reflect a wide range of practical modeling scenarios.

As established in Theorem 1, the analytical expression for $f_{Y_t^{\left(m\right)}}^{\left(\beta \right)}(y,t\mid {\mathit{x}}_{t_0},t_0)$ relies on computing the Laguerre expansion coefficients ${\widehat{b}}_k^{\left(m\right)}$ recursively via Equations (18)–(21). The convergence behavior of these coefficients, which is fundamental to the validity and numerical stability of the series expansion, is guaranteed by our theoretical framework and is further supported by the numerical evidence illustrated in Figure 1. As shown in the figure, the coefficients decay rapidly to zero for both cases $m=3$ and $m=10$, with the rate of decay being noticeably faster in the lower-dimensional setting $m=3$. This aligns with the intuition that as dimensionality increases, more terms are required in the expansion to achieve a similar level of accuracy.

Once the coefficients are computed, the TPDF is evaluated via the Laguerre series expansion in Equation (17). Figure 2 presents a visual comparison between the analytical density and a histogram obtained from MC simulations using $5\times {10}^6$ sample paths. The close alignment between the series-based TPDF and the MC histogram highlights the high fidelity of our formula even in relatively high-dimensional settings ($m=10$) and under non-trivial time-dependent dynamics.

In summary, this example demonstrates that the Laguerre series expansion proposed in this paper provides a highly accurate and computationally efficient method for evaluating the TPDF of a conic combination of squared Bessel processes with time-dependent dimensions. The agreement between the theoretical density and empirical distribution from MC simulations strongly supports the validity and robustness of the proposed analytical approach.

5.2. The Performance of Our Analytical Formula for the Conditional Moments

Example 2. 

In this example, we evaluate the performance and accuracy of the analytical formulas for the conditional moments of the conic combination of independent squared Bessel processes, as developed in Theorem 2 and Corollary 1. We use the same conic process $Y_t^{\left(m\right)}$ defined in Equation (103), with parameters set according to Equations (104)–(106), and consider two cases: $m=3$ and $m=10$.

The objective is twofold: (i) to validate the correctness of the closed-form expressions for both fractional and integer moments and (ii) to assess the convergence behavior and computational efficiency relative to standard MC simulations. Specifically, we compute the conditional expectations $\mathbb{E}[{\left(Y_t^{\left(m\right)}\right)}^{\gamma }\mid {\mathit{x}}_{t_0}]$ for $\gamma =0.5,2$ when $m=3$ and for $\gamma =0.2,1.2$ when $m=10$ and compare them with empirical estimates obtained via MC simulations with up to $5\times {10}^6$ sample paths. The accuracy of our formulas is clearly demonstrated in Figure 3, where the MC results closely converge to the analytical benchmarks.

In addition, we examine the truncation error of the Laguerre series expansion for positive real moments.

Table 1. Absolute values of the truncation errors defined in Example 2 for the Laguerre series expansion of the conditional moments of $Y_t^{\left(m\right)}$ with $\gamma =0.5,1,2$ and $m=3,10$. The results demonstrate rapid convergence of the series as the truncation level k increases. In particular, for integer orders $\gamma =1$ and $\gamma =2$, the truncation error vanishes exactly when $k\ge \gamma$, consistent with Corollary 1.

k	$\left|{\mathit{V}}_3^{(0.5,\mathit{k})}\right|$	$\left|{\mathit{V}}_3^{(1,\mathit{k})}\right|$	$\left|{\mathit{V}}_3^{(2,\mathit{k})}\right|$	$\left|{\mathit{V}}_{10}^{(0.5,\mathit{k})}\right|$	$\left|{\mathit{V}}_{10}^{(1,\mathit{k})}\right|$	$\left|{\mathit{V}}_{10}^{(2,\mathit{k})}\right|$
0	$2.98\times {10}^{-1}$	$5.10$	$779.30$	$5.44\times {10}^{-1}$	$15.15$	$6013.68$
1	$2.57\times {10}^{-2}$	0	$160.15$	$3.27\times {10}^{-2}$	0	$661.830$
2	$2.46\times {10}^{-3}$	0	0	$2.94\times {10}^{-3}$	0	0
3	$5.40\times {10}^{-4}$	0	0	$5.36\times {10}^{-4}$	0	0
4	$1.08\times {10}^{-4}$	0	0	$9.20\times {10}^{-5}$	0	0
5	$3.07\times {10}^{-5}$	0	0	$2.25\times {10}^{-5}$	0	0
6	$8.57\times {10}^{-6}$	0	0	$5.16\times {10}^{-6}$	0	0
7	$2.88\times {10}^{-6}$	0	0	$1.50\times {10}^{-6}$	0	0
8	$9.86\times {10}^{-7}$	0	0	$4.07\times {10}^{-7}$	0	0
9	$3.76\times {10}^{-7}$	0	0	$1.31\times {10}^{-7}$	0	0
10	$1.48\times {10}^{-7}$	0	0	$4.04\times {10}^{-8}$	0	0

