Actuarial Valuation and Hedging of Life Insurance Liabilities in the Presence of Stochastic Mortality Risk under the Locally Risk-Minimizing Hedging Approach

Abstract

The paper examines the valuation and hedging of life insurance obligations in the presence of mortality risk using the local risk-minimizing hedging approach. Roughly speaking, it is assumed that the lifetime of policyholders in an insurance portfolio is modeled by a point process whose stochastic intensity is controlled by a diffusion process. The stock price process is assumed to be a regime-switching Lévy process with non-zero regime-switching drift, where the parameters are assumed to depend on the economic states. Using the Föllmer–Schweizer decomposition, the main valuation and hedging results for a conditional payment process are determined. Some specific situations have been considered in which the local risk-minimizing strategies for a stream of liability payments or a unit-linked contract are presented.

1. Introduction

In insurance businesses, the timing claims and the value of liabilities are linked not only to non-market risk factors, as claim propensity or mortality risk, but are related as well to the financial market. Modern insurance products are characterized by complicated structures, and non-trivial risks are involved when running an insurance business and call for a sound asset–liability management and derivation of hedging strategies.

In this work, the streams of liabilities investigated are related to the financial market and to a non-tradable factor in the case of insurance products. The payment process considered is similar to the processes considered in [1,2,3], taking into account the systematic and unsystematic insurance risk and equity-linked liabilities. The mortality risk is the most-important example of systematic insurance risk. We mention that the integration of these three risk components is very crucial in insurance risk management. We understand by unsystematic insurance risk the randomness of insurance claims (i.e., number of accidents) compared to their associated net premiums (expected loss), and by systematic insurance risk, we mean unpredictable variability in the underlying claim intensity (i.e., population mortality intensity or clients’ claim frequency). For further reading, we recall that, in [4], local risk minimization hedging was considered for an equity-linked pure endowment only under unsystematic mortality risk, in the exponential Lévy market model, whereas in [5], the authors extended the model and analysis of [4] by assuming that the parameters of the Lévy process that models the dynamics of risky assets depend on a finite-state Markov chain, and the local risk-minimizing hedging strategies for some unit-linked life insurance products including both the term insurance contract and the pure endowment policy were obtained under the regime-switching Lévy model. In the paper of [2], risk minimization hedging under the martingale measure was investigated without a financial component–in traditional liabilities–under systematic and unsystematic mortality risk, in a financial model consisting of a bond, mortality derivative, and risk-free return driven by a CIR process. Lastly, the study conducted in [6] explored risk minimization hedging within the framework of the martingale measure and mean-variance optimization. This analysis pertains to a sequence of liabilities in the context of insurance against unsystematic risk within the continuous Black–Scholes financial model.

Changes in environmental factors and economic conditions frequently lead to shifts in the behavior of the economy’s states. Undoubtedly, economic states exert a significant impact on the dynamics of asset prices. The Markovian regime-switching model, which involves coefficients in stochastic differential equations (SDEs) governed by a continuous-time Markov chain, provides a means to incorporate shifts in economic trends. Subsequently, as introduced by [7], regime-switching time series models gained prominence in the fields of economics and business cycle analysis. These models have since found widespread application in actuarial science and finance literature.

The origins of regime-switching modeling can be traced back to recent contributions in the field. For instance, Ref. [8] utilized a hidden Markov chain modulated by geometric Brownian motion to price dynamic fund protection plans. In a related context, Ref. [9] assessed equity-annuities incorporating stochastic mortality risk within a double-regime-switching framework. Additionally, Ref. [10] explored the pricing of variance swaps, applying a Markovian regime-switching model based on the quadratic criterion.

The notion of quadratic hedging, characterized by the selection of an investment portfolio that minimizes the conditional mean-squared error process in a mean-squared sense, has found applications in both insurance and finance. For instance, Ref. [11] explored its relevance in the context of dynamic strategies for asset allocations. Similarly, Ref. [12] conducted investigations into its implications for annuity risk within defined contribution pension schemes.

In the presence of an incomplete market, risk minimization serves as a valuable framework for quadratic hedging and pricing contingent claims. This theory was initially introduced by [13], who discussed risk minimization when the underlying discounted price process was a local martingale. Subsequently, Ref. [13] extended this concept to local risk minimization in the case of a semi-martingale discounted price process.

The adoption of two-stage market valuations Ref. [14] examined well-known actuarial premium principles such as the variance principle and the standard deviation principle and investigated their extension in time- and market-consistent directions. The fair valuation of liabilities associated with an insurance policy or portfolio that is both market-consistent (mark-to-market for each hedgeable portion of a claim) and actuarial (mark-to-model for each claim that is independent of financial market developments) was discussed in [15]. In [16], the authors proposed a general approximation framework for the valuation of (path-dependent) options under time-varying Markov processes, which can be used for the valuation of equivalent insurance policies such as excess of loss reinsurance. In [17], an innovative three-stage projection algorithm was presented, which was developed for the valuation of hybrid contracts. In this approach, the payout of the contract is decomposed into three different components: a financial, hedgeable segment, a diversifiable actuarial component, and a remainder that is not hedgeable and not diversifiable.

In the literature, there are two approaches to price financial instruments: the martingale approach and benchmark approach. Indeed, the martingale approach is a mathematical framework used to price financial derivatives, refer to [18]. In this context, the martingale approach often involves constructing a risk-neutral probability measure. Under this measure, the expected return on the derivative is the risk-free rate, simplifying the pricing problem. The martingale approach is widely used in options pricing, where it helps derive the famous Black–Scholes formula. The benchmark approach, developed by [19], considers the relationship between the derivative and a set of benchmark securities or factors. It is rooted in the idea that the price of a derivative is influenced by various underlying economic factors. The Arbitrage Pricing Theory (APT) assumes that the expected return of a financial instrument is related to various macroeconomic factors, and any deviation from this relationship can be exploited through arbitrage. By identifying a set of factors that influence asset prices, the APT aims to capture the risk premium associated with these factors. This approach is particularly useful when dealing with derivatives that are not easily replicated by a portfolio of standard securities.

In a recent publication authored by Deelstra and Hieber [20], the emphasis was on death-linked contingent claims, Guaranteed Minimum Death Benefits (GMDBs). These claims entail a random financial return occurring at a stochastic time of death, within the broader context where financial returns adhere to a regime-switching model featuring two-sided phase-type jumps.

This research builds upon the framework proposed by [4], extending it by introducing a novel dimension, where the parameters of the financial model dynamically respond to changes in the economic state. This dynamic framework incorporates transitions between economic states, modeled through a finite-state Markov chain. Within the context of a regime-switching Lévy model, we derive local risk-minimizing strategies tailored to a diverse range of life insurance payment processes. Notably, our approach takes into account both systematic and unsystematic insurance risks.

To enhance the robustness of our valuation and risk management framework, we integrated the benchmark approach outlined in [19]. This method, renowned in quantitative finance, serves as a guiding principle for pricing and hedging insurance contracts, providing a comprehensive and rigorous foundation for our analyses.

In summary, our contribution culminates in the provision of local risk-minimizing hedging strategies specifically designed for payment processes associated with liability streams and unit-linked contracts. This research not only extends the existing framework by incorporating dynamic parameters tied to economic states, but also advances the understanding of risk management strategies in the context of life insurance by integrating a proven benchmark approach.

The paper’s structure is as follows: In Section 2, we introduce the framework and outline the modeling assumptions. In Section 3, we establish foundational results related to the hedging problem within the context of local risk-minimizing hedging under the original measure, subsequently deriving the corresponding risk process. Section 4 and Section 5 are dedicated to addressing scenarios involving general liability stream payment processes and unit-linked contracts, both of which pertain to two distinct types of life insurance. Finally, our concluding section offers some insights and remarks.

2. Modeling and Assumption

Consider a stochastic framework $(\Omega ,\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbb{P})$ with a filtration that fulfills the completeness condition (where ${\mathcal{F}}_0$ includes all sets with $\mathbb{P}$-measure zero from $\mathcal{F}$) and is right-continuous (denoted as $({\mathcal{F}}_t={\mathcal{F}}_{t+})$). We also assume a finite-time horizon $T<\infty$, adhering to standard assumptions. Unless stated otherwise, all expected values are calculated under the real-world objective probability measure $\mathbb{P}$. Here, $\mathcal{B}\left(A\right)$ represents the Borel subsets of set A. The filtration is generated by the following mutually independent processes.

Consider the following components within the framework:

• Two Brownian motions, B and W, adapted to the filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$.
• An integer-valued random measure ℵ derived from an $\mathcal{F}$-adapted step process ${\left(J\left(t\right)\right)}_{0\le t\le T}$.

Certain assumptions about the integer-valued random measure and its compensator need to be considered. Indeed, these assumptions are crucial for the analysis:

$\left({\mathbf{A}}_{\mathbf{1}}\right)$

The compensator defined on $\Omega \times \mathcal{B}\left(\right[0,T\left]\right)\times \mathcal{B}(\mathbb{R}\setminus \{0\left\}\right)$ takes the form: $\mathfrak{Q}(t,dz)\gamma \left(t\right)dt$, where $\gamma :\Omega \times [0,T]\to [0,\infty )$ represents a predictable process, $\mathfrak{Q}(t,·)$ is a probability measure on $\mathcal{B}\left(\mathbb{R}\right)$ for a fixed $(\omega ,t)\in \Omega \times [0,T]$, and $\mathfrak{Q}(·,A):\Omega \times [0,T]\to [0,1]$ is an $\mathcal{F}$-predictable process for $A\in \mathcal{B}\left(\mathbb{R}\right)$ and $\mathfrak{Q}:(\Omega \times [0,T],\mathbb{P})\to (\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$ satisfying:

$${\int }_0^T{\int }_{\mathbb{R}}{z}^2\mathfrak{Q}(t,dz)\gamma \left(t\right)dt<\infty ,{\int }_0^T\gamma \left(t\right)dt<\infty \mathbb{P}\mathrm{a}.\mathrm{s}.,$$

(1)

and

$$\aleph \left(\right[0,T],\{0\left\}\right)=\aleph \left(\right\{0\},\mathbb{R}\setminus \{0\left\}\right)=0.$$

We assume throughout this paper that any random measure ℵ satisfies the condition $\left({\mathbf{A}}_{\mathbf{1}}\right)$.

In the following subsections, we will introduce the financial market, the insurance portfolio, and a payment process representing the stream of liabilities for the insurance company.

2.1. Financial Modeling

Consider a continuous-time Markov chain $X_{·}:={\left(X_t\right)}_{0\le t\le T}$ defined on the stochastic basis $(\Omega ,\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbb{P})$, where $X_t$ takes values in a finite state space $\mathcal{S}:=\{1,2,3,\dots ,n\}$. This process $X_{·}$ represents the state of the economy. We assume that the risk-free interest rate and stock volatility depend on the states of the economy. The Markov chain $X_{·}$ is subject to the following condition:

$$\mathbb{P}\left(X_{t+\Delta t}=j|X_t=i\right)=C_{ij}\Delta t+o(\Delta t),i\ne j$$

(2)

where $C_{ij}$ is given by:

$$C_{ij}=-\sum _{k\ne j,k=1}^n{C}_{ik}11_{\{i=j\}}.$$

Here, the matrix $C_{ij}$ represents the generating matrix of the Markov chain.

Alternatively, it is more convenient to express Equation (2) $X_t$ as a stochastic integral involving a Poisson random measure. Considering lexicographic ordering on $\mathcal{S}\times \mathcal{S}$, let ${\Delta }_{ij}$ represent consecutive left-closed, right-open intervals on the real line, each with a length of $C_{ij}$ (see [21]). Define the function $g:\mathcal{S}\times \mathbb{R}⟶\mathbb{R}$ by

$$g(i,z)=(j-i)11_{\{z\in {\Delta }_{ij}\}}.$$

Then, the evolution of $X_{·}$ can be expressed as

$$d{X}_t={\int }_{\mathbb{R}}g(X_{t-},z)\aleph (dt,dz),$$

where $\aleph (dt,dz)$ is a Poisson random measure with intensity $dt\times \nu \left(dz\right)$ and $\nu \left(dz\right)$ represents the Lévy measure on $\mathbb{R}$.

The financial market under consideration comprises two continuously tradable underlying instruments: a risk-free asset $S_{·}^0={\left(S_t^0\right)}_{0\le t\le T}$ and a risky asset $S_{·}={\left(S_t\right)}_{0\le t\le T}$. Here, $r_{·}:\mathcal{S}\to [0,\infty )$ represents the risk-free interest rate. If the regime $X_t=i$, the instantaneous interest rate takes the value $r\left(i\right)=r_{X_{t-}}$. In other words, $r_{X_{t-}}$ represents an irreducible Markov chain with values in $\left(r\right(1),r(2),\dots ,r(n\left)\right)$. Consequently, the price of a risk-free asset is governed by the ordinary differential equation:

$${\displaystyle \frac{d{S}_t^0}{S_t^0}}=r_{X_{t-}}dt,S_0^0=1.$$

(3)

We consider the dynamics of the risky asset price as described by the following stochastic differential equation (SDE):

$${\displaystyle \frac{d{S}_t}{S_{t-}}}={\mu }_{X_{t-}}dt+{\sigma }_{X_{t-}}d{L}_t$$

(4)

where ${\mu }_{X_{t-}}$ represents the expected return on the risky asset, ${\sigma }_{X_{t-}}$ denotes the volatility, and $L:={\left(L_t\right)}_{0\le t\le T}$ is a general zero-mean Lévy process that is independent of ${\left(X_t\right)}_{t\ge 0}$. We assume L is $\mathcal{F}$-adapted, and we consider its càdlàg modification as described in Theorem 2.1.7 in [22]. It is worth noting that the models (3) and (4) have become standard in financial mathematics in recent years. Here, a Lévy process refers to a continuous-time stochastic process with independent and stationary increments, as outlined in Section 1.3 of [22], and possesses càdlàg trajectories.

Additionally, we assume that the Lévy measure $\nu \left(dz\right)$ of L satisfies the condition:

$${\int }_{\left\{\right|z|>1\}}{\left|z\right|}^3\nu \left(dz\right)<\infty .$$

(5)

This technical requirement allows us to decompose the zero-mean process L (refer to [4,23] and Theorem 2.4.16 in [22]). This decomposition ensures that, for $0\le t\le T$, $L_t$ can be expressed as $L_t=W_t+K_t$, where $W_{·}:={\left(W_t\right)}_{0\le t\le T}$ represents a standard Brownian motion:

$$K_{·}:={\left({\int }_0^t{\int }_{\mathbb{R}}z(\aleph (ds,dz)-\nu \left(dz\right)ds)\right)}_{0\le t\le T}$$

is a square integrable martingale, and $\aleph (ds,dz)$ denotes a Poisson random measure. Importantly, ℵ is independent of W, and this measure is defined on $\mathcal{B}\left(\right(0,T\left]\right)\otimes \mathcal{B}(\mathbb{R}\setminus \{0\left\}\right)$ by

$$\aleph (dt,dz)=\sum _{0<s\le T}11_{\{s,\Delta L_s\}}(dt,dz)11_{\{\Delta L_s\ne 0\}}\left(s\right),$$

where, for a fixed s, $\Delta L_s=L_s-L_{s-}$ quantifies the number of jumps. This measure, denoted by $\nu$, is $\sigma$-finite and defined on $\mathcal{B}(\mathbb{R}\setminus \{0\left\}\right)$, the compensator, representing the predictable dual-projection of ℵ, is $dt\nu \left(dz\right)$. The adjusted random measure is given by:

$$\tilde{\aleph }(ds,dz)=\aleph (ds,dz)-ds\nu \left(dz\right).$$

We establish the conditions $\nu \left(\right\{0\left\}\right)=\aleph \left(\right\{0\},\mathbb{R}\setminus \{0\left\}\right)=\aleph \left(\right(0,T],\{0\left\}\right)=0$.

Regarding the coefficients and compensator, the following assumptions are made:

$\left({\mathbf{A}}_{\mathbf{2}}\right)$

The values of ${\sigma }_{X_{t-}}$, ${\mu }_{X_{t-}}$ and $r_{X_{t-}}$ are non-negative and vary according to the state $X_{t-}$.

$\left({\mathbf{A}}_{\mathbf{3}}\right)$

The measure $\nu$ of L is defined over $(-1,\infty )$ and meets the condition (5).

Condition $\left({\mathbf{A}}_{\mathbf{2}}\right)$ is self-evident from an economic perspective and requires no further explanation. Notably, when ${\mu }_{X_{t-}}=r_{X_{t-}}=0$, this signifies the scenario where the risky asset process S is already a martingale under $\mathbb{P}$. On the other hand, Condition $\left({\mathbf{A}}_{\mathbf{3}}\right)$ ensures that the jumps of the Lévy process L are bounded from below by $-1$, guaranteeing the positivity of the price process S, as demonstrated in Proposition 5.1 [22]. Additionally, the integrability of $\nu$ implies that

$${\left(K_t={\int }_0^t{\int }_{\mathbb{R}}z\tilde{\aleph }(ds,dz)\right)}_{0\le t\le T}$$

The measure $K_{·}$ is a square-integrable martingale measure, as proven in Theorem 4.2.3 of [22]. It is important to note that the integral is zero on the interval $(-\infty ,-1]$, and integration always occurs over the entire real line. Consequently, Equation (4) can be expressed as

$${\displaystyle \frac{d{S}_t}{S_{t-}}}={\mu }_{X_{t-}}dt+{\sigma }_{X_{t-}}d(W_t+K_t)$$

(6)

By applying Itô’s formula, the discounted price of the risky asset, denoted as ${\tilde{S}}_t=\frac{S_t}{S_t^0}$, can be represented as a special semi-martingale. Consequently, ${\tilde{S}}_t$ can be decomposed as follows:

$${\tilde{S}}_t={\tilde{S}}_0+Z_t+A_t,$$

where the process $Z_{·}:={\left(Z_t\right)}_{0\le t\le T}$ is a square-integrable martingale starting from 0 and $A_{·}:={\left(A_t\right)}_{0\le t\le T}$ is an integrable stochastic process with finite variation $\left|A\right|$ (i.e., ${\sup }_{\tau }{\sum }_i\left(\right|A_{t_i}-A_{t_{i-1}}\left|\right)<\infty$) for every partition $\tau$ of $[0,T]$) with $A_0=0$. Consequently, the decomposition can be expressed as follows:

$$\begin{array}{ccc}\hfill d{\tilde{S}}_t& =& \frac{d{S}_{t-}}{S_t^0}-\frac{S_{t-}{r}_{X_{t-}}}{S_t^0}dt\hfill \\ & =& {\tilde{S}}_t(({\mu }_{X_{t-}}-r_{X_{t-}})dt+{\sigma }_{X_{t-}}d{W}_t+{\sigma }_{X_{t-}}{\int }_{\mathbb{R}}z\tilde{\aleph }(dt,dz).\hfill \end{array}$$

(7)

Therefore, the martingale component Z and the finite variation component A of $\tilde{S}$ are determined by:

$$\begin{array}{ccc}\hfill Z_t& =& {\int }_0^t{\tilde{S}}_{s-}{\sigma }_{X_{s-}}d{W}_s+{\int }_0^t{\tilde{S}}_{s-}{\sigma }_{X_{s-}}{\int }_{\mathbb{R}}z\tilde{\aleph }(ds,dz)\hfill \\ & =& {\int }_0^t{\tilde{S}}_{s-}{\sigma }_{X_{s-}}d(W_s+K_s)\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill A_t& =& {\int }_0^t{\tilde{S}}_{s-}({\mu }_{X_{s-}}-r_{X_{s-}})ds.\hfill \end{array}$$

(9)

The equivalent martingale measure $\mathbb{Q}$ can be characterized by the following Girsanov density:

$${\displaystyle \frac{d\mathbb{Q}}{d\mathbb{P}}}{|}_{{\mathcal{F}}_t}=1+{\int }_0^t{S}_{s-}{\Phi }_{s}d{W}_s+{\int }_0^t{\int }_{\mathbb{R}}{S}_{s-}{\sigma }_{X_{t-}}({\kappa }_{s,z}-1)\tilde{\aleph }(ds,dz)$$

(10)

where ${\Phi }_{·}:={\left({\Phi }_t\right)}_{0\le t\le T}$ and ${\kappa }_{·,·}:={\left({\kappa }_{t,z}\right)}_{0\le t\le T,z\in \mathbb{R}}$ are ${\mathcal{F}}_t$- adapted and predictable processes that satisfy Equation (14) below and also satisfy:

$${\int }_0^T\mathbb{E}\left[{\left|{\Phi }_t\right|}^2\right]dt<\infty ,{\int }_0^T{\int }_{\mathbb{R}}\mathbb{E}\left[{\left|{\kappa }_{t,z}\right|}^2\mathfrak{Q}(t,dz)\gamma \left(t\right)\right]dt<\infty \mathrm{and}{\kappa }_{t,z}>-1,$$

for all $0\le t\le T,z\in \mathbb{R}$.

For such processes ${\Phi }_{·}$ and ${\kappa }_{·,·}$, we set

$$W_t^{\mathbb{Q}}=W_t-{\int }_0^t{\Phi }_{u}du.$$

(11)

and

$$d{K}_t^{\mathbb{Q}}:={\int }_{\mathbb{R}}z{\aleph }^{\mathbb{Q}}(dt,dz)={\int }_{\mathbb{R}}z\left(\aleph (dt,dz)-\underset{{\nu }^{\mathbb{Q}}(dt,dz)}{\underbrace{{\kappa }_{t,z}dt\nu \left(dz\right)}}\right)$$

(12)

The price (7) of the risky asset can be written as follows:

$$\begin{array}{ccc}\hfill d{\tilde{S}}_t& =& {\tilde{S}}_t\left(({\mu }_{X_{t-}}-r_{X_{t-}})dt+{\sigma }_{X_{t-}}d{W}_t^{\mathbb{Q}}+{\sigma }_{X_{t-}}{\Phi }_{t}dt\right)\hfill \\ & & +{\tilde{S}}_{t-}{\sigma }_{X_{t-}}{\int }_{\mathbb{R}}z(\aleph (dt,dz)-{\kappa }_{t,z}dt\nu \left(dz\right))\hfill \\ & & +\left({\tilde{S}}_{t-}{\sigma }_{X_{t-}}{\int }_{\mathbb{R}}z({\kappa }_{t,z}-1)\nu \left(dz\right)\right)dt\hfill \end{array}$$

(13)

Note that $W^{\mathbb{Q}}$ is a standard Brownian motion under $\mathbb{Q}$ and $K^{\mathbb{Q}}$ is a martingale under $\mathbb{Q}$. For further details, interested readers can refer to Section II of [24]. Thus, the discounted risky asset price process $\tilde{S}$ is a martingale under $\mathbb{Q}$ if and only if

$${\sigma }_{X_{t-}}{\int }_{\mathbb{R}}z({\kappa }_{t,z}-1)\nu \left(dz\right)+{\sigma }_{X_{t-}}{\Phi }_t+({\mu }_{X_{t-}}-r_{X_{t-}})=0.$$

(14)

Thus, we have, under $\mathbb{Q}$, that

$$\begin{array}{ccc}\hfill d{\tilde{S}}_t& =& {\tilde{S}}_{t-}{\sigma }_{X_{t-}}{\int }_{\mathbb{R}}z{\aleph }^{\mathbb{Q}}(dt,dz)+{\tilde{S}}_{t-}{\sigma }_{X_{t-}}d{W}_t^{\mathbb{Q}}\hfill \\ & =& {\tilde{S}}_{t-}{\sigma }_{X_{t-}}(d{W}_t^{\mathbb{Q}}+d{K}_t^{\mathbb{Q}})\hfill \end{array}$$

(15)

which means that ${\tilde{S}}_{·}$ is a square integrable $\mathbb{Q}$-martingale. The compensator of $\aleph (dt,dz)$ under the equivalent martingale measure $\mathbb{Q}$ is denoted as ${\nu }^{\mathbb{Q}}(dt,dz)={\kappa }_{t,z}dt\nu \left(dz\right).$ In the following subsection, we outline the life insurance portfolio and the setup for the stream of liabilities.

2.2. Insurance Modeling

We examined an insurance portfolio comprising ${\ell }_x$ insured lives aged x over a T-year term. The life-spans of policyholders at time 0 are represented by identically distributed non-negative random variables $\left({\tau }_i\right)$ for $i=1,\dots ,{\ell }_x$. These variables depend on the age x. We denote ${\left({\mathcal{H}}_t^i\right)}_{t\ge 0}$ for $i=1,\dots ,{\ell }_x$ as the smallest $\sigma$-algebra with respect to which ${\tau }_i$ is a stopping time. Assuming that the distribution of ${\tau }_i$ is absolutely continuous with an intensity ${\mu }_{x+t}^i$ (i.e., the intensity of $\left({\tau }_i\right)$ at time t, for an individual of age x at time 0), this intensity is also referred to as the mortality rate. It is a non-negative predictable process defined on the filtration ${\left({\mathcal{G}}_t\right)}_{t\ge 0}$, where ${\mathcal{G}}_t=\sigma ({\mu }_u^i,i=1,\dots ,{\ell }_{x}0\le u\le t)\vee \mathcal{N}$, which provides sufficient information about the mortality intensity’s evolution. This ensures that ${\int }_0^T{\mu }_{x+s}^{i}ds$ is finite almost surely, where $\mathcal{N}$ is the set of all $\mathbb{P}$-negligible sets. The survival function for all $0\le t\le T$ is denoted as

$${}_{t}p_x^i=\mathbb{P}\left({\tau }_i\ge t|{\mathcal{G}}_t\right)=\exp \left(-{\int }_0^t{\mu }_{x+s}^{i}ds\right).$$

(16)

This entails that conditionally to ${\left({\mathcal{G}}_t\right)}_{t\ge 0}$, the remaining lifetimes follow independent inhomogeneous Poisson processes. In actuarial mathematics, it is common to model the number of deaths or claims using step processes, as outlined in References such as [25,26]. In our case, we represent insurance claims using the step process J.

Definition 1.

A process J is termed a step process if its paths are right-continuous with left limit (RCLL) step functions, displaying a finite number of jumps within any finite time interval. The $\mathcal{F}$-adapted step process J with $J_0=0$ can be expressed as follows:

$$J_t=\sum _{n=1}^{{\ell }_x}{\mathrm{a}}_{n}11_{\{{\tau }_n\le t\}},$$

(17)

where ${\tau }_n$ represents a sequence of $\mathcal{F}$-stopping times, signifying the $n\mathrm{th}$ jump time of the process J and ${\mathrm{a}}_n$ corresponds to its jump intensity at time ${\tau }_n$. The jump measure of a step process is a finite random measure. When ${\mathrm{a}}_n=1$ for all $n\ge 1$, $J_t={\sum }_{n=1}^{{\ell }_x}11_{\{{\tau }_n\le t\}}$ signifies the number of deaths until time t, within the interval $0\le t\le T$. Additionally, the $\mathbb{P}$-martingale $\overline{J}:={\left({\overline{J}}_t\right)}_{0\le t\le T}$ is defined as

$${\overline{J}}_t=J_t-{\int }_0^t{\eta }_{s}ds,\mathrm{with}{\eta }_s=\left({\ell }_x-J_{s-}\right){\mu }_{x+s}.$$

Using the process J, we can define the random measure as follows:

$$\aleph (dt,dz)=\sum _{s\in (0,T]}11_{\{s,\Delta J_s\}}(dt,dz)11_{\{\Delta J_s\ne 0\}}\left(s\right).$$

(18)

satisfying $\left({\mathbf{A}}_{\mathbf{1}}\right)$.

We also take into account a systematic risk factor that cannot be mitigated, represented by the process: $Z_{·}:={\left(Z_t\right)}_{0\le t\le T}$, with the dynamics described by the forward stochastic differential equation (SDE), where ${\left(B_t\right)}_{0\le t\le T}$ is an $\mathcal{F}$-adapted Brownian motion:

$$d{Z}_t=b^Z\left(Z_t\right)dt+{\sigma }^Z\left(Z_t\right)d{B}_t,{\mu }_0=\mu >0.$$

(19)

The longevity risk is modeled by a stochastic process that solves the forward SDE (19) describing the time evolution of the mortality intensity. Such stochastic dynamics are advocated in the actuarial literature; see [27,28]; we assume that:

$\left({\mathbf{A}}_{\mathbf{4}}\right)$

There exists an $\mathcal{F}$-adapted positive unique solution on $[0,T]$ of (19);

$\left({\mathbf{A}}_{\mathbf{5}}\right)$

The processes W and B are independent.

Note that a positive solution can be guaranteed by means of the comparison principle for the SDEs, for example whenever the drift $b_Z>0.$

In this research, we explored a wide array of payment scenarios commonly found in reinsurance, insurance, and pensions. Our analysis included a variety of insurance liabilities and explored their potential links with financial dynamics.

In this study, we explored the insurance payment process in-depth ${\left(\mathrm{P}\left(t\right)\right)}_{0\le t\le T}$ arising from the liabilities stream, as expressed by:

$$\mathrm{P}\left(t\right)={\int }_0^t\hat{A}\left(s\right)ds+{\int }_0^t{\int }_{\mathbb{R}}\hat{D}(s,y)\aleph (ds,dy)+\hat{S}11_{\{t=T\}}$$

where J represents the death counting process for a life insurance portfolio comprising n policies and ℵ is its corresponding jump measure defined in (18). The general stream of liabilities is captured by the process $\mathrm{P}$, encompassing continuous annuity payments denoted by $\hat{A}$, death benefits associated with the claim $\hat{D}$ occurring randomly due to jumps in the step process J, and the liability $\hat{S}$ settled at the end of the contract, also known as the survival benefit. By considering the representation (18), we deduce that:

$${\int }_0^t{\int }_{\mathbb{R}}\hat{D}(s,y)\aleph (ds,dy)=\sum _{s\in [0,t]}\hat{D}(s,\Delta J_s)11_{\{\Delta J_s\ne 0\}}0\le t\le T,\mathbb{P}-\mathrm{a}.\mathrm{s}.$$

with regard to the random measure ℵ, this stochastic integral models claims occurring at various times corresponding to jumps in the step process $J_{·}$.

It is important to highlight the following:

$({\mathbf{A}}_{\mathbf{6}}$)

The process of the annuity $\hat{A}:\Omega \times [0,T]\times \mathbb{R}⟶\mathbb{R}$ is an $\mathcal{F}$-predictable product measurable and satisfying

$$\mathbb{E}\left[{\int }_0^T{\left|\hat{A}\left(t\right)\right|}^{2}dt\right]<\infty .$$

The process representing the death benefit $\hat{D}:\Omega \times [0,T]⟶\mathbb{R}$ is $\mathcal{F}$-predictable satisfying

$$\mathbb{E}\left[{\int }_0^T{\int }_{\mathbb{R}}{\left|\hat{D}(t,y)\right|}^2{\max (\gamma \left(t\right),|\gamma \left(t\right)|}^2)\mathfrak{Q}(t,dy)dt\right]<\infty .$$

The random variable $\hat{S}:\Omega ⟶\mathbb{R}$ represents the survival benefit, which is ${\mathcal{F}}_T$-measurable and satisfying $\mathbb{E}\left[\right|\hat{S}{|}^2]<\infty$;

$\left({\mathbf{A}}_{\mathbf{7}}\right)$

The quadratic variation ${[J,L]}_t=0$, $\forall t\in [0,T]$, i.e., the process J and the Lévy process L never jump together.

By the predictability of the mapping $\Omega \times [0,T]\times \mathbb{R}⟶\mathbb{R}$, we mean measurability with respect to the product $\mathcal{P}\otimes \mathcal{B}\left(\mathbb{R}\right)$, and for quadratic optimization purposes, square integrability is essential. Given the assumptions $\left({\mathbf{A}}_{\mathbf{6}}\right)$, we assume that the process $\mathrm{P}$ is square-integrable. The Cauchy–Schwarz inequality and Doob’s martingale inequality imply:

$$\begin{array}{ccc}\hfill \mathbb{E}\left[\underset{t\in [0,T]}{\sup }{\left|\mathrm{P}\left(t\right)\right|}^2\right]& \le & C\{\mathbb{E}\left[\underset{t\in [0,T]}{\sup }{\left|{\int }_0^t{\int }_{\mathbb{R}}\hat{D}(s,y)\aleph (ds,dy)\right|}^2\right]\hfill \\ & & +\mathbb{E}\left[{\left|{\int }_0^t{\int }_{\mathbb{R}}\left|\hat{D}(s,y)\right|\mathfrak{Q}(s,dy)\gamma \left(s\right)ds\right|}^2\right]\hfill \\ & & +\mathbb{E}\left[{\int }_0^T{\left|\hat{A}\left(t\right)\right|}^{2}dt\right]+\mathbb{E}\left[|\hat{S}11_{\{t=T\}}{|}^2\right]\}\hfill \\ & \le & C\{\mathbb{E}\left[{\int }_0^T{\int }_{\mathbb{R}}{\left|\hat{D}(s,y)\right|}^2\mathfrak{Q}(s,dy)\gamma \left(s\right)ds\right]\hfill \\ & & +\mathbb{E}\left[{\int }_0^T{\int }_{\mathbb{R}}{\left|\hat{D}(s,y)\gamma \left(s\right)\right|}^2\mathfrak{Q}(s,dy)ds\right]\hfill \\ & & +\mathbb{E}\left[{\int }_0^T{\left|\hat{A}\left(t\right)\right|}^{2}dt\right]\mathbb{E}\left[|\hat{S}11_{\{t=T\}}{|}^2\right]\}<\infty \hfill \end{array}$$

where we used the fact that the process

$${\left({\int }_0^t{\int }_{\mathbb{R}}\hat{D}(s,y)(\aleph (ds,dy)-\mathfrak{Q}(s,dy)\gamma \left(s\right)ds\right)}_{0\le t\le T}$$

is a square-integrable martingale with quadratic variation process $[·,·]$ (see Definition 11.16 in [29]) satisfying

$$\mathbb{E}\left[{\left[{\int }_0^{·}{\int }_{\mathbb{R}}\hat{D}(s,y)\aleph (ds,dy)\right]}_T\right]=\mathbb{E}\left[{\int }_0^T{\int }_{\mathbb{R}}{\left|\hat{D}(s,y)\right|}^2\mathfrak{Q}(s,dy)\gamma \left(s\right)ds\right],$$

where specific assumption on $\mathfrak{Q}(s,dy)$ is given by (1). Moreover, the right-hand side of the above inequality is finite thanks to $\left({\mathbf{A}}_{\mathbf{6}}\right)$.

3. Local Risk-Minimizing Hedging Strategies

We suppose that $\mathrm{P}\left(t\right)$ is an ${\mathcal{F}}_t$-measurable claim satisfying

$$\underset{t\in [0,T]}{\sup }{\mathbb{E}}_{\mathbb{Q}}\left[{\left(e^{-{\int }_0^t{r}_{X_{u-}}du}\mathrm{P}\left(t\right)\right)}^2\right]<\infty ,$$

The arbitrage-free price $F(t,S_t,s)$, based on risk-neutral valuation, is expressed as follows:

$$F(t,S_t,s)=\left\{\begin{array}{cc}{\mathbb{E}}_{\mathbb{Q}}\left[{\displaystyle \frac{S_t^0}{S_s^0}}\mathrm{P}\left(s\right)|{\mathcal{F}}_t\right]\hfill & \mathrm{if}t<s\le T\hfill \\ {\displaystyle \frac{S_t^0}{S_s^0}}\mathrm{P}\left(s\right)\hfill & \mathrm{if}0\le s<t.\hfill \end{array}\right.$$

We used $F_t$, the derivative of F with respect to time (first variable), and $F_x$ for the derivative with respect to space (second variable). Regarding the second variable, we assumed that the first derivative of F is almost surely bounded under $\mathbb{Q}$. Now, let us introduce a class of admissible strategies: A trading strategy, denoted as $\varphi =(\pi ,\overline{\pi })$, comprises $\pi :={\left({\pi }_t\right)}_{0\le t\le T}$, indicating the number of risky assets, and $\overline{\pi }:={\left({\overline{\pi }}_t\right)}_{0\le t\le T}$, representing the amount invested in riskless assets. The value of the discounted portfolio at time t is then given by:

$$V_t\left(\varphi \right)={\pi }_t{\tilde{S}}_t+{\overline{\pi }}_t$$

Definition 2 (Strategy).

1. 

A strategy $\pi :={\left({\pi }_t\right)}_{0\le t\le T}$ is considered admissible, denoted by $\pi \in {\mathcal{A}}_{loc}^s$, if it fulfills the following criteria:

(i) 

$\pi :\Omega \times [0,T]⟶\mathbb{R}$ is an $\mathcal{F}$-predictable process;

(ii) 

$\mathbb{E}\left[{\int }_0^T\right|{\pi }_t{\tilde{S}}_t\sigma \left(t\right){|}^{2}dt]<\infty$.

2. 

A strategy $\overline{\pi }:={\left({\overline{\pi }}_t\right)}_{0\le t\le T}$ is deemed admissible, denoted as $\overline{\pi }\in {\mathcal{A}}_{loc}^b$, if it meets the following conditions:

(i) 

$\overline{\pi }:\Omega \times [0,T]⟶\mathbb{R}$ is an $\mathcal{F}$-adapted process;

(ii) 

The process $V_t\left(\varphi \right):={\pi }_t{\tilde{S}}_t+{\overline{\pi }}_t$ is considered admissible if it is both square-integrable and right-continuous.

3. 

A couple ${\varphi }_t=({\pi }_t,{\overline{\pi }}_t)$ is called an admissible strategy for local risk minimization written as ${\varphi }_t\in {\mathcal{A}}_{loc}$, if $\pi \in {\mathcal{A}}_{loc}^s$ and $\overline{\pi }\in {\mathcal{A}}_{loc}^b$.

The strategy $\pi$ represents the quantity of risky assets held in the investment portfolio; $\overline{\pi }$ indicates the position in the riskless asset (such as a bank account); $V\left(\varphi \right)$ represents the discounted value process of the portfolio investment corresponding to the strategy $\varphi =(\pi ,\overline{\pi })$. It is important to note that the investment portfolio $V\left(\varphi \right)$ is not self-financing under $\varphi \in {\mathcal{A}}_{loc}$.

Next, we introduce the risk and cost processes associated with the hedging strategy.

Definition 3 (Risk process).

The accumulative cost process of an admissible strategy $\varphi \in {\mathcal{A}}_{loc}$ associated with the hedging insurance payment process $\mathrm{P}$ for $0\le t\le T$ is modeled by

$$C^{\varphi }\left(t\right)=V_t\left(\varphi \right)-{\int }_0^t{\pi }_{u}d{\tilde{S}}_u,$$

(20)

The risk associated with an admissible strategy $\varphi \in {\mathcal{A}}_{loc}$ related to the process $\mathrm{P}$ for $0\le t\le T$ is given by:

$$R^{\varphi }(t,T)=\mathbb{E}\left[{\left|C^{\varphi }\left(T\right)-C^{\varphi }\left(t\right)\right|}^2|{\mathcal{F}}_t\right]$$

(21)

The integral term ${\int }_0^t{\pi }_{s}d{\tilde{S}}_s$ is well-defined and referred to as the gain process, $C^{\varphi }$ square-integrable due to the fact that $\varphi =(\pi ,\overline{\pi })\in {\mathcal{A}}_{loc}$.

Therefore, the process $C^{\varphi }\left(t\right)$ represents the cumulative costs incurred by the hedger over the period $[0,t]$, encompassing the payment process $\mathrm{P}$. It captures the total discounted cash inflow or outflow for an insurer who adopts the investment strategy $\varphi$ and maintains the investment portfolio $V_{·}\left(\varphi \right)$ while also fulfilling the insurance benefit $\mathrm{P}$. The process $V_t\left(\varphi \right)$ signifies the value of the portfolio ${\varphi }_t$ held at time t after accounting for the payment $\mathrm{P}\left(t\right)$. Notably, the random variable $V_T\left(\varphi \right)$ represents the portfolio’s value ${\varphi }_T$ upon the settlement of all liabilities. To confine the analysis to zero-admissible strategies, it is necessary to satisfy the condition:

$$V_T\left(\varphi \right)=\mathrm{Survival}\mathrm{benefit}\mathbb{P}-\mathrm{a}.\mathrm{s}.$$

Remark 1.

A trading strategy pair ${\varphi }_t=({\pi }_t,{\overline{\pi }}_t)$ is labeled as self-financing if the cost process remains constant over time. It earns the label mean self-financing if the cost process behaves as a martingale under the probability measure$\mathbb{P}$, meaning that, on average, there is no net gain or loss in the value of the portfolio.

Definition 4 (Föllmer–Schweizer decomposition).

A random variable $\mathrm{P}$, which is measurable with respect to ${\mathcal{F}}_T$ and belongs to the square integrable space $L_{\mathbb{P}}^2$, is said to have a Föllmer–Schweizer decomposition if it can be expressed as:

$$\mathrm{P}={\mathrm{P}}_0+{\int }_0^T{\phi }_{t}d{S}_t+{\varrho }_T\mathbb{P}-\mathrm{a}.\mathrm{s}.$$

Here, ${\mathrm{P}}_0$ is a square-integrable ${\mathcal{F}}_0$-measurable random variable, φ is an admissible strategy, and $\varrho :={\left({\varrho }_t\right)}_{0\le t\le T}$ represents a right-continuous square-integrable $\mathbb{P}$-martingale, which is strongly orthogonal to the martingale component of S with ${\varrho }_0=0$.

If the claim has a Föllmer–Schweizer decomposition, the local risk-minimizing strategy can be directly determined by $\varphi$:

$${\varphi }_t=\left({\pi }_t,{\mathrm{P}}_0+{\int }_0^T{\phi }_{t}d{S}_t+{\varrho }_t-{\pi }_t{S}_t\right)$$

In the paper by Riesner [30], the author extended the computations from his previous work in [23] to derive the local risk-minimizing hedging strategy for claims to payment processes. He established the Föllmer–Schweizer decomposition of $\frac{1}{S_t^0}F(t,S_t,s)$, with $0\le t<s\le T$, provided by:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{S_t^0}}F(t,S_t,s)& =& F(0,S_0,s)+{\int }_0^t{\phi }_{u,s}d{\tilde{S}}_u+{\varrho }_{t,s}.\hfill \end{array}$$

(22)

However, he did not specify under which measure this decomposition holds. He assumed that finding the Galtchouk–Kunita–Watanabe decomposition under the minimal martingale measure is equivalent to the Föllmer–Schweizer decomposition under the real-world measure $\mathbb{P}$. It is important to note that, when referring to the Föllmer–Schweizer decomposition, the martingale parts of $\tilde{S}$ and $\varrho (·,s)$ should be orthogonal under $\mathbb{P}$. In [4], it was demonstrated that checking this orthogonality condition under the real-world measure is challenging, as $\varrho (·,s)$ is only a $\mathbb{P}$-semi-martingale, not a $\mathbb{P}$-local martingale.

Our approach involves finding the Föllmer–Schweizer decomposition of the portfolio under the measure $\mathbb{P}$. At maturity, the portfolio’s value should be a martingale under an equivalent martingale measure, which will be specified later, and it must equal the required amount. We imposed all necessary conditions on the unknown decomposition (see (i)–(v) in the proof of Theorem 1) to obtain such a portfolio. Specifically, we describe all equivalent probability measures using the Girsanov density (10)). We assumed that the functions $\Phi$ and $\kappa$ are defined in a way that $\tilde{S}$ becomes a $\mathbb{Q}$-martingale, leading to the condition (14). We derive the process V described in (24), which is a martingale under $\mathbb{Q}$. Finally, we determine the Föllmer–Schweizer decomposition of V under $\mathbb{P}$ and find the functions $\phi$, $\varrho$ (see the proof of Theorem 2). Simultaneously, we provide an explicit expression for $\Phi$ and $\kappa$. Our results are presented in Equations (11)–(14) and are summarized in the following theorem:

Theorem 1.

Under $\left({\mathbf{A}}_{\mathbf{1}}\right)$–$\left({\mathbf{A}}_{\mathbf{7}}\right)$, the $\mathbb{Q}$-martingale value process $(V{(t,s)}_{0\le t\le s}$ defined by

$$\begin{array}{ccc}\hfill V(t,s):& =& \frac{1}{S_t^0}F(t,S_t,s)={\mathbb{E}}_{\mathbb{Q}}\left[\frac{\mathrm{P}\left(s\right)}{S_s^0}|{\mathcal{F}}_t\right]fort<s\le T\hfill \end{array}$$

(23)

under the new measure $\mathbb{Q}$ evolves according to the following dynamics:

$$\begin{array}{ccc}\hfill V(t,s)& =& V(0,s)+{\int }_0^t{F}_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}d{W}_u^{\mathbb{Q}}+{\int }_0^t{\int }_{\mathbb{R}}h(u,z,s){\aleph }^{\mathbb{Q}}(du,dz).\hfill \end{array}$$

(24)

where $W^{\mathbb{Q}}$ (respectively) ${\aleph }^{\mathbb{Q}}(du,dz)$ are characterized by (11) (respectively) (12) and

$$h(u,z,s)={\displaystyle \frac{1}{S_u^0}}\left(F_x(u,S_{u-}(1+{\sigma }_{X_{u-}}z),s)-F(u,S_{u-},s)\right)$$

Proof. 

By applying the differential form, with respect to the variable t, of the Itô’s formula to $\frac{1}{S_t^0}F(t,S_t,s)$, given in (23), we obtain the dynamics below for $V(t,s)$:

$$\begin{array}{ccc}\hfill dV(t,s)& =& -r_{X_{t-}}{\displaystyle \frac{1}{S_t^0}}F(t,S_{t-},s)dt+{\displaystyle \frac{1}{S_t^0}}[F_t(t,S_{t-},s)dt\hfill \\ & & +F_x(t,S_{t-},s)d{S}_t+\frac{1}{2}{F}_{xx}(t,S_{t-},u)d{[S,S]}_t^c]\hfill \\ & & +{\int }_{\mathbb{R}}{\displaystyle \frac{1}{S_t^0}}\left[F(t,S_{t-}(1+{\sigma }_{X_{t-}}z),s)-F(t,S_{t-},s)\right]\aleph (dt,dz)\hfill \\ & & -{\int }_{\mathbb{R}}{\displaystyle \frac{1}{S_t^0}}{F}_x(t,S_{t-},s){\sigma }_{X_{t-}}{S}_{t-}z\aleph (dt,dz).\hfill \end{array}$$

Now, we drive the relationship between $S_t$ and ${\tilde{S}}_t$:

$$d{\tilde{S}}_t=d\left({\displaystyle \frac{S_t}{S_t^0}}\right)=\frac{d{S}_t}{S_t^0}-\frac{S_{t-}{r}_{X_{t-}}}{S_t^0}dt.$$

Therefore, $\frac{d{S}_t}{S_t^0}=d{\tilde{S}}_t+{\tilde{S}}_t{r}_{X_{t-}}dt$. Let $S_t^c$ be the continuous part of $S_t$; we need the quadratic variation process of the martingale component in (6):

$${[S,S]}_t^c={\left[S^c,S^c\right]}_t={\left[{\int }_0^{·}{\sigma }_{X_{s-}}{S}_{s-}d{W}_s,{\int }_0^{·}{\sigma }_{X_{s-}}{S}_{s-}d{W}_s\right]}_t={\int }_0^t{\sigma }_{X_{s-}}^2{S}_{s-}^{2}ds.$$

the dynamics of $V(t,s)$ becomes

$$\begin{array}{ccc}\hfill dV(t,s)& =& (-r_{X_{t-}}{\displaystyle \frac{1}{S_t^0}}F(t,S_{t-},s)+{\displaystyle \frac{1}{S_t^0}}{F}_t(t,S_{t-},s)\hfill \\ & & +{\tilde{S}}_{t-}{r}_{X_{t-}}{F}_x(t,S_{t-},s)+\frac{1}{2}{F}_{xx}(t,S_{t-},s){\displaystyle \frac{1}{S_t^0}}{\sigma }_{X_{t-}}^2{S}_{t-}^2)dt\hfill \\ & & +{\int }_{\mathbb{R}}\{{\displaystyle \frac{1}{S_t^0}}\left[F(t,S_{t-}(1+{\sigma }_{X_{t-}}z),s)-F(t,S_{t-},s)\right]\hfill \\ & & -{\displaystyle \frac{1}{S_t^0}}{F}_x(t,S_{t-},s){\sigma }_{X_{t-}}{S}_{t-}z\}{\kappa }_{t,z}\nu \left(dz\right)dt+F_x(t,S_{t-},s)d{\tilde{S}}_t\hfill \\ & & +{\int }_{\mathbb{R}}\{{\displaystyle \frac{1}{S_t^0}}\left[F(t,S_{t-}(1+{\sigma }_{X_{t-}}z),s)-F(t,S_{t-},s)\right]\hfill \\ & & -{\displaystyle \frac{1}{S_t^0}}{F}_x(t,S_{t-},s){\sigma }_{X_{t-}}{S}_{t-}z\}\left[\aleph (dt,dz)-{\kappa }_{t,z}\nu \left(dz\right)dt\right].\hfill \end{array}$$

This expression is a martingale under the measure $\mathbb{Q}$, with a zero drift term. Substituting (15) for the discounted risky asset under the $\mathbb{Q}$-measure and utilizing (12), we obtain:

$$\begin{array}{ccc}\hfill dV(t,s)& =& F_x(t,S_{t-},s){\sigma }_{X_{t-}}{\tilde{S}}_{t-}d{W}_t^{\mathbb{Q}}\hfill \\ & & +F_x(t,S_{t-},s){\sigma }_{X_{t-}}{\tilde{S}}_{t-}{\int }_{\mathbb{R}}z{\aleph }^{\mathbb{Q}}(dt,dz)\hfill \\ & & +{\int }_{\mathbb{R}}\left(h(t,z,s)-F_x(t,S_{t-},s){\sigma }_{X_{t-}}{\tilde{S}}_{t-}z\right){\aleph }^{\mathbb{Q}}(dt,dz)\hfill \\ & =& F_x(t,S_{t-},s){\sigma }_{X_{t-}}{\tilde{S}}_{t-}d{W}_t^{\mathbb{Q}}+{\int }_{\mathbb{R}}h(t,z,s){\aleph }^{\mathbb{Q}}(dt,dz),\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill h(t,z,s)& =& {\displaystyle \frac{1}{S_t^0}}\left[F(t,S_{t-}(1+{\sigma }_{X_{t-}}z),s)-F(t,S_{t-},s)\right].\hfill \end{array}$$

This last equation yields the differential form of the formula (24). □

Recall that

$$\begin{array}{ccc}\hfill \left[{\aleph }^{\mathbb{Q}}(dt,dz),{\aleph }^{\mathbb{Q}}(dt,dz)\right]& =& {\nu }^{\mathbb{Q}}(dt,dz),\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill \left[d{W}_t^{\mathbb{Q}},d{W}_t^{\mathbb{Q}}\right]& =& dt.\hfill \end{array}$$

(26)

Moreover, for the compensated random measure ${\aleph }^{\mathbb{Q}}(dt,dz)$ we have

$${\left({\aleph }^{\mathbb{Q}}(dt,dz)\right)}^2={(\aleph (dt,dz))}^2=\aleph (dt,dz).$$

Additionally, $[d{W}_t^{\mathbb{Q}},\int {\aleph }^{\mathbb{Q}}(dt,dz)]=0$, since $W^{\mathbb{Q}}$ is a continuous square-integrable martingale and ${\aleph }^{\mathbb{Q}}$ is a discontinuous processes. For the semi-martingale $\tilde{S}$ defined by Equation (7), we demonstrate that the conditions $\left({\mathbf{B}}_1\right)$ and $\left({\mathbf{B}}_2\right)$ are satisfied. For notational simplicity, we set $a=1+{\int }_{\mathbb{R}}{z}^2\nu \left(dz\right)$:

$\left({\mathbf{B}}_1\right)$

In the martingale part $Z_t$ given by (8), the square bracket process $\left[Z\right]$ is $\mathbb{P}$-a.s. strictly increasing on $[0,T]$, which can be proven by using the formulas of (25) and (26).

$$\begin{array}{ccc}\hfill {\left[Z\right]}_t& =& {\left[{\int }_0^{·}{\tilde{S}}_{u-}{\sigma }_{X_{u-}}d\left(W_u+K_u\right)\right]}_t\hfill \\ & =& {\int }_0^t\left({\tilde{S}}_{u-}^2{\sigma }_{X_{u-}}^2+{\int }_{\mathbb{R}}{z}^2{\tilde{S}}_{u-}^2{\sigma }_{X_{u-}}^2\nu \left(dz\right)\right)du\hfill \\ & =& {\int }_0^t{\tilde{S}}_{u-}^2{\sigma }_{X_{u-}}^{2}adu.\hfill \end{array}$$

(27)

So, ${\left[Z\right]}_{·}$ is $\mathbb{P}$-a.s. strictly increasing on $[0,T]$ if and only if ${\tilde{S}}_{u-}^2{\sigma }_{X_{u-}}^{2}a>0,\forall u\in [0,T]$, which is satisfied since $a>0$.

$\left({\mathbf{B}}_2\right)$

Due to the null Lebesgue measure of the jump part, the finite variation part presented in (9) is continuous.

In the subsequent phase of the procedure, we impose the following conditions to determine the local risk-minimizing hedging strategy. For a chosen portfolio $\varphi$ we use the notation:

(i)

$V_{t,t}\left(\varphi \right):=V_{t,t}=\frac{\mathrm{P}\left(t\right)}{S_t^0}$;

(ii)

Föllmer–Schweizer decomposition

$$\begin{array}{ccc}\hfill V_{t,s}\left(\varphi \right):=V_{t,s}& =& V_{0,s}\left(\varphi \right)+{\int }_0^t\phi (u,s)d{\tilde{S}}_u+\varrho (t,s),\forall t<s.\hfill \end{array}$$

(28)

(iii)

The process $\varrho (·,s)$ is a $\mathbb{P}$-martingale and is $\mathbb{P}$-orthogonal to the martingale part Z of the discounted risky asset. The functions $\phi (u,s)$ and $\varrho (t,s)$ are unspecified in the Föllmer–Schweizer decomposition (28). As the assumptions $\left({\mathbf{B}}_1\right)$ and $\left({\mathbf{B}}_2\right)$ are met, this strategy qualifies as local risk-minimizing. To define the equivalent probability measure $\mathbb{Q}$ such that $\tilde{S}$ is a $\mathbb{Q}$-martingale, we need to satisfy the following condition and explicitly establish the Föllmer–Schweizer decomposition:

(iv)

$V_{·,s}\left(\varphi \right)$ is a martingale under $\mathbb{Q}$.

In this particular segment of our work, we incorporated a novel methodology to assess and determine the pricing of life insurance policies. Our significant contribution lies in the application of the benchmark approach, a framework meticulously developed as outlined in the book by Platen and Heath [19]. This approach serves as the cornerstone of our pricing strategy, facilitating a robust and comprehensive evaluation of life insurance products.

By seamlessly integrating the benchmark approach, we aimed to enhance the precision and reliability of our pricing models. This methodological framework has proven to be highly effective in the framework of quantitative finance, offering a systematic and structured approach to valuing financial instruments. Our adaptation of this approach specifically caters to the intricacies of life insurance, leveraging quantitative finance techniques to arrive at more-accurate and informed pricing decisions.

By drawing on established quantitative finance principles and building upon the benchmark approach, we strive to bring a higher level of sophistication and accuracy to the evaluation of life insurance policies, ultimately benefiting both insurers and policyholders alike. This fusion of quantitative finance techniques with the benchmark approach represents a noteworthy stride towards refining and optimizing the pricing landscape within the life insurance sector.

Here is a crucial result in which we shall match the expressions (22) and (24).

Theorem 2.

Assume that $\left({\mathbf{A}}_{\mathbf{1}}\right)$–$\left({\mathbf{A}}_{\mathbf{7}}\right)$ hold true. If the payment process $\mathrm{P}\left(u\right)$ is ${\mathcal{F}}_u$-measurable and covers all the claims, then the Föllmer–Schweizer decomposition of $\mathrm{P}\left(u\right)$ can be determined through a change of measure, as shown in Equation (10). This change of measure ensures that, $\forall t\in [0,T]$,

$$\begin{array}{ccc}\hfill {\kappa }_{t,z}& =& 1+{\displaystyle \frac{{\mu }_{X_{t-}}-r_{X_{t-}}}{a{\sigma }_{X_{t-}}}}z,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& {\displaystyle \frac{r_{X_{t-}}-{\mu }_{X_{t-}}}{a{\sigma }_{X_{t-}}}}.\hfill \end{array}$$

(30)

For the corresponding portfolio that fulfills the conditions (i)–(iv), the Föllmer–Schweizer decomposition under the real-world measure $\mathbb{P}$ for all $0\le t<s\le T$ is given by:

$$\begin{array}{ccc}\hfill V_{t,s}\left(\varphi \right)& =& V_{0,s}\left(\varphi \right)+{\int }_0^t\phi (u,s)d{\tilde{S}}_u+{\int }_0^t{\psi }^{\left(1\right)}(u,s)d{W}_u\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}{\psi }^{\left(2\right)}(u,z,s)\tilde{\aleph }(du,dz),\hfill \end{array}$$

(31)

such that

$$\begin{array}{ccc}\hfill \phi (u,s)& =& \frac{F_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}+{\int }_{\mathbb{R}}h(u,z,s)z\nu \left(dz\right)}{a{\sigma }_{X_{u-}}{\tilde{S}}_{u-}}\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill {\psi }^{\left(1\right)}(u,s)& =& {\sigma }_{X_{u-}}{\tilde{S}}_{u-}\left(F_x(u,S_{u-},s)-\phi (u,s)\right)\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill {\psi }^{\left(2\right)}(u,z,s)& =& h(u,z,s)-z{\sigma }_{X_{u-}}{\tilde{S}}_{u-}\phi (u,s).\hfill \end{array}$$

(34)

At time u, the strategy $\varphi (u,s)=(\phi (u,s),V_{u,s}\left(\varphi \right)-\phi (u,s){\tilde{S}}_u)$.

Proof. 

With the conditions (i)–(iv) and the result of Theorem 1, we show that

$$V_{t,s}\left(\varphi \right)={\mathbb{E}}_{\mathbb{Q}}\left[\frac{\mathrm{P}\left(s\right)}{S_s^0}|{\mathcal{F}}_t\right]={\mathbb{E}}_{\mathbb{Q}}\left[V_{s,s}\left(\varphi \right)|{\mathcal{F}}_t\right].$$

Employing the expression provided in (ii) and utilizing the decomposition of $V(t,s)$ in (24), along with the discounted risky asset decomposition in (7) and (12), we obtain the following representation for $\varrho (t,s)$:

$$\begin{array}{ccc}\hfill \varrho (t,s)& =& {\int }_0^t{F}_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}d{W}_u^{\mathbb{Q}}-{\int }_0^t\phi (u,s)d{\tilde{S}}_u\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}h(u,z,s)(\aleph (du,dz)-{\kappa }_{u,z}du\nu \left(dz\right)).\hfill \end{array}$$

We aimed to characterize the process $\varrho (·,s)$ under the original measure $\mathbb{P}$. To achieve this, we employed the formula given in (13), which describes the dynamics of the risky asset.

$$\begin{array}{ccc}\hfill \varrho (t,s)& =& {\int }_0^t{F}_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}d{W}_u-{\int }_0^t{F}_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}{\Phi }_{u}du\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}h(u,z,s)\left(\aleph (du,dz)-\nu \left(dz\right)du\right)\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}h(u,z,s)\left(1-{\kappa }_{u,z}\right)\nu \left(dz\right)du\hfill \\ & & -{\int }_0^t\phi (u,s){\tilde{S}}_{u-}\left({\mu }_{X_{u-}}-r_{X_{u-}}\right)du-{\int }_0^t\phi (u,s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}d{W}_u\hfill \\ & & -{\int }_0^t{\int }_{\mathbb{R}}\phi (u,s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}\left(\aleph (du,dz)-\nu \left(dz\right)du\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \varrho (t,s)& =& {\int }_0^t\left(F_x(u,S_{u-},s)-\phi (u,s)\right){\sigma }_{X_{u-}}{\tilde{S}}_{u-}d{W}_u\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}\left(h(u,z,s)-{\sigma }_{X_{u-}}{\tilde{S}}_{u-}\phi (u,s)z\right)\tilde{\aleph }(du,dz)\hfill \\ & & -{\int }_0^t\phi (u,s)\left({\mu }_{X_{u-}}-r_{X_{u-}}\right){\tilde{S}}_{u-}du\hfill \\ & & -{\int }_0^t{F}_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}{\Phi }_{u}du\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}h(u,z,s)\left(1-{\kappa }_{u,z}\right)\nu \left(dz\right)du.\hfill \end{array}$$

It becomes evident that, for $\varrho (·,s)$ to be a martingale under the original measure $\mathbb{P}$, the drift term in the previous expression should be equal to zero for all u in the interval $[0,t]$ almost surely under $\mathbb{P}$. Therefore,

$$\begin{array}{ccc}\hfill F_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}{\Phi }_u& =& {\int }_{\mathbb{R}}h(u,z,s)(1-{\kappa }_{u,z})\nu \left(dz\right)\hfill \\ & & -\phi (u,s)({\mu }_{X_{u-}}-r_{X_{u-}}){\tilde{S}}_{u-}.\hfill \end{array}$$

(35)

The process $\varrho (·,s)$ can also be described by the martingale representation theorem as

$$\begin{array}{ccc}\hfill \varrho (t,s)& =& {\int }_0^t\left(F_x(u,S_{u-},s)-\phi (u,s)\right){\sigma }_{X_{u-}}{\tilde{S}}_{u-}d{W}_u\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}\left(h(u,z,s)-{\sigma }_{X_{u-}}{\tilde{S}}_{u-}\phi (u,s)z\right)\tilde{\aleph }(du,dz)\hfill \\ & =& {\int }_0^t{\psi }^{\left(1\right)}(u,s)d{W}_u+{\int }_0^t{\int }_{\mathbb{R}}{\psi }^{\left(2\right)}(u,z,s)\tilde{\aleph }(du,dz).\hfill \end{array}$$

(36)

The processes ${\psi }^{\left(1\right)}$ and ${\psi }^{\left(2\right)}$ are defined in (33) and (34), respectively. Additionally, the requirement for $\varrho (·,s)$ in (36) and the martingale part Z in (8) of the discounted risky asset has to be orthogonal under $\mathbb{P}$. This is the case if the product $\varrho (·,s)Z_{·}$ is a $\mathbb{P}$-martingale. Therefore, the dynamics of $\varrho (·,s)Z_{·}$ forms a martingale under $\mathbb{P}$ if and only if the drift term is zero for all u in the interval $[0,t]$ almost surely under $\mathbb{P}$.

$$\begin{array}{ccc}\hfill 0& =& {\int }_{\mathbb{R}}(h(u,z,s)-{\sigma }_{X_{u-}}{\tilde{S}}_{u-}\phi (u,s)z){\sigma }_{X_{u-}}{\tilde{S}}_{u-}z\nu \left(dz\right)\hfill \\ \hfill & & +(F_x(u,S_{u-},s)-\phi (u,s)){\sigma }_{X_{u-}}^2{\tilde{S}}_{u-}^2.\hfill \end{array}$$

Solving this equation for $\phi (u,s)$, and to obtain the equation introduced in (32).

Let us recall that the condition (14) ensures that $\tilde{S}$ is a $\mathbb{Q}$-martingale, and multiplying by $F_x(t,S_{t-},s){\tilde{S}}_{t-}$ leads to:

$$F_x(t,S_{t-},s){\tilde{S}}_{t-}\left[{\sigma }_{X_{t-}}{\int }_{\mathbb{R}}({\kappa }_{t,z}-1)z\nu \left(dz\right)+{\sigma }_{X_{t-}}{\Phi }_t({\mu }_{X_{t-}}-r_{X_{t-}})\right]=0.$$

(37)

Now, from the condition (35), which ensures that $\varrho (·,s)$ is a $\mathbb{P}$-martingale, we substitute the value of $\phi (u,s)$ obtained in (32) to get:

$$\begin{array}{ccc}\hfill F_x(t,S_{t-},s){\sigma }_{X_{t-}}{\tilde{S}}_{t-}{\Phi }_t& =& {\int }_{\mathbb{R}}h(t,z,s)(1-{\kappa }_{t,z})\nu \left(dz\right)\hfill \\ & & -\left({\displaystyle \frac{F_x(t,S_{t-},s){\sigma }_{X_{t-}}{\tilde{S}}_{t-}+{\int }_{\mathbb{R}}h(t,z,s)z\nu \left(dz\right)}{a{\sigma }_{X_{t-}}}}\right)\left({\mu }_{X_{t-}}-r_{X_{t-}}\right).\hfill \end{array}$$

(38)

Using Equations (37) and (38) to eliminate $\Phi$ we end-up with:

$${\int }_{\mathbb{R}}\left({\kappa }_{t,z}-1+{\displaystyle \frac{({\mu }_{X_{t-}}-r_{X_{t-}})z}{a{\sigma }_{X_{t-}}}}\right)\left({\sigma }_{X_{t-}}{\tilde{S}}_{t-}{F}_x(t,S_{t-},s)z-h(t,z,s)\right)\nu \left(dz\right)=0.$$

(39)

Notice that we cannot always back-up ${\kappa }_{t,z}$ from Equation (39), in particular when

$${\sigma }_{X_{t-}}{\tilde{S}}_{t-}{F}_x(t,S_{t-},s)z-h(t,z,s)=0,\nu \left(dz\right)-\mathrm{a}.\mathrm{e}.$$

Consequently, we deduce the expression of ${\Phi }_t$ from Equation (14)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& {\int }_{\mathbb{R}}\left({\displaystyle \frac{{\mu }_{X_{t-}}-r_{X_{t-}}}{a{\sigma }_{X_{t-}}}}\right)z\nu \left(dz\right)-{\displaystyle \frac{{\mu }_{X_{t-}}-r_{X_{t-}}}{{\sigma }_{X_{t-}}}}\hfill \\ & =& \left({\displaystyle \frac{{\mu }_{X_{t-}}-r_{X_{t-}}}{a{\sigma }_{X_{t-}}}}\right)\left({\int }_{\mathbb{R}}{z}^2\nu \left(dz\right)-1-{\int }_{\mathbb{R}}{z}^2\nu \left(dz\right)\right)\hfill \\ & =& {\displaystyle \frac{r_{X_{t-}}-{\mu }_{X_{t-}}}{a{\sigma }_{X_{t-}}}}.\hfill \end{array}$$

which gives the required expression in (30). □

Remark 2.

The minimal martingale measure coincides with the equivalent martingale measure $\mathbb{Q}$ constructed via the change of measure in Equation (10).

We introduce the notations: ${\beta }_{t,s}:={\int }_{\mathbb{R}}h(t,z,s)z\nu \left(dz\right)$ then for the associated risk process, we obtain making use of Föllmer–Schweizer decomposition (28), and the expressions (20), (21), (32) and (36) for all $0\le t<t^{\prime }<s<T$, we have

$$\begin{array}{ccc}\hfill R^{\varphi }(t,t^{\prime },s)& =& {\mathbb{E}}_{\mathbb{P}}\left[{\left|C^{\varphi }(t^{\prime },s)-C^{\varphi }(t,s)\right|}^2|{\mathcal{F}}_t\right]\hfill \\ & =& {\mathbb{E}}_{\mathbb{P}}\left[{\left(V_{t^{\prime },s}\left(\varphi \right)-{\int }_0^{t^{\prime }}\phi (u,s)d{\tilde{S}}_u-V_{t,s}\left(\varphi \right)+{\int }_0^t\phi (u,s)d{\tilde{S}}_u\right)}^2|{\mathcal{F}}_t\right]\hfill \\ & =& {\mathbb{E}}_{\mathbb{P}}\left[{(\varrho (t^{\prime },s)-\varrho (t,s))}^2|{\mathcal{F}}_t\right]\hfill \\ & =& {\mathbb{E}}_{\mathbb{P}}\left[{\left({\int }_t^{t^{\prime }}{\psi }^1(u,s)d{W}_u+{\int }_t^{t^{\prime }}{\int }_{\mathbb{R}}{\psi }^2(u,z,s)\tilde{\aleph }(du,dz)\right)}^2|{\mathcal{F}}_t\right]\hfill \\ & =& {\int }_t^{t^{\prime }}{\mathbb{E}}_{\mathbb{P}}\left[{\left(F_x(u,S_{u-},s)-\phi (u,s)\right)}^2{\sigma }_{X_{u-}}^2{\tilde{S}}_{u-}^2|{\mathcal{F}}_t\right]du\hfill \\ & & +{\int }_t^{t^{\prime }}{\mathbb{E}}_{\mathbb{P}}\left[{\int }_{\mathbb{R}}{\left(h(u,z,s)-{\sigma }_{X_{u-}}{\tilde{S}}_{u-}\phi (u,s)z\right)}^2\nu \left(dz\right)|{\mathcal{F}}_t\right]du\hfill \end{array}$$

Now, substituting expressions of $\phi$, ${\psi }^{\left(1\right)}$ and ${\psi }^{\left(2\right)}$ given by (32)–(34), we obtain

$$\begin{array}{ccc}\hfill R^{\varphi }(t,t^{\prime },s)& =& {\int }_t^{t^{\prime }}{\mathbb{E}}_{\mathbb{P}}\left[{\left(F_x(u,S_{u-},s)-\frac{1}{a}{F}_x(u,S_{u-},s)-{\displaystyle \frac{{\beta }_{u,s}}{a{\sigma }_{X_{u-}}{\tilde{S}}_{u-}}}\right)}^2{\sigma }_{X_{u-}}^2{\tilde{S}}_{u-}^2|{\mathcal{F}}_t\right]du\hfill \\ & & +{\int }_t^{t^{\prime }}{\mathbb{E}}_{\mathbb{P}}\left[{\int }_{\mathbb{R}}{\left(h(u,z,s)-\frac{z}{a}\left(F_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}-{\beta }_{u,s}\right)\right)}^2\nu \left(dz\right)|{\mathcal{F}}_t\right]du\hfill \\ & =& {\int }_t^{t^{\prime }}{\mathbb{E}}_{\mathbb{P}}\left[{\left(F_x(u,S_{u-},s){\displaystyle \frac{a-1}{a}}-{\displaystyle \frac{{\beta }_{u,s}}{a{\sigma }_{X_{u-}}{\tilde{S}}_{u-}}}\right)}^2{\sigma }_{X_{u-}}^2{\tilde{S}}_{u-}^2|{\mathcal{F}}_t\right]du\hfill \\ & & +{\int }_t^{t^{\prime }}{\mathbb{E}}_{\mathbb{P}}\left[{\int }_{\mathbb{R}}\right(h^2(u,z,s)+\frac{z^2}{a^2}{\left(F_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}-{\beta }_{u,s}\right)}^2\hfill \\ & & -2\frac{z}{a}h(u,z,s)\left(F_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}-{\beta }_{u,s}\right)\left)\nu \left(dz\right)\right|{\mathcal{F}}_t]du.\hfill \end{array}$$

In the particular case where $0\le t<t^{\prime }=s=T$, the associated risk process reads $R^{\varphi }(t,T,T)=R^{\varphi }(t,T)$

$$\begin{array}{ccc}\hfill R^{\varphi }(t,T)& =& {\int }_t^T{\mathbb{E}}_{\mathbb{P}}\left[{\left(F_x(u,S_{u-},T){\displaystyle \frac{a-1}{a}}-{\displaystyle \frac{{\beta }_{u,T}}{a{\sigma }_{X_{u-}}{\tilde{S}}_{u-}}}\right)}^2{\sigma }_{X_{u-}}^2{\tilde{S}}_{u-}^2|{\mathcal{F}}_t\right]du\hfill \\ & & +{\int }_t^T{\mathbb{E}}_{\mathbb{P}}\left[{\int }_{\mathbb{R}}\right(h^2(u,z,T)+\frac{z^2}{a^2}{\left(F_x(u,S_{u-},T){\sigma }_{X_{u-}}{\tilde{S}}_{u-}-{\beta }_{u,T}\right)}^2\hfill \\ & & -2\frac{z}{a}h(u,z,T)\left(F_x(u,S_{u-},T){\sigma }_{X_{u-}}{\tilde{S}}_{u-}-{\beta }_{u,T}\right)\left)\nu \left(dz\right)\right|{\mathcal{F}}_t]du.\hfill \end{array}$$

Notice that all processes considered in [23] are formulated under the equivalent measure $\mathbb{Q}$. However, in our analysis, we obtained this above decomposition under the original probability measure $\mathbb{P}$.

4. Local Risk-Minimizing Strategy for Stream of Liability Payment Process

In the context of life insurance, the portfolio typically comprises n identical policyholders over a term of T years. Here, J represents the number of deaths within the portfolio and ${\mu }_{·}$ denotes the stochastic mortality intensity of the individual policyholders. Following the approach outlined in [2,3], we analyze the payment process defined for $0\le t\le T$ as

$$\begin{array}{ccc}\hfill \mathrm{P}(t,S_t)& =& {\int }_0^t(n-J_u)\hat{A}(u,S_{u-})du+{\int }_0^t\hat{D}(u,S_{u-})d{J}_u\hfill \\ & & +(n-J_T)\hat{S}(T,S_T)11_{\{t=T\}},\hfill \end{array}$$

(40)

with the step process J exhibiting the specified characteristics:

$$\gamma \left(t\right)=(n-J(t-\left)\right)\lambda \left(t\right),\mathrm{and}\mathfrak{Q}(t,\{1\left\}\right)=1,\mathrm{for}0\le t\le T$$

Additionally, we assumed the following conditions:

$\left({\mathbf{B}}_3\right)$

The number of deaths in the portfolio, mortality intensity, and financial market are independent (i.e., J, $\Theta$, and L are independent).

$\left({\mathbf{B}}_4\right)$

The functions $\hat{A}$, $\hat{D}$, and $\hat{S}$: $[0,T]\times (0,\infty )⟶\mathbb{R}$ are measurable.

$\left({\mathbf{B}}_5\right)$

The future lifetime as described in Equation (16) of individuals and the number of deaths in the portfolio are independent of the financial market dynamics.

By considering (40) and (41), we incorporated the systematic mortality risk introduced by ${\Theta }_{·}$. This approach enables the derivation of explicit hedging strategies for mortality-dependent products, such as guaranteed annuity options, by linking the value of benefits to the process ${\Theta }_{·}$. For further details on quadratic hedging of guaranteed annuity options under longevity risk, interested readers can refer to [31]. Adopting the market-consistent valuation and complying with the conditions (i)–(iv) and the result (24), the value of insurance liabilities at time t is expressed as:

$$\begin{array}{ccc}\hfill V(t,s)& =& {\mathbb{E}}_{\mathbb{Q}}\left[{\int }_t^s{e}^{-r(u-t)}d\mathrm{P}(u,S_u)|{\mathcal{F}}_t\right]\hfill \\ & =& {\mathbb{E}}_{\mathbb{Q}}\left[{\int }_t^s{e}^{-r(u-t)}(n-J_u)\hat{A}(u,S_u)du+{\int }_t^s{e}^{-r(u-t)}\hat{D}(u,S_{u-})d{J}_u+e^{-r(t-s)}(n-J_s){\hat{S}}_s|{\mathcal{F}}_t\right]\hfill \\ & =& V(0,s)+{\int }_0^t{F}_x(u,S_{u-},s){\sigma }_{X_{u-}}{\tilde{S}}_{u-}d{W}_u^{\mathbb{Q}}+{\int }_0^t{\int }_{\mathbb{R}}h(u,z,s){\aleph }^{\mathbb{Q}}(du,dz).\hfill \end{array}$$

It is essential to note that market-consistent valuation is recommended by regulations such as Solvency II and several accounting standards. Referring to the outcomes presented in Equation (31) and considering the conditions $\left({\mathbf{A}}_{\mathbf{6}}\right)$ and $\left({\mathbf{A}}_{\mathbf{7}}\right)$, the Föllmer–Schweizer decomposition of the discounted payment process under the original measure takes the following form:

$$\begin{array}{ccc}\hfill V_{t,s}\left(\varphi \right)& =& V_{0,s}\left(\varphi \right)+{\int }_0^t\phi (u,s)d{\tilde{S}}_u+{\int }_0^t{\psi }^{\left(1\right)}(u,s)d{W}_u\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}{\psi }^{\left(2\right)}(u,z,s)\tilde{\aleph }(du,dz),\hfill \end{array}$$

(41)

In this expression, the optimal strategy invested in the risky asset $\phi$ and the processes ${\psi }^{\left(1\right)}$ and ${\psi }^{\left(2\right)}$ are defined by (32)–(34). The stochastic integral involving W and $\tilde{\aleph }$ represents the cash inflow (or outflow), ensuring that, for all $0\le t\le T$:

$$\begin{array}{ccc}\hfill V_{t,s}\left(\varphi \right)& =& {\mathbb{E}}_{\mathbb{Q}}\left[{\int }_t^s{e}^{-r(u-t)}d\mathrm{P}(u,S_u)|{\mathcal{F}}_t\right].\hfill \end{array}$$

If the expected value of the cash inflow (or outflow) is zero, then

$$\begin{array}{ccc}\hfill \mathbb{E}\left[{\int }_0^t{\psi }^{\left(1\right)}(u,s)d{W}_u+{\int }_0^t{\int }_{\mathbb{R}}{\psi }^{\left(2\right)}(u,z,s)\tilde{\aleph }(du,dz)\right]& =& 0.\hfill \end{array}$$

In summary, the hedging portfolio (41) is mean self-financing, indicating that the cost process is a $\mathbb{P}$-martingale.

The processes $(\kappa ,\Phi )$ represent the risk premiums required by investors for hedging, reflecting the unsystematic and systematic insurance risk, respectively. The pricing of a liability is associated with a corresponding hedging portfolio. The cost involved in constructing the hedging portfolio establishes the price of the liability, from which insurance premiums can be determined.

To implement the investment strategy (32), accurate estimation of the drift term, ${\mu }_{X_{t-}}$, is essential. Estimating the drift can be a complex task. However, the benefit of employing local quadratic risk-minimizing strategies is that, in this context, the drift term ${\mu }_{X_{t-}}$ of the stock price process does not influence the optimal strategy.

5. Local Risk-Minimizing Strategy for Unit-Linked Contracts

The insurer issues insurance contracts for each of the N individuals, specifying premiums and benefit payments contingent on the policyholders’ remaining lifetime and linked to the financial market’s performance. The number of deaths until time t, denoted by $N_t^d={\sum }_{i=1}^{N}11_{\{T_i\le t\}}$, represents a crucial factor in these contracts. In this section, we explore two types of insurance contracts: term insurance and pure endowment contracts.

5.1. The Term Insurance

We introduce a time-dependent contract function $g_t=g(t,S_t)$. In the case of term insurance, the entire sum insured becomes due immediately upon the policyholder’s death before time T. According to the contract’s definition, payments can occur at any time within the interval $[0,T]$, and obligations arising from such contracts do not naturally form T-claims without introducing specific assumptions. To transform these obligations into a T-claim, one approach is to assume that all payments are deferred and accumulated until the contract’s term with the risk-free interest rate r. Under this arrangement, the heirs of a policyholder who dies at time t would receive the benefit $S_{T-t}^{0}g(t,S_t)$. Alternatively, deferred payments could be accumulated differently, such as following a predefined strategy by investing $g(t,S_t)$. Although these modifications might seem most appropriate for contracts with shorter time frames, like one year, we assumed that benefits are indeed deferred until the end of the insurance period.

Therefore, the present value of the insurance company’s liabilities, considering a portfolio of term insurance contracts with deferred and accumulated payments using the risk-free asset $S_t^0$, can be described as the discounted general T-claim.

$$H_T=\frac{1}{S_T^0}\sum _{i=1}^N\frac{S_T^0}{S_{T_i}^0}g(T_i,S_T)11_{\{T_i\le T\}}={\int }_0^T\frac{g(s,S_{s-})}{S_s^0}d{N}_s^d.$$

(42)

Denote by $d{\tilde{N}}_s^d$ the compensated measure of $d{N}_s^d$. The Föllmer–Schweizer decomposition of the T-claim $H_T$ is given by

$$\begin{array}{ccc}\hfill V_{t,T}\left(\varphi \right)& =& V_{0,T}\left(\varphi \right)+{\int }_0^t(N-N_{s-}^d)\overline{\phi }(s,T)d{\tilde{S}}_s+{\varrho }^{H_T}(t,T)\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \overline{\phi }(s,T)& =& {\int }_s^T{\phi (s,r)}_{u-s}{p}_{x+s}{\mu }_{x+r}dr,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\varrho }^{H_T}(t,T)& =& {\int }_0^t(N-N_{s-}^d)E^{\left(1\right)}(s,T)d{W}_s+{\int }_0^{t}e(s,T)d{\tilde{N}}_s^d\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}(N-N_{s-}^d)E^{\left(2\right)}(s,z,T)\tilde{\aleph }(ds,dz).\hfill \end{array}$$

Here, also,

$$\begin{array}{ccc}\hfill E^{\left(1\right)}(s,T)& =& {\int }_s^T{ }_{u-s}{p}_{x+s}{\mu }_{x+r}{\psi }^{\left(1\right)}(s,r)dr,\hfill \\ \hfill e(s,T)& =& \frac{1}{S_s^0}\left(\mathrm{P}(s,S_s)-{\int }_s^{T}F(s,S_{s-},r){ }_{r-s}{p}_{x+s}{\mu }_{x+r}dr\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill E^{\left(2\right)}(s,z,T)& =& {\int }_s^T{ }_{r-s}{p}_{x+s}{\mu }_{x+r}{\psi }^{\left(2\right)}(s,z,r)dr.\hfill \end{array}$$

For the definitions of ${}_{t}p_x$ and ${\mu }_x$, we refer to Section 2.2.

The optimal strategy portfolio for all t, $0\le t\le T$ is characterized by the following:

• ${\varphi }^{★}(t,T)=(N-N_{t-}^d)\overline{\phi }(t,T)$ is invested in a risky asset;
• $(N-N_{t-}^d){\int }_t^T\frac{1}{S_t^0}F(t,S_{t-},s){ }_{s-t}{p}_{x+t}{\mu }_{x+s}ds+{\int }_0^t\frac{1}{S_s^0}\mathrm{P}(s,S_s)d{N}_s^d-{\varphi }^{★}(t,T){\tilde{S}}_t$ is invested in a riskless asset.

The corresponding risk process can be expressed as follows:

$$\begin{array}{ccc}\hfill R^{{\varphi }^{★}}(t,T)& =& {\int }_t^T\left\{{(N-N_t^d)}^2\mathbb{E}\left[{\left(E^{\left(1\right)}(s,T)\right)}^2+{\int }_{\mathbb{R}}{\left(E^{\left(2\right)}(s,z,T)\right)}^2\nu \left(dz\right)|{\mathcal{F}}_t\right]\right\}ds\hfill \\ & & +(N-N_t^d){\int }_t^T\left\{\mathbb{E}\left[e^2(s,T)|{\mathcal{F}}_t\right]{ }_{s-t}{p}_{x+t}{\mu }_{x+s}\right\}ds.\hfill \end{array}$$

5.2. The Pure Endowment

In the case of pure endowment contracts, the insured sum is payable at the maturity time T if the policyholder is still alive. The payout, determined by a continuous function g outlined in the contract, takes the form $g\left(S_T\right)$, thereby relