Advanced Operator Theory for Energy Market Trading: A New Framework

Abstract

This paper analyzes a parabolic operator $\mathcal{L}$ that generalizes several well-known operators commonly used in financial mathematics. We establish the existence and uniqueness of the Feller semigroup associated with $\mathcal{L}$ and derive its explicit analytical representation. The theoretical framework developed in this study provides a robust foundation for modeling stochastic processes relevant to financial markets. Furthermore, we apply these findings to energy market trading by developing specialized simulation algorithms and forecasting models. These methodologies were tested across all assets comprising the S&P 500 Energy Index, evaluating their predictive accuracy and effectiveness in capturing market dynamics. The empirical analysis demonstrated the practical advantages of employing generalized semigroups in modeling non-Gaussian market behaviors and extreme price fluctuations.

1. Introduction

The pricing of financial derivatives in energy markets is a complex and multifaceted challenge that requires advanced mathematical tools capable of capturing market uncertainties and stochastic behaviors. Traditional models, such as the Black–Scholes framework and its extensions, often rely on assumptions that may not fully reflect the realities of energy market dynamics, particularly in the presence of heavy-tailed distributions and skewed return profiles. To address these limitations, we analyze the role of generalized semigroups of operators in option pricing, extending classical methodologies based on Feller semigroups Ethier and Kurtz (2009); Peskir and Shiryaev (2006). Specifically, we focus on a parabolic operator $\mathcal{L}$ that generalizes well-known differential operators commonly used in financial modeling, thereby providing a more flexible and comprehensive approach to modeling asset price evolution.

Semigroup theory provides a rigorous and powerful framework for studying the temporal evolution of financial processes, establishing a crucial link between stochastic differential equations (SDEs) and partial differential equations (PDEs).

Our approach is based on the theoretical tool provided by Bufalo et al. (2019), which is valid for any linear or nonlinear parabolic operator $\mathcal{L}$ (as well as that in Equation (13)). In particular, we use the techniques of (Bufalo et al. 2019, Appendix A) to show the generation results and the related analytic representation.

The Feynman–Kac formula Karatzas and Shreve (1998); Shreve (2004) plays an essential role in this connection, enabling the representation of option prices as expectations of stochastic processes. This approach not only facilitates analytical tractability but also enhances computational efficiency in derivative pricing. The fundamental existence and uniqueness of results for Feller semigroups associated with $\mathcal{L}$ ensure the mathematical robustness of our proposed framework.

Beyond its theoretical significance, our study has direct applications to energy market trading, where price dynamics exhibit significant volatility and non-Gaussian characteristics. We develop numerical algorithms that leverage the properties of generalized semigroups to simulate asset price movements, providing more accurate forecasting models tailored to real-world market conditions. Our empirical validation involved an extensive analysis of the S&P 500 energy index components, evaluating the predictive performance of our model against standard benchmarks such as the Black–Scholes model Black and Scholes (1973) and skew-geometric Brownian motion models Barndorff-Nielsen (2001). The results underscore the advantages of employing generalized semigroups in capturing intricate market behaviors, such as extreme price fluctuations and structural shifts in volatility patterns.

Furthermore, our research contributes to the broader field of financial mathematics by demonstrating the applicability of semigroup theory to derivative pricing in energy markets. By incorporating a more general class of differential operators, we bridge the gap between traditional financial models and the complex stochastic processes observed in commodity and energy trading. The insights gained from this study can inform risk management strategies, portfolio optimization, and pricing methodologies for a wide range of financial instruments beyond energy markets.

This paper is structured as follows: Section 2 introduces the mathematical framework of semigroup theory, emphasizing its relevance to finance and its connections to stochastic analysis. Section 2.2 formulates the generalized operator $\mathcal{L}$ and derives its corresponding semigroup properties, highlighting its implications for option pricing. Section 3 presents numerical validations, including simulations and forecasting techniques, with a comparative analysis against established financial models. Finally, Section 4 concludes with potential directions for future research, particularly concerning the existence of risk-neutral measures, the development of more refined numerical schemes, and broader applications of generalized semigroups in financial modeling.

2. Mathematical Frameworks

2.1. Preliminaries on Semigroup Theory

Semigroup theory plays a fundamental role in the study of linear operators, particularly in connection with stochastic processes.

Let $(\mathcal{X},\parallel ·\parallel )$ be a Banach space on $\mathbb{R}$ and ${\mathcal{L}}_{\mathbb{R}}\left(\mathcal{X}\right)$ the set of linear and bounded operators $T:\mathcal{X}\to \mathcal{X}$.

Definition 1.

($\left(C_0\right)$ Semigroup of Operators)

The functional ${\left(T\left(t\right)\right)}_{t\ge 0}\in {\mathcal{L}}_{\mathbb{R}}\left(\mathcal{X}\right)$, is said to be a strongly continuous, or $\left(C_0\right)$, semigroup of operators on $\mathcal{X}$, if the following properties hold:

i 

$T\left(0\right)$ is the identity function,

ii 

for all $t,s\in {\mathbb{R}}_{+},T(s+t)=T\left(t\right)·T\left(s\right)$,

iii 

fixed $t_0\in {\mathbb{R}}_{+}$ one has ${lim}_{t\to t_0}\parallel T\left(t\right)f-T\left(t_0\right)f\parallel .$ for all $f\in \mathcal{X}$.

Definition 2.

(Semigroup Generator)

Let ${\left(T\left(t\right)\right)}_{t\ge 0}$ be a $\left(C_0\right)$ semigroup on $\mathcal{X}$. The operator $(A,D(A\left)\right)$, defined as

$$Af=\underset{t\to 0^{+}}{lim}{\displaystyle \frac{T\left(t\right)f-f}{t}},D\left(A\right)=\{f\in \mathcal{X}\left|\exists \underset{t\to 0^{+}}{lim}{\displaystyle \frac{T\left(t\right)f-f}{t}}\in \mathcal{X}\right\},$$

is said to be the generator of ${\left(T\left(t\right)\right)}_{t\ge 0}$.

Theorem 1.

(Existence and Uniqueness of a Solution of (ACP)) Given $A:D\left(A\right)\to \mathcal{X}$ a linear operator, consider the following abstract Cauchy problem (ACP):

$$\left\{\begin{array}{c}{u}_t(t,x)=Au(t,x),\hfill \\ u(0,x)=u_0\left(x\right),\hfill \end{array}\right.$$

(1)

with $u_0\in D\left(A\right)$. If A is the generator of a $\left(C_0\right)$ semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on $\mathcal{X}$, then the (ACP) admits a unique solution given by

$$u(t,x)=T\left(t\right)u_0\left(x\right).$$

(2)

Proof. 

See (Engel 2000, chap. II, Proposition 6.2). □

Another crucial characterization for the generation is provided by Feller theory, which links semigroup theory with stochastic differential equations (SDE) (for more details see, e.g., Böttcher et al. (2013), Taira (1984)). Denote by $C\left(J\right)$ the space of all real-valued continuous functions f defined on $J\subseteq \mathbb{R}$. In particular, if f vanishes at infinity, this will be denoted by $C_0\left(J\right)$.

Definition 3.

(Feller Semigroup)

A $\left(C_0\right)$ semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}\in {\mathcal{L}}_{\mathbb{R}}\left(C\left(J\right)\right)$ is a Feller semigroup on $C\left(J\right)$ if it satisfies the following property:

$$(f\in C(J),0\le f\le 1\mathit{on}J)\iff (0\le T(t)f\le 1,t\ge 0\mathit{on}J,T(t)1=1,t\ge 0).$$

1 represents the constant function equal to 1, in these cases. Note that every Feller semigroup is a positive semigroup, i.e., $T\left(t\right)f\ge 0$ for all $f\ge 0,t\ge 0$.

If we leave out the $\left(C_0\right)$ hypothesis above, we obtain a Markov semigroup.

The link with SDE theory can be summarized in this way. Let $(\mathrm{\Omega },\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbb{P})$ be a filtered probability space, and consider ${\left(X_t\right)}_{t\ge 0}$ a Markov process with state space $(J,\mathcal{B}(J\left)\right)$ (for more details see, for instance, Øksendal and Øksendal (2003), Shiryaev (2004)), with dynamics

$$d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{W}_t,X_0=x,$$

(3)

where $W_t$ denotes a standard Brownian motion, and $\mu ,\sigma$ are the drift and diffusion functions, respectively.

Let ${\mathbb{P}}^x(X_t\in B)=\mathbb{P}(X_t\in B|X_0=x)$ for any $B\in \mathcal{B}\left(J\right)$, and

$${\mathbb{E}}^{\mathbb{P}}[f\left(X_t\right)\mid X_0=x]={\int }_{J}f\left(y\right){\mathbb{P}}^x(X_t\in dy),$$

(4)

for any $f\in C_0\left(J\right)$. A Feller process is a Markov process whose transition semigroup $T\left(t\right)f\left(x\right)={\mathbb{E}}^x\left[f\left(X_t\right)\right]$ is a Feller semigroup. Clearly, the generator $(A,D(A\left)\right)$ of ${\left(T\left(t\right)\right)}_{t\ge 0}$ coincides with the infinitesimal generator of the process ${\left(X_t\right)}_{t\ge 0}$, which is defined by

$$Au\left(x\right)=\underset{t\to 0^{+}}{lim}{\displaystyle \frac{{\mathbb{E}}^{\mathbb{P}}[u\left(X_t\right)\mid X_0=x]-u\left(x\right)}{t}},$$

and $D\left(A\right)$ is the set of all u for which the above limit exists for any $x\in J$. One has that

$$Au={\displaystyle \frac{1}{2}}{\sigma }^2{u}^{″}+\mu u^{\prime },$$

for all $u\in D\left(A\right)$. So, we can conclude that if ${\left(X_t\right)}_{t\ge 0}$ is a process which satisfies the SDE (3), then a function $u(t,x)$ defined by (4) for all $x\in J,t\in {\mathbb{R}}_{+}$ has to solve the PDE $u_t\left(x\right)=Au\left(x\right)$.

In particular, if we consider the following (ACP)

$$\left\{\begin{array}{c}{u}_t(t,x)=Au(t,x)-ru(t,x),(t,x)\in {\mathbb{R}}_{+}\times J\hfill \\ u(0,x)=f\left(x\right),x\in J\hfill \end{array}\right.$$

for all $r\ge 0$, its solution is given by the Feller semigroup

$$u(t,x)={\mathbb{E}}^{\mathbb{P}}[e^{-rt}f\left(X_t\right)\mid X_0=x].$$

(5)

The above result is well known as the Feynman–Kac Theorem (see, e.g., (Shiryaev 2004, chap. VI, sct. 4)).

Now, we recall the so-called $Feller$ $classification$ for the generation theorem (see (Engel 2000, chap. VI, sct. 4)).

Definition 4.

(Feller Classification)

If $J=(r_1,r_2)$ is a real interval, with $-\infty \le r_1<r_2\le +\infty$, let A be a second-order differential operator of the type $Au=a\left(x\right)u^{″}+b\left(x\right)u^{\prime }$, where $a,b$ are real continuous functions on J such that $a\left(x\right)>0$ for any $x\in J$. Define the following functions:

$$W\left(x\right)=exp\left(-{\int }_{x_0}^x{\displaystyle \frac{b\left(s\right)}{a\left(s\right)}}ds\right),$$

(6)

$$Q\left(x\right)={\displaystyle \frac{1}{a\left(x\right)W\left(x\right)}}{\int }_{x_0}^{x}W\left(s\right)ds,$$

(7)

$$R\left(x\right)=W\left(x\right){\int }_{x_0}^x{\displaystyle \frac{1}{a\left(s\right)W\left(s\right)}}ds,$$

(8)

where $x\in J,x_0$ is fixed in J. The boundary point $r_2$ is said to be

i. 

regular if $Q\in L^1(x_0,r_2)$ and $R\in L^1(x_0,r_2)$;

ii. 

exit if $Q\notin L^1(x_0,r_2)$ and $R\in L^1(x_0,r_2)$;

iii. 

entrance if $Q\in L^1(x_0,r_2)$ and $R\notin L^1(x_0,r_2)$;

iv. 

natural if $Q\notin L^1(x_0,r_2)$ and $R\notin L^1(x_0,r_2)$.

Analogous notations can be given for $r_1$ by considering the interval $(r_1,x_0)$ instead of $(x_0,r_2)$.

Theorem 2.

The operator A with the so-called maximal domain

$$D_M\left(A\right)=\{u\in C\left(\overline{J}\right)\cap C^2\left(J\right)|Au\in C\left(\overline{J}\right)\},$$

generates a Feller semigroup on $C\left(\overline{J}\right)$1 if and only if $r_1$ and $r_2$ are of entrance or natural type.

Moreover, the operator A with the so called Wentzell domain

$$D_W\left(A\right)=\{u\in C\left(\overline{J}\right)\cap C^2\left(J\right)\left|\underset{\underset{x\to r_2}{x\to r_1}}{lim}Au\left(x\right)=0\right\},$$

generates a Feller semigroup on $C\left(\overline{J}\right)$ if and only if both the endpoints $r_1$ and $r_2$ are not of entrance type.

Proof. 

See (Engel 2000, chap. VI, Theorem 4.15, Theorem 4.18). □

The next results are useful for studying the behavior of the boundary points according to the Feller classification (see, also (Karlin and Taylor 1981, chap. XV, para. 6.1), and Bufalo et al. (2019)).

Lemma 1.

Fix $x_0\in J=(r_1,r_2)$, then we have

(i) 

$W\notin L^1(x_0,r_2)$ implies $R\notin L^1(x_0,r_2)$;

(ii) 

$R\in L^1(x_0,r_2)$ implies $W\in L^1(x_0,r_2)$;

(iii) 

${\left(mW\right)}^{-1}\notin L^1(x_0,r_2)$ implies $Q\notin L^1(x_0,r_2)$;

(iv) 

$Q\in L^1(x_0,r_2)$ implies ${\left(mW\right)}^{-1}\in L^1(x_0,r_2)$.

Analogously for $r_1$.

Proof. 

See (Engel 2000, chap. VI, Remark 4.10). □

Remark 1.

By some interchanges between the integration variables, the integrals of Q and R may be written as

$$\begin{array}{cc}\hfill {\int }_{x_0}^{r_2}Q\left(x\right)dx& ={\int }_{x_0}^{r_2}\left({\int }_{x_0}^{x}W\left(s\right)ds\right){\left(m\left(x\right)W\left(x\right)\right)}^{-1}dx\hfill \\ \hfill & ={\int }_{x_0}^{r_2}\left({\int }_x^{r_2}{\left(m\left(s\right)W\left(s\right)\right)}^{-1}ds\right)W\left(x\right)dx\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill {\int }_{x_0}^{r_2}R\left(x\right)dx& ={\int }_{x_0}^{r_2}\left({\int }_{x_0}^x{\left(m\left(s\right)W\left(s\right)\right)}^{-1}ds\right)W\left(x\right)dx\hfill \\ \hfill & ={\int }_{x_0}^{r_2}\left({\int }_x^{r_2}W\left(s\right)ds\right){\left(m\left(x\right)W\left(x\right)\right)}^{-1}dx\hfill \end{array}$$

(10)

for any fixed $x_0\in (r_1,r_2)$. The analogous facts hold for $r_1$.

Finally, we give some results concerning the explicit representation of the semigroup, which give an analytic representation of the solution of (ACP) (1). The most famous result is provided by Romanov’s formula (see (Goldstein 2017, chap. II, sct. 8)).

Definition 5.

A $\left(C_0\right)$ cosine function on a Banach space $\mathcal{X}$ is a family ${\left(C\left(t\right)\right)}_{t\in \mathbb{R}}$ of linear bounded operators on $\mathcal{X}$ satisfying the following properties:

i. 

$C\left(0\right)=I$;

ii. 

for all $t,s\in \mathbb{R}$, $C(t+s)+C(t-s)=2C\left(t\right)·C\left(s\right)$;

iii. 

for all $f\in \mathcal{X}$, $C(·)f\in C(\mathbb{R},\mathcal{X})$.

The generator $(A,D(A\left)\right)$ of a cosine function C is given by the operator $A=C^{″}\left(0\right)$, with domain

$$D\left(A\right)=\{f\in \mathcal{X}|C(·)f\in C^2(\mathbb{R},\mathcal{X})\}.$$

Theorem 3.

If $(A,D(A\left)\right)$ is the generator of a $\left(C_0\right)$ group ${\left(S\left(t\right)\right)}_{t\in \mathbb{R}}$ on $\mathcal{X}$, then $(A^2,D\left(A^2\right))$ generates a $\left(C_0\right)$ cosine function ${\left(C\left(t\right)\right)}_{t\in \mathbb{R}}$ and generates a $\left(C_0\right)$ semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ given by

$$T\left(t\right)f=2{\int }_0^{+\infty }C\left(y\right)fp(t,y)dy,$$

(11)

($T\left(0\right)=I$, if $t=0$) for any $f\in X$, where

$$C\left(t\right)={\displaystyle \frac{S\left(t\right)+S(-t)}{2}}.$$

In particular,

$$p(t,y)={\displaystyle \frac{1}{\sqrt{4\pi t}}}{e}^{-{\displaystyle \frac{y^2}{4t}}},t>0,$$

(12)

is the density distribution function of a normally distributed random variable with zero mean and variance $2t$.

Proof. 

See (Goldstein 2017, chap. II, sct. 8). □

Remark 2.

Notice that the integral in (11) may be rewritten as

$${\int }_0^{+\infty }(S\left(y\right)+S(-y))f\left(x\right)p(t,y)dy$$

$$={\int }_0^{+\infty }S\left(y\right)f\left(x\right)p(t,y)dy+{\int }_0^{+\infty }S(-y)f\left(x\right)p(t,y)dy,$$

for all $x\in J,t\ge 0$. By substituting the variable $w=-y$ in the above second integral, and by using (12), the semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ has the following explicit representation:

$$T\left(t\right)f\left(x\right)={\int }_{-\infty }^{+\infty }S\left(y\right)f\left(x\right)p(t,y)dy.$$

2.2. A Generalized Operator for an Option Pricing Model, and Its Semigroup

This subsection is based on Goldstein et al. (2007). As a generalization of this paper, we prove the existence and uniqueness of results on the Wentzell and maximal domain, in a more general way, based on (Bufalo et al. 2019, Appendix A). Moreover, an explicit representation is also given in the case $a=\{0,1\}$. Finally, we provide an extensive numerical application to financial markets in Section 3.

Let $(\mathrm{\Omega },\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbb{P})$ a filtered probability space. Let us consider the following operator:

$$\mathcal{L}u\left(x\right)={\theta }_2{x}^{2a}{u}^{″}\left(x\right)+({\theta }_{2}a{x}^{2a-1}+{\theta }_1{x}^a)u^{\prime }\left(x\right),$$

(13)

with ${\theta }_1,{\theta }_2,a\in \mathbb{R}$ and $x\in {\mathbb{R}}_{+}$. This is the infinitesimal generator of the following SDE:

$$d{X}_t=({\theta }_{2}a{X}_t^{2a-1}+{\theta }_1{X}_t^a)dt+\sqrt{2{\theta }_2}{X}_t^{a}d{W}_t,$$

(14)

where $W_t$ is a $\mathbb{P}$-Wiener process. The drift and diffusion coefficients are given by $({\theta }_{2}a{X}_t^{2a-1}+{\theta }_1{X}_t^a)$ and $\sqrt{2{\theta }_2}{X}_t^a$, respectively. The parameter a represents the elasticity of the model, and controls the relationship between the process and its volatility. When $a<1$, we see an effect commonly observed in equity markets, where the volatility of a stock increases as its price falls and the leverage ratio increases. Conversely, in commodity markets, we often observe $a\ge 1$, whereby the volatility of the price of a commodity tends to increase when its price increases and the leverage ratio decreases, as typically occurs for short-term interest rates. We observe that

• if $a={\displaystyle \frac{1}{2}}$, the SDE (14) becomes

  $$d{X}_t=\left({\displaystyle \frac{{\tau }^2}{4}}-k\sqrt{X_t}\right)dt+\tau \sqrt{X_t}d{W}_t,$$

  (15)

  where we let $\tau =\sqrt{2{\theta }_2}>0$ and $k=-{\theta }_1\in \mathbb{R}$. When $k>0$, the SDE (15) represents Longstaff’s model in the financial literature, which describes the dynamics of the short-term interest rate of zero-coupon bonds. The parameter k represents the speed of adjustment to the process long mean, while $\tau$ is the volatility. It belongs to the class of the so-called “double square-root” processes and is a modified version of the Cox–Ingersoll–Ross (CIR) process;
• if $a=1$, the SDE (14) becomes

  $$d{X}_t=\alpha X_{t}dt+\beta X_{t}d{W}_t,$$

  (16)

  where we let $\alpha =({\theta }_1+{\theta }_2)\in \mathbb{R}$ and $\beta =\sqrt{2{\theta }_2}>0$. The SDE (16) is used in economics and genetics as a model to describe the evolution of certain population growth processes with environmental effects varying randomly over time. It is clear that, if $\alpha =\mu >0$ and $\beta =\sigma$ represent the drift and the volatility of $X_t$, respectively, it reduces to the classic Black–Scholes (BS) model.

Now, let $\mathcal{X}=C[0,+\infty ]$ be a Banach space and consider the following (ACP):

$$\left\{\begin{array}{c}u(t,x)=\mathcal{L}u(t,x),t\ge 0,x\ge 0,\hfill \\ u(0,x)=u_0\left(x\right),x\ge 0,\hfill \end{array}\right.$$

(17)

with $\mathcal{L}$ defined in (13) and $u_0\in \mathcal{X}$. First of all, we give the generation result which provides the existence and uniqueness of the solution of (17).

Theorem 4.

The operator $\mathcal{L}$ defined in (13) with the Wentzell domain

$$D_W\left(\mathcal{L}\right)=\{u\in C[0,+\infty ]\cap C^2(0,+\infty )\left|\underset{\underset{x\to +\infty }{x\to 0^{+}}}{lim}\mathcal{L}u\left(x\right)=0\right\},$$

generates a Feller semigroup on $\mathcal{X}$.

Proof. 

According to Theorem 2, we have to prove that the endpoints $0,+\infty$ are both not of entrance type, i.e., $Q\notin L^1(x_0,+\infty )$ or $R\in L^1(x_0,+\infty )$ for an arbitrary $x_0\in (0,+\infty )$ (analogously for the endpoint 0). Let $x_0=1$ and evaluate the Wronksian

$$W\left(x\right)=exp\left(-{\int }_1^x{\displaystyle \frac{{\theta }_{2}a{s}^{2a-1}+{\theta }_1{s}^a}{{\theta }_2{s}^{2a}}}ds\right)=exp\left(-{\int }_1^x{\displaystyle \frac{a}{s}}+{\displaystyle \frac{\theta }{s^a}}ds\right),$$

(18)

where $\theta ={\theta }_1/{\theta }_2$. Distinguish three cases for the parameter a.

Case 1. If $a=1$, we have

$$Q\left(x\right)={\displaystyle \frac{1}{{\theta }_2{x}^{1-\theta }}}{\int }_1^x{s}^{-(1+\theta )}ds=\left\{\begin{array}{c}{\displaystyle \frac{lnx}{{\theta }_{2}x}},if\theta =0,\hfill \\ {\displaystyle \frac{1}{{\theta }_1}}\left(x^{\theta -1}-{\displaystyle \frac{1}{x}}\right),if\theta \ne 0.\hfill \end{array}\right.$$

So, we can compute

$${\int }_0^{1}Q\left(x\right)dx=\left\{\begin{array}{c}{\displaystyle \frac{1}{2{\theta }_2}}{\left[l{n}^{2}x\right]}_0^1=\infty ,if\theta =0,\hfill \\ {\displaystyle \frac{1}{{\theta }_1}}({\left[x^{\theta }\right]}_0^1-{\left[lnx\right]}_0^1)=\infty ,if\theta \ne 0,\hfill \end{array}\right.$$

and

$${\int }_1^{+\infty }Q\left(x\right)dx=\left\{\begin{array}{c}{\displaystyle \frac{1}{2{\theta }_2}}{\left[l{n}^{2}x\right]}_1^{+\infty }=\infty ,if\theta =0,\hfill \\ {\displaystyle \frac{1}{{\theta }_1}}({\left[x^{\theta }\right]}_1^{+\infty }-{\left[lnx\right]}_1^{+\infty })={\displaystyle \frac{1}{{\theta }_1}}{\left[{\displaystyle \frac{1-\frac{lnx}{\theta /x^{\theta }}}{\theta /x^{\theta }}}\right]}_1^{+\infty }=\infty ,if\theta \ne 0.\hfill \end{array}\right.$$

Hence, $Q\notin L^1(0,1)$ and $Q\notin L^1(1,+\infty )$ that is equivalent to say 0 and $+\infty$ are both not of entrance type.

Case 2. If $a>1$, we have

$$W\left(x\right)=\left\{\begin{array}{c}{x}^{-a},if\theta =0,\hfill \\ exp\left(-{\displaystyle \frac{\theta }{1-a}}(1-x^{1-a})\right),if\theta \ne 0.\hfill \end{array}\right.$$

Then,

$$Q\left(x\right)={\displaystyle \frac{x^a}{{\theta }_2{x}^{2a}}}{\int }_1^x{s}^{-a}ds={\displaystyle \frac{1}{{\theta }_2{x}^a}}{\left[{\displaystyle \frac{s^{1-a}}{1-a}}\right]}_1^x={\displaystyle \frac{1}{{\theta }_2}}\left({\displaystyle \frac{x^{1-2a}}{1-a}}-x^{-a}\right),$$

if $\theta =0$, and

$$Q\left(x\right)={\displaystyle \frac{x^{a}exp\left(\frac{\theta }{1-a}(1-x^{1-a})\right)}{{\theta }_2{x}^{2a}}}{\int }_1^x{\displaystyle \frac{exp\left(-\frac{\theta }{1-a}(1-s^{1-a})\right)}{s^{2a}}}ds$$

$${\displaystyle \frac{exp\left(-\frac{\theta x^{1-a}}{1-a}\right)}{{\theta }_2{x}^a}}{\int }_1^x{\displaystyle \frac{exp\left(\frac{\theta s^{1-a}}{1-a}\right)}{s^a}}ds={\displaystyle \frac{exp\left(-\frac{\theta x^{1-a}}{1-a}\right)}{{\theta }_2{x}^a}}{\left[{\displaystyle \frac{exp\left(\frac{\theta s^{1-a}}{1-a}\right)}{\theta }}\right]}_1^x$$

$$={\displaystyle \frac{1}{{\theta }_1{x}^a}}-exp\left({\displaystyle \frac{\theta }{1-a}}\right){\displaystyle \frac{exp\left(-\frac{\theta x^{1-a}}{1-a}\right)}{{\theta }_1{x}^a}},$$

if $\theta \ne 0$. It is easy to verify with similar calculations that $R\left(x\right)=Q\left(x\right)$ if $\theta =0$ and

$$R\left(x\right)=-{\displaystyle \frac{1}{{\theta }_1{x}^a}}+exp\left(-{\displaystyle \frac{\theta }{1-a}}\right){\displaystyle \frac{exp\left(\frac{\theta x^{1-a}}{1-a}\right)}{{\theta }_1{x}^a}},$$

if $\theta \ne 0$. Now, we compute

$${\int }_0^{1}R\left(x\right)dx=\left\{\begin{array}{c}{\int }_0^{1}Q\left(x\right)dx={\displaystyle \frac{x^{1-a}}{{\theta }_2(1-a)}}{\left[{\displaystyle \frac{x^{1-a}}{2(1-a)}}-1\right]}_0^1=\infty ,if\theta =0,\hfill \\ {\left[-{\displaystyle \frac{x^{1-a}}{{\theta }_1(1-a)}}\right]}_0^1-exp\left(-{\displaystyle \frac{\theta }{1-a}}\right){\left[{\displaystyle \frac{exp\left(\frac{\theta x^{1-a}}{1-a}\right)}{{\theta }_1\theta }}\right]}_0^1<+\infty ,if\theta \ne 0,\hfill \end{array}\right.$$

and analogously ${\int }_1^{+\infty }R\left(x\right)dx<+\infty$, for any $\theta \in \mathbb{R}$. Hence, we can conclude that 0 and $+\infty$ are both not of entrance type. Case 3. If $a<1$, the expression of $W\left(x\right),Q\left(x\right),R\left(x\right)$ and their integrals are computed as in Case 2. In a similar way, we can conclude that $R\in L^1(0,1)$ and $Q\notin L^1(1,+\infty )$ for any $\theta \in \mathbb{R}$, i.e., 0 and $+\infty$ are both not of entrance type. □

Now, we are able to give an explicit representation in the case $0\le a\le 1$, which refers to the financial models described above. We note that if $a<0\cup a>1$, one can always use the Trotter–Kato approximation formulas (see, e.g., (Engel 2000, chap. III, Corollary 5.8)).

Theorem 5.

Fixed $u_0\in \mathcal{X}$ and for any $(x,t)\in {[0,+\infty )}^2$ the explicit solution to the problem (17) is given by

• Case 1: if $a=0$,

  $$u(t,x)={\int }_{-\infty }^{+\infty }{u}_0(x+{\theta }_{1}t+\sqrt{{\theta }_2}y)p(t,y)dy;$$

  (19)
• Case 2: if $a=1$,

  $$u(t,x)={\int }_{-\infty }^{+\infty }{u}_0\left(x{e}^{{\theta }_{1}t+\sqrt{{\theta }_2}y}\right)p(t,y)dy;$$

  (20)
• Case 3: if $0<a<1$,

  $$u(t,x)={\int }_{-\infty }^{+\infty }{u}_0\left({(x^{1-a}+(1-a)({\theta }_{1}t+\sqrt{{\theta }_2}y))}^{{\displaystyle \frac{1}{1-a}}}\right)p(t,y)dy;$$

  (21)

where $p(t,y)$ is defined in (12).

Proof. 

Case 1: if $a=0$, the operator (13) becomes

$$\mathcal{L}u\left(x\right)={\theta }_2{u}^{″}\left(x\right)+{\theta }_1{u}^{\prime }\left(x\right).$$

(22)

Let us define

$$Gu=\sqrt{{\theta }_2}{u}^{\prime },D\left(G\right)=\{u\in C[0,+\infty ]|u^{\prime }\in C[0,+\infty ]\},$$

then, the square of G is given by

$$G^{2}u={\theta }_2{u}^{″},$$

with domain

$$D\left(G^2\right)=\{u\in D\left(G\right)|Gu\in D\left(G\right)\}=\{u\in C[0,+\infty ]\cap C^2[0,+\infty )|u^{\prime },u^{″}\in C[0,+\infty ]\}.$$

Thus, for any $u\in D\left(G^2\right)$, we can rewrite the operator (22) as

$$\mathcal{L}u=G^{2}u+\theta Gu,$$

with $\theta ={\displaystyle \frac{{\theta }_1}{\sqrt{{\theta }_2}}}$. Notice that $(G,D(G\left)\right)$ generates the $\left(C_0\right)$ contraction semigroup $e^{tG}f\left(x\right)=f(x+\sqrt{{\theta }_2}t)$ for all $f\in \mathcal{X}$; and $\theta G$ is a bounded perturbation. Hence, using Theorem 3 and (Engel 2000, chap. III, sct. 1), the operator $(\mathcal{L},D\left(G^2\right))$ generates the $\left(C_0\right)$ contraction semigroup

$$T\left(t\right)u_0\left(x\right)={\int }_0^{+\infty }(u_0(x+{\theta }_{1}t+\sqrt{{\theta }_2}y)+u_0(x+{\theta }_{1}t-\sqrt{{\theta }_2}y))p(t,y)dy,$$

and by Remark 2, Formula (19) is proved.

Case 2: if $a=1$, the operator (13) becomes

$$\mathcal{L}u\left(x\right)={\theta }_2{x}^2{u}^{″}\left(x\right)+({\theta }_1+{\theta }_2)x{u}^{\prime }\left(x\right).$$

(23)

Let us define

$$Gu=\sqrt{{\theta }_2}x{u}^{\prime },D\left(G\right)=\{u\in C[0,+\infty ]|u^{\prime },x{u}^{\prime }\in C[0,+\infty ]\},$$

then, the square of G is given by

$$G^{2}u={\theta }_2{x}^2{u}^{″}+{\theta }_{2}x{u}^{\prime },$$

with domain

$$D\left(G^2\right)=\{u\in D\left(G\right)|Gu\in D\left(G\right)\}=\{u\in C[0,+\infty ]\cap C^2[0,+\infty )|u^{\prime },x{u}^{\prime },x{\left(x{u}^{\prime }\right)}^{\prime }\in C[0,+\infty ]\}.$$

Thus, for any $u\in D\left(G^2\right)$, we can rewrite the operator (23) as

$$\mathcal{L}u=G^{2}u+\theta Gu,$$

with $\theta ={\displaystyle \frac{{\theta }_1}{\sqrt{{\theta }_2}}}$. Notice that $(G,D(G\left)\right)$ generates the $\left(C_0\right)$ contraction semigroup $e^{tG}f\left(x\right)=f\left(x{e}^{\sqrt{{\theta }_2}t}\right)$ for all $f\in X$; and $\theta G$ is a bounded perturbation. Hence, by Theorem 3 and (Engel 2000, chap. III, sct. 1), the operator $(\mathcal{L},D\left(G^2\right))$ generates the $\left(C_0\right)$ contraction semigroup

$$T\left(t\right)u_0\left(x\right)={\int }_0^{+\infty }(u_0\left(x{e}^{{\theta }_{1}t+\sqrt{{\theta }_2}y}\right)+u_0\left(x{e}^{{\theta }_{1}t-\sqrt{{\theta }_2}y}\right))p(t,y)dy,$$

and by Remark 2, Formula (20) is proved.

Case 3: if $0<a<1$, let us define

$$Gu=\sqrt{{\theta }_2}{x}^a{u}^{\prime },D\left(G\right)=\{u\in C[0,+\infty ]|u^{\prime },x^a{u}^{\prime }\in C[0,+\infty ]\},$$

then, the square of G is given by

$$G^{2}u={\theta }_2{x}^{2a}{u}^{″}+{\theta }_{2}a{x}^{2a-1}{u}^{\prime },$$

with domain

$$D\left(G^2\right)=\{u\in D\left(G\right)|Gu\in D\left(G\right)\}=\{u\in C[0,+\infty ]\cap C^2[0,+\infty )|u^{\prime },x^a{u}^{\prime },x^a{\left(x^a{u}^{\prime }\right)}^{\prime }\in C[0,+\infty ]\}.$$

Thus, for any $u\in D\left(G^2\right)$, we can rewrite the operator $\mathcal{L}$ as

$$\mathcal{L}u=G^{2}u+\theta Gu,$$

with $\theta ={\displaystyle \frac{{\theta }_1}{\sqrt{{\theta }_2}}}$. Notice that $(G,D(G\left)\right)$ generates the $\left(C_0\right)$ contraction semigroup

$$e^{tG}f\left(x\right)=f\left({(x^{1-a}+\sqrt{{\theta }_2}(1-a)t)}^{{\displaystyle \frac{1}{1-a}}}\right)$$

for all $f\in \mathcal{X}$ (see (Goldstein et al. 2007, sct. 4) and (Goldstein et al. 2016, Lemma 3.1)); and $\theta G$ is a bounded perturbation. Hence, by Theorem 3 and (Engel 2000, chap. III, sct. 1) the operator $(\mathcal{L},D\left(G^2\right))$ generates the $\left(C_0\right)$ contraction semigroup

$$T\left(t\right)u_0\left(x\right)=$$

$${\int }_0^{+\infty }\left(u_0\left({(x^{1-a}+(1-a)({\theta }_{1}t+\sqrt{{\theta }_2}y))}^{{\displaystyle \frac{1}{1-a}}}\right)+u_0\left({(x^{1-a}+(1-a)({\theta }_{1}t-\sqrt{{\theta }_2}y))}^{{\displaystyle \frac{1}{1-a}}}\right)\right)p(t,y)dy,$$

and by Remark 2, Formula (21) is proved. □

From a practical point of view, Theorem 4 ensures the existence and uniqueness of a mild solution of SDE (14) and of a (general) related option pricing problem. In addition, Theorem 5 will be used to make future predictions on the process (14) (see Section 3.5).

We emphasize that the aim of the theoretical results provided in this paper is making simulations and predictions with the process (14) on real market data. In this sense, the explicit representation semigroup given by Theorem 5 and Formula (36) needs to perform future simulations of the process under the objective measure $\mathbb{P}$. With regard to the pricing framework, in a forthcoming work, we will extend our findings to a $\mathbb{Q}$-world. To do this, we give some hints below (according to (Shreve 2004, chap. V)):

• first of all, find a closed solution of SDE (14)
• then, search for a drift coefficient function such that ${\mathbb{E}}^{\mathbb{Q}}[e^{-r\left(t\right)}{X}_t\mid X_s]=e^{-rs}{X}_s$ for any $0\le s<t$,
• finally, try Novikov’s condition, which ensures that a Radon–Nikodym derivative (in Girsanov’s theorem) is a martingale.

Remark 3.

Observe that it is always possible derive the general Feynman–Kac solution, which is given by

$$\mathbb{E}\left[u_0\left(X_T\right)\right|{\mathcal{F}}_t],$$

(24)

for the (ACP) (17). In particular, if we consider a Call option with strike price K, i.e.,

$$u_0\left(x\right)={(x-K)}^{+},$$

where $x\equiv X_t,t\ge 0$, Formula (24) becomes $\mathbb{E}\left[{(X_T-K)}^{+}\right|{\mathcal{F}}_t]$. We observe that in this Feynman–Kac formula, the discount factor is absent, since the PDE of (17) is homogeneous.

So, we can compare the analytic solution provided by Theorem 5 with the Feynman–Kac solution (24), i.e., by a Monte Carlo simulation, where the process $X_t$ is defined by (14) for all $t\ge 0$, and simulated as in Equation (32). In Figure 1, we compare these two alternative forms of solution $u(t,x)$ for different values of $a\in [0,1]$.

3. Numerical Validation

In this section, we evaluated the model proposed in Section 2.2 on the S&P500 Energy (SPNY) index. The data were retrieved from Yahoo Finance and consist of weekly observations taken from 25 December 2023 to 23 December 2024. More precisely, we first tested the behavior of the returns of every (21) SPNY component in terms of distribution.

Table 1. Mean, standard deviation, skewness, and kurtosis of asset returns of SPNY index.

Asset	Mean	Standard Dev.	Skewness	Kurtosis
Chevron Corporation (CVX)	6.3758$·{10}^{-5}$	0.0272	0.7121	3.231
Conoco Phillips (COP)	−9.2635$·{10}^{-4}$	0.0323	−0.1026	3.2221
EOG Resources, Inc. (EOG)	−4.6655$·{10}^{-4}$	0.0358	−0.4703	3.3402
Williams Companies, Inc. (WMB)	−9.2619$·{10}^{-3}$	0.0284	−0.5748	3.8821
Kinder Morgan, Inc. (KMI)	−9.4588$·{10}^{-3}$	0.0289	−1.1115	5.7160
ONEOK, Inc. (OKE)	−7.9725$·{10}^{-3}$	0.0348	−1.1504	6.3662
Schlumberger N.V. (SLB)	5.7053$·{10}^{-3}$	0.0418	−0.0357	2.5529
Diamondback Energy, Inc. (FANG)	2.8406$·{10}^{-3}$	0.0384	0.3411	3.0976
Phillips 66 (PSX)	−1.1050$·{10}^{-3}$	0.0460	−1.1130	5.5000
Occidental Petroleum Corporation (OXY)	3.8091$·{10}^{-3}$	0.0330	−0.0863	3.5804
Marathon Petroleum Corporation (MPC)	1.4819$·{10}^{-3}$	0.0397	−9.8637$·{10}^{-5}$	2.6393
Baker Hughes Company (BKR)	−3.9330$·{10}^{-3}$	0.0379	−0.5770	3.7861
Hess Corporation (HES)	1.6281$·{10}^{-3}$	0.0328	0.8284	3.9164
Targa Resources, Inc. (TRGP)	−0.0142	0.0383	−0.9581	5.3760
Valero Energy Corporation (VLO)	1.1847$·{10}^{-3}$	0.0410	−0.1199	2.7115
EQT Corporation (EQT)	−0.0157	0.0757	−0.8760	5.3174
Texas Pacific Land Corporation (TPL)	−2.909$·{10}^{-3}$	0.0477	−0.3243	3.7044
Halliburton Company (HAL)	5.352$·{10}^{-3}$	0.0444	0.1826	3.0245
Devon Energy Corporation (DVN)	6.5625$·{10}^{-3}$	0.0396	0.1687	2.4979
Coterra Energy, Inc. (CTRA)	2.5539$·{10}^{-4}$	0.0348	−0.0010	2.7696
APA Corporation (APA)	8.8974$·{10}^{-3}$	0.0501	0.1248	2.0983

shows the first four moments of such returns. Then, we simulated these assets through the stochastic process of our model (see Equation (14)), in order to assess the goodness of fit. Finally, we forecast the next level of each stock price, comparing our results with standard benchmarks proposed in the current literature.

3.1. Benchmark Models

We compared our process described in Equation (14) with the following alternatives:

• the geometric Brownian motion (GBM), used in the classic Black–Scholes model Black and Scholes (1973), in which the underlying asset is a geometric-Brownian motion, i.e.,

  $$d{X}_t=\mu X_{t}dt+\sigma X_{t}d{W}_t$$

  (25)

  where $\sigma$ is the diffusion coefficient, $\mu$ the drift coefficient, and $W_t$ a (standard) $\mathbb{P}$-Brownian motion.
  The (conditioned) expectation of such a process is given by

  $${\mathbb{E}}^{\mathbb{P}}[X_t\mid X_s]=X_s{e}^{\mu (t-s)}$$

  (26)

  for any $X_s>0(0\le s<t)$.
• the skew-geometric Brownian motion (sGBM) recently proposed in Ascione et al. (2024b), Bufalo and Fanelli (2024), Bufalo et al. (2022), Zhu and He (2018), for instance. As is well known, asset returns are generally not normally distributed in financial markets, but show a significant amount of skewness and extra-kurtosis. For this reason, the authors hypothesized that the underlying asset is a skew-geometric Brownian motion, i.e.,

  $$d{X}_t=\zeta X_{t}dt+\sigma X_{t}d{Z}_t$$

  (27)

  with $\omega$ being the scale parameter, $\zeta$ the location parameter, and $Z_t$ a (standard) $\mathbb{P}$-skew-Brownian motion (see, e.g., Ito et al. (2012)). In particular, according to (Corns and Satchell 2007, Proposition 2.1), we adopt the following stochastic representation for $Z_t$:

  $$Z_t=\sqrt{1-{\delta }^2}{B}_{1,t}+\delta \left|B_{2,t}\right|,$$

  with $B_{i,t}$ being two independent Brownian motions, and $\delta$ the rescaled shape parameter.
  The (conditioned) expectation of this process is given by (see (Ascione et al. 2024b, Proposition 3.3))

  $${\mathbb{E}}^{\mathbb{P}}[X_t\mid X_s]=X_{s}exp\left(\zeta -{\displaystyle \frac{{\omega }^2}{2}}\right)-ln\left({\displaystyle \frac{X_s}{X_0}}\right)·$$

  $$\{{\int }_{\mathbb{R}}{\int }_0^{t-s}{\displaystyle \frac{\left|z\right|(1+\delta sign(z\left)\right)}{2\pi {(t-s-u)}^{\frac{3}{2}}}}exp\left(-{\displaystyle \frac{z^2}{2(t-s-u)}}-{\displaystyle \frac{G^2\left(X_s\right)}{2u}}+\sigma z\right)dudz+$$

  $${\displaystyle \frac{1}{\sqrt{2\pi (t-s)}}}{\int }_{\mathbb{R}}\left[exp\left(-{\displaystyle \frac{{(z-G\left(X_s\right))}^2}{2(t-s)}}\right)-exp\left(-{\displaystyle \frac{{(z+G\left(X_s\right))}^2}{2(t-s)}}\right)\right]e^{\omega z}{\mathbf{1}}_{{\mathbb{R}}_{+}}\left(G\left(X_s\right)z\right)dz\},$$

  (28)

  for any $X_s>0(0\le s<t)$, where $G\left(x\right)={\displaystyle \frac{1}{\omega }}\left(ln\left({\displaystyle \frac{x}{X_0}}\right)-\left(\zeta -{\displaystyle \frac{{\omega }^2}{2}}\right)\right)$.

3.2. Distributions of Market Returns

According to Table 1, the mean of returns was always very close to zero, which opens the possibility of the returns being normally distributed. Moreover, while the standard deviation was, on average, close to 0.04, the kurtosis was highly variable. Moreover, the returns were nearly always skewed on the left. However, in order to assess whether the returns are normally distributed, more specific tests are needed. The Shapiro–Wilk, Jarque–Bera, and Anderson–Darling tests were chosen to check the normality of price returns, due to their reliability and precision. We briefly give a description of these tests below.

• The Shapiro–Wilk (SW) test uses the null hypothesis that a sample ${\left(x_h\right)}_{h\in [1,n]\cap \mathbb{N}}$ is normally distributed. It is based on comparing how far the asymmetry and kurtosis measures are from the values of the (standard) normal distribution. The SW test statistic is defined as

  $${\mathcal{S}}^{SW}={\displaystyle \frac{{\left({\sum }_h=1^n{\alpha }_h{x}_{\left(h\right)}\right)}^2}{{\sum }_{h=1}^n{(x_h-\overline{x})}^2}},$$

  where $x_{\left(h\right)}$ is the h-th order statistic, $\overline{x}$ is the sample mean, and the vector $\mathit{\alpha }=({\alpha }_1,...,{\alpha }_n)$ is given by $\mathit{\alpha }={\displaystyle \frac{{\mathit{m}}^{\prime }·{\mathit{V}}^{-1}}{{\mathit{m}}^{\prime }·{\mathit{V}}^{-1}{\mathit{V}}^{-1}·\mathit{m}}}$, where ${\mathit{m}}^{\prime }$ indicates the transposed of the array $(m_1,...m_n)$, where $m_h$ is the expected values of the order statistics of i.i.d. (standard) normal variables, and $\mathit{V}$ is the covariance matrix of these order statistics.
• The Jarque–Bera (JB) test is a goodness-of-fit test of whether a sample ${\left(x_h\right)}_{h\in [1,n]\cap \mathbb{N}}$ has skewness and kurtosis matching a normal distribution. The JB test statistic is defined as

  $${\mathcal{S}}^{JB}={\displaystyle \frac{n}{6}}\left(s+{\displaystyle \frac{{(k-3)}^2}{4}}\right),$$

  where $s,k$ represent the sample skewness and kurtosis, respectively, i.e.,

  $$s={\displaystyle \frac{n^{-1}{\sum }_h=1^n{(x_h-\overline{x})}^3}{{\left(n^{-1}{\sum }_{h=1}^n{(h_h-\overline{x})}^2\right)}^{3/2}}},k={\displaystyle \frac{n^{-1}{\sum }_h=1^n{(x_h-\overline{x})}^4}{{\left(n^{-1}{\sum }_{h=1}^n{(h_h-\overline{x})}^2\right)}^2}}.$$
• The Anderson–Darling (AD) test is a modification of the Kolmogorov–Smirnov (KS) test, used to assess whether a sample ${\left(x_h\right)}_{h\in [1,n]\cap \mathbb{N}}$ has a specific distribution. Differently from the KS test, it gives more weight to the tails. The AD test is defined as

  $${\mathcal{S}}^{AD}=-n-\sum _{h=1}^n{\displaystyle \frac{2h-1}{n}}\left[ln\left(\mathrm{\Phi }\left(y_h\right)\right)-ln(1-\mathrm{\Phi }\left(y_{n-h+1}\right))\right]$$

  where $y_h$ represents the ordered data.

The null hypothesis that the returns are normally distributed was tested at a 1% confidence level and the results are shown in

Table 2. SW, JB, and AD normality tests for asset returns of SPNY index.

Asset	SW Test		JB Test		AD Test
	resp.	p-value	resp.	p-value	resp.	p-value
Chevron Corporation (CVX)	0	0.0802	0	0.0586	0	0.8273
Conoco Phillips (COP)	0	0.2288	0	0.5000	0	0.8227
EOG Resources, Inc. (EOG)	0	0.3825	0	0.1972	0	0.2766
Williams Companies, Inc. (WMB)	0	0.0345	0	0.0578	0	0.8616
Kinder Morgan, Inc. (KMI)	1	0.0029	1	0.0018	0	0.9686
ONEOK, Inc. (OKE)	1	0.0020	1	0.0010	0	0.8412
Schlumberger N.V. (SLB)	0	0.6420	0	0.5000	0	0.1448
Diamondback Energy, Inc. (FANG)	0	0.7933	0	0.5000	0	0.0842
Phillips 66 (PSX)	1	0.0024	1	0.0022	0	0.9206
Occidental Petroleum Corporation (OXY)	0	0.5401	0	0.5000	0	0.3130
Marathon Petroleum Corporation (MPC)	0	0.9560	0	0.5000	0	0.0100
Baker Hughes Company (BKR)	0	0.1556	0	0.0656	0	0.2815
Hess Corporation (HES)	0	0.0142	0	0.0240	0	0.9250
Targa Resources, Inc. (TRGP)	1	0.0070	1	0.0035	0	0.8107
Valero Energy Corporation (VLO)	0	0.6678	0	0.5000	0	0.4595
EQT Corporation (EQT)	1	0.0015	1	0.0043	0	0.9989
Texas Pacific Land Corporation (TPL)	0	0.4769	0	0.2276	0	0.1786
Halliburton Company (HAL)	0	0.3004	0	0.5000	0	0.8474
Devon Energy Corporation (DVN)	0	0.5736	0	0.5000	0	0.5407
Coterra Energy, Inc. (CTRA)	0	0.3866	0	0.5000	0	0.7395
APA Corporation (APA)	0	0.213	0	0.2457	0	0.9141

. One can see that, globally, the null hypothesis was rejected (i.e., there was no asset with three responses equal to 1) with no exceptions. Hence, it is possible to infer that the Black–Scholes assumption did not give a confirmation with real data taken from the market: the prices did not follow a geometric Brownian motion, and the returns were not normally distributed. A careful analysis of returns, shows the presence of heavy tails, which shows a high probability of large price movements that standard models do not take into consideration.

3.3. Calibration

In this section, we explain the parameter calibration procedure. Let ${\left(x_h\right)}_{h\in \mathcal{I}}$ be the time series observations over a given time horizon $\mathcal{I}$. With regard to the Geometric and skew-Geometric Brownian motions described in Section 3.1, one may use the maximum likelihood estimation (MLE). The estimated parameters of process (25) are given, in closed form, by

$$\widehat{\mu }={\displaystyle \frac{1}{\parallel \mathcal{I}\parallel }}\sum _{h\in \mathcal{I}}{x}_h,\widehat{\sigma }=\sqrt{{\displaystyle \frac{1}{(\parallel \mathcal{I}\parallel -1)}}{\sum }_{h\in \mathcal{I}}{(x_h-\widehat{\mu })}^2},$$

(29)

with $\parallel \mathcal{I}\parallel$ denoting the size of interval $\mathcal{I}$; while

$$(\widehat{\zeta },\widehat{\omega },\widehat{\delta })=arg\left\{\underset{(\zeta ,\omega ,\delta )}{max}\sum _{h\in \mathcal{I}}ln\left(f_h\left(x_h\right)\right)\right\}$$

(30)

(see, e.g., (Bufalo et al. 2022, sct. 5.1)) where

$$f_h\left(x_h\right)={\displaystyle \frac{2}{\omega \sqrt{h}}}\phi \left({\displaystyle \frac{x_h-\mu h}{{\omega }_h\sqrt{h}}}\right)\mathrm{\Phi }\left(\beta {\displaystyle \frac{x_h-\mu h}{\omega \sqrt{h}}}\right),$$

where $\phi ,\mathrm{\Phi }$ denote the (standard) normal probability density (PDF) and cumulative distribution function (CDF), while $\beta$ is the shape parameter, such that $\delta ={\displaystyle \frac{\beta }{\sqrt{1+{\beta }^2}}}$.

Finally, observe that the parameters $({\theta }_1,{\theta }_2,a)$ of process (14) must be estimated through an ordinary least squared (OLS) approach, i.e.,

$$({\widehat{\theta }}_1,{\widehat{\theta }}_2,\widehat{a})=arg\left\{\underset{({\theta }_1,{\theta }_2,a)}{min}\sum _{h\in \mathcal{I}}{\left(x_h-{\widehat{x}}_h\right)}^2\right\}$$

(31)

where ${\widehat{x}}_h$ can be calculated as in Equation (32). In forthcoming research, we study the probability density function of the process $X_t$ defined by Equation (14), and use it to estimate the parameters.

3.4. In Sample Simulations

In this section, we illustrate how to simulate our process price $X_t$ over a time interval $[0,n]\cap \mathbb{N}$. In order to do this, we use an (explicit) Eulr discretization algorithm with (second-order) Milshtein’s scheme Mil’shtein (1979). We first partition our interval into N subintervals of length $\mathrm{\Delta }$, i.e., $0={\tau }_0<{\tau }_1<\dots <{\tau }_N=T$, with ${\tau }_N=n\mathrm{\Delta }$ and $\mathrm{\Delta }=n/N$. Then, starting from ${\widehat{x}}_0=X_0$, we recursively define, for any $1\le h\le (N-1)$:

$${\widehat{x}}_{h+1}={\widehat{x}}_h+({\theta }_1{\widehat{x}}_h^a+{\theta }_2{\widehat{x}}_h^{2a-1})\mathrm{\Delta }+\sqrt{2{\theta }_2}{\widehat{x}}_h^a\sqrt{\mathrm{\Delta }}{\epsilon }_h+a{\theta }_2{\widehat{x}}_h^{2a-1}\left[{\left(\sqrt{\mathrm{\Delta }}{\epsilon }_h\right)}^2-\mathrm{\Delta }\right]$$

(32)

with ${\epsilon }_h$ being a (standard) normal variable. Obviously, one can calibrate the parameters on any subinterval, as in Section 3.3 (see also Table 5).

Analogously, we can simulate the process defined in Equation (25) using the scheme (32) with the following substitutions:

$$\left\{\begin{array}{c}({\theta }_1+{\theta }_2)\to \mu \hfill \\ \sqrt{2{\theta }_2}\to \sigma \hfill \\ a\to 1\hfill \end{array}\right.$$

(33)

and the process defined in Equation (27) using the same scheme with the substitutions

$$\left\{\begin{array}{c}({\theta }_1+{\theta }_2)\to \zeta \hfill \\ \sqrt{2{\theta }_2}\to \omega \hfill \\ {\epsilon }_h\to \overline{\epsilon }\sim SN(0,1,\delta )\hfill \\ a\to 1\hfill \end{array}\right.$$

(34)

Observe that a more precise (but slower) discretization algorithm for the skew-process of Equation (27) is provided in Bourza (2023).

The measure of the goodness of fit of the proposed simulations with respect to the real data could be tested through the following statistics:

• The root mean squared error (RMSE):

  $$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _h{(x_h-{\widehat{x}}_h)}^2}.$$
• The mean absolute percentage error (MAPE):

  $$MAPE={\displaystyle \frac{1}{n}}\sum _{h=1}^n\left|{\displaystyle \frac{x_h-{\widehat{x}}_h}{x_h}}\right|$$
• The index of directionality (IDX) introduced in Orlando and Bufalo (2021):

  $$IDX={\displaystyle \frac{1}{(n-1)}}\sum _{h=1}^{n-1}{\mathbf{1}}_{\{{\alpha }_{h+1}={\beta }_{h+1}\}}$$

  where ${\alpha }_{h+1}=x_{h+1}-x_h$ and ${\beta }_{h+1}={\widehat{x}}_{h+1}-x_h$. The IDX indicates the percentage success (in sign) of prediction directionality, i.e., the percentage of times where the simulated values rise or fall coherently with the real data.

In addition, we compared our model performance with that of the proposed benchmarks using the Diebold—Mariano (DM) test (see Diebold and Mariano (2002)) for the equality of the predictive accuracy of two models. The null hypothesis of this test is that the two compared forecasts have the same accuracy, i.e., their errors are statistically similar.

Figure 2 shows the RMSE (first bar plot), MAPE (second bar plot), and IDX (third bar plot) statistics calculated from the geometric Brownian motion (yellow bar), skew-geometric Brownian motion (green bar), and our process (14) (blu bar), for assets (y axis) of the SPNY index. The results demonstrate that our process always achieved a better fit (i.e., smaller RMSE and MAPE, and higher IDX) to real data, while the sGBM process was globally superior with respect to the GBM process.

Remark 4.

Since the original time series exhibited both skewness and kurtosis (see Table 1), and our process (14) better fit these data in terms of accuracy statistics, we can conclude that its probability density is skewed. However, it is simple to verify that the skew-normal density does not solve the Folker–Plank equation related to SDE (14)

$${\displaystyle \frac{\partial }{\partial t}}f(x,t)+{\displaystyle \frac{\partial }{\partial x}}({\theta }_{2}a{x}^{2a-1}+{\theta }_1{x}^a)f(x,t)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}\left({\theta }_2{x}^{2a}\right)f(x,t)=0.$$

Therefore, we hypothesize that our process has a more general skewed probability density function than a skew-normal one. This topic will be treated in a forthcoming work.

3.5. Forecasts

In this section, we analyze the prediction power of our model. As is usual, we identified the out of sample simulations as the (conditioned) expectations of the process $X_t$. Thus, for the geometric Brownian motion and its skew version, defined by Equations (25) and (27), respectively, we let

$$x_{h+1}^F={\mathbb{E}}^{\mathbb{P}}[X_{h+1}\mid X_k](0\le k\le h)$$

(35)

be the above expectation given by Equations (26) and (28), respectively. Observe that the (conditioned) expectation of the process $X_t$ defined in (14) by the semigroup of Theorem 5, i.e.,

$$x_{h+1}^F=T(h+1)\mathit{id}\left(x\right)(x=X_k,0\le k\le h)$$

is $\mathit{id}$ the identity function. In particular, when $a\in (0,1)$, it is simple to see that

$$T(h+1)\mathit{id}\left(x\right)={\int }_{-\infty }^{+\infty }{\left(x^{1-a}+(1-a)({\theta }_1(h+1-k)+\sqrt{2{\theta }_2(h+1-k)}y)\right)}^{{\displaystyle \frac{1}{1-a}}}\phi \left(y\right)dy.$$

(36)

As observed in Remark 3, Equation (36) coincides with the Feynman–Kac formula, but has the advantage of being in closed form (with a considerable reduction in computational cost).

Figure 3 shows the RMSE (first bar plot), MAPE (second bar plot), and IDX (third bar plot) statistics calculated from the geometric Brownian motion (yellow bar), skew-geometric Brownian motion (green bar) and our process (14) (blu bar), for any asset (y axis) of SPNY index. The results demonstrate that our process always achieved a better fit (i.e., smaller RMSE and MAPE, and higher IDX) to real data, while the sGBM process was globally superior with respect to the GBM process.

Table 3. p-values from the Diebold–Mariano test carried out with our model versus the GBM and sGBM processes, respectively, with a significance level of 95%. When the p-values were greater than 0.05, the null hypothesis was not rejected.

Asset	Our Process vs. GBM	Our Process vs. sGBM
Chevron Corporation (CVX)	0.0062$·{10}^{-4}$	0.1217$·{10}^{-4}$
Conoco Phillips (COP)	0.0036$·{10}^{-4}$	0.1687$·{10}^{-4}$
EOG Resources, Inc. (EOG)	0.0170$·{10}^{-3}$	0.1656$·{10}^{-3}$
Williams Companies, Inc. (WMB)	0.2063$·{10}^{-4}$	0.0761$·{10}^{-4}$
Kinder Morgan, Inc. (KMI)	0.1350$·{10}^{-3}$	0.0496$·{10}^{-3}$
ONEOK, Inc. (OKE)	0.2011$·{10}^{-3}$	0.1346$·{10}^{-3}$
Schlumberger N.V. (SLB)	0.0068	0.0791
Diamondback Energy, Inc. (FANG)	0.0001	0.0011
Phillips 66 (PSX)	0.0155$·{10}^{-3}$	0.1070$·{10}^{-3}$
Occidental Petroleum Corporation (OXY)	0.0006	0.0193
Marathon Petroleum Corporation (MPC)	0.0895$·{10}^{-3}$	0.4491$·{10}^{-3}$
Baker Hughes Company (BKR)	0.0923$·{10}^{-3}$	0.5454$·{10}^{-3}$
Hess Corporation (HES)	0.0265$·{10}^{-3}$	0.5025$·{10}^{-3}$
Targa Resources, Inc. (TRGP)	0.1550$·{10}^{-3}$	0.0848$·{10}^{-3}$
Valero Energy Corporation (VLO)	0.0296$·{10}^{-3}$	0.2587$·{10}^{-3}$
EQT Corporation (EQT)	0.1677$·{10}^{-4}$	0.0223$·{10}^{-4}$
Texas Pacific Land Corporation (TPL)	0.0003	0.0047
Halliburton Company (HAL)	0.0047	0.0907
Devon Energy Corporation (DVN)	0.0008	0.0203
Coterra Energy, Inc. (CTRA)	0.0005	0.0089
APA Corporation (APA)	0.1056$·{10}^{-4}$	0.5400$·{10}^{-4}$

displays the Diebold-Mariano p-values when one compares our model with the GBM and the sGBM processes, respectively, with a significance level of 95%. We can observe that just two assets (SLB and HAL) achieved a p-value greater than 0.05 (i.e., did not reject the null hypothesis), while the others exhibited forecasts with different levels of accuracy.

Table 4. Maximum and minimum variation for the RMSE, MAPE, and IDX statistics (with respect to Figure 3) when the rolling window size ranged from 4 to 24 weeks.

Statistics	Max. Variation	Min. Variation
RMSE	0.1204$·{10}^{-1}$	0.4721$·{10}^{-3}$
MAPE	0.2145	0.8651$·{10}^{-2}$
IDX	0.3945$·{10}^{-2}$	0.3629$·{10}^{-3}$

reports the maximum and minimum (absolute) variation in the RMSE, MAPE, and IDX statistics (with respect to those of Figure 3) when the rolling window size ranged from 4 to 24 weeks. Larger variations were obtained for larger windows, but on average, we observed minor changes (i.e., of order ${10}^{-2}$), as one would expect.

Table 5. Pseudocode of the proposed model.

1. Let n be the length of the observed series ${\left(x_h\right)}_{h\in [1,n]\cap \mathbb{N}}$.

2. Consider a rolling window of fixed size $m=8$ weeks.

3. Choose the predictive horizon $u\ge 1$. Start from $h=1$.

4. while $h\le n-u$

5. Take the observations of $x_h$, with $h\in [h,h+m-1]$.

6. Calibrate the parameters of the selected model, as explained in Section 3.3.

7. Compute the out of sample values $x_{h+m-1+u}^F$ through Formula (36)

8. Compute the in of sample values ${\widehat{x}}_{h+m-1}$ through scheme (32)

(with substitutions (33) or (34) for the benchmarks).

9. Update $h=h+1$;

10. end

describes the steps of our algorithm to compute the out of sample (and in sample) simulations with the proposed model.

Remark 5.

Notice that if we consider high-frequency data, i.e., daily or hourly data, our model should incorporate an autocorrelation component, as is well-known in finance (see, e.g., Hwang and Vogelsang (2024)). A possible way to solve this issue when working with stochastic differential equations is addressed by Ceci et al. (2024) and Ascione et al. (2024a). This approach correlates the process with its mean and/or variance process, i.e.,

$$\left\{\begin{array}{c}d{X}_t=({\theta }_{2}a{X}_t^{2a-1}+{\theta }_1{X}_t^a)dt+\sqrt{2{\theta }_2}{X}_t^{a}d{W}_t\hfill \\ d{m}_t={\mu }^m\left(m_t\right)dt+{\sigma }^m\left(m_t\right)d{W}_t^m\hfill \\ d{\nu }_t={\mu }^{\nu }\left({\nu }_t\right)dt+{\sigma }^{\nu }\left({\nu }_t\right)d{W}_t^{\nu }\hfill \end{array}\right.$$

for suitable drift and volatility functions ${\mu }^m,{\mu }^{\nu },{\sigma }^m,{\sigma }^{\nu }$; with $W_t^m,W_t^{\nu }$ being two Brownian motions correlated with $W_t$.

4. Conclusions

In this work, we considered a generalized operator $\mathcal{L}$, defined by Equation (13), which encompasses several well-known financial operators, such as the Cox–Ingersoll–Ross (CIR) model and the Black–Scholes (BS) model. Using Feller semigroup theory, we established the existence and uniqueness of a Feller semigroup generated by $\mathcal{L}$. Furthermore, we derived an explicit formulation of this semigroup, which can be applied to solve certain abstract Cauchy problems (ACPs) arising in finance.

From a probabilistic perspective, our results were obtained under the objective measure $\mathbb{P}$, under which the infinitesimal generator $X_t$ of $\mathcal{L}$ satisfies the stochastic differential equation (SDE) (14). However, this does not necessarily allow us to directly apply our results to general option pricing problems, which typically require a risk-neutral measure $\mathbb{Q}$. Nevertheless, the semigroup formulas (under $\mathbb{P}$) remain useful for making predictions about the underlying process $X_t$, thanks to the Feynman–Kac theorem. In this context, we analyzed a case study of the S&P 500 Energy (SPNY) index and its constituent assets. Our research included both in-sample and out-of-sample simulations of these assets, whose accuracy was compared against benchmarks from the literature.

A forthcoming study will further investigate the solution of the SDE (14), its properties, the existence of a martingale measure $\mathbb{Q}$, and its application to general option pricing problems.