reports the decay behavior of the absolute values of the truncation errors, as defined in Equation (55), associated with the conic process $Y_t^{\left(m\right)}$ for $k\ge 0$ and moment orders $\gamma =0.5$, $\gamma =1$, and $\gamma =2$. The results indicate that the series converges rapidly to zero as k increases. Notably, for integer moment orders γ, the truncation error vanishes exactly when $k\ge \gamma$, consistent with Corollary 1.

As illustrated in Figure 3, the results obtained from MC simulations converge closely to those computed using our analytical formulas, thereby confirming the accuracy and validity of the closed-form expressions. A natural question then arises: why not rely solely on MC simulations? The answer lies in computational efficiency. Although MC methods are widely applicable, they are computationally expensive. For example, simulating $5\times {10}^6$ paths takes over 8000 s when $m=3$ and nearly 32,000 s for $m=10$. In contrast, our analytical formulas evaluate the same conditional moments in just 0.3 and 2.37 s, respectively.

Table 2. Comparison of computational time and accuracy for evaluating $\mathbb{E}[{\left(Y_t^{\left(m\right)}\right)}^{0.5}\mid {\mathit{x}}_{t_0}]$ using the analytical formula versus MC simulations as illustrated in Example 2. The table reports error percentages, computational times for both methods, and the corresponding speedup factors (reduction folds).

${\mathit{N}}_{\mathit{p}}$	Error (%)	TMC (s)	${\mathit{T}}_{\mathit{c}}$ (s)	Reduction (Folds)
Case: $m=3$, $\gamma =0.5$
$1\times {10}^4$	$0.18$	$15.55$	$0.3$	52
$1\times {10}^5$	$0.05$	$154.66$	$0.3$	516
$1\times {10}^6$	$0.038$	$1617.98$	$0.3$	5393
$5\times {10}^6$	$0.026$	$8018.85$	$0.3$	$26,730$
Case: $m=10$, $\gamma =0.5$
$1\times {10}^4$	$0.1$	$65.28$	$2.37$	28
$1\times {10}^5$	$0.014$	$698.97$	$2.37$	295
$1\times {10}^6$	$0.007$	$5995.61$	$2.37$	2530
$5\times {10}^6$	$0.005$	$31,850.05$	$2.37$	$13,439$

provides a comprehensive comparison between the two methods, reporting error percentages, computational times, and speedup factors. These results underscore the substantial performance advantage of the analytical approach, achieving up to a $26,000$-fold reduction in computation time without compromising accuracy.

In summary, this example strongly supports the theoretical findings by confirming both the rapid convergence of the derived series expansions and their substantial advantages in computational cost. These results underscore the potential of our analytical approach as a highly effective alternative to traditional simulation-based methods in stochastic modeling applications.

6. Conclusions

This paper has presented a comprehensive analytical framework for characterizing the transition dynamics and statistical moments of conic combinations of independent squared Bessel processes with time-dependent dimensions. By deriving closed-form expressions for the TPDF using Laguerre series expansions and formulating conditional moments through generalized hypergeometric functions, we have significantly advanced the classical theory of squared Bessel processes into the time-inhomogeneous regime. These contributions not only generalize well-established results but also open new pathways for modeling a wide array of non-stationary diffusion systems that arise in complex stochastic environments.

The theoretical development is further elevated by its practical relevance. Through the lens of interest rate swap valuation, we have demonstrated how the proposed formulas offer a viable alternative to conventional numerical methods. The ability to obtain closed-form solutions in such a setting underlines the strength of the analytical framework, especially in terms of speed, accuracy, and interpretability—properties crucial in real-time financial applications.

To ensure the practical feasibility of the proposed approach, we have conducted a detailed analysis of truncation errors associated with the series expansions, yielding explicit convergence guarantees and quantifiable error bounds. The empirical results, benchmarked against MC simulations, affirm the accuracy and robustness of our methods while highlighting substantial improvements in computational efficiency—achieving speedups by several orders of magnitude.

Beyond the specific case of squared Bessel processes, the methodologies developed in this paper provide foundational tools that can be adapted to other classes of time-inhomogeneous stochastic processes. Future work may explore the extension of this framework to conic combinations involving correlated squared Bessel components, multidimensional coupled systems, and broader applications in stochastic control, nonlinear filtering, and dynamic risk assessments. These avenues not only promise to enhance theoretical understanding but also hold substantial promise for real-world implementation across finance, physics, and engineering domains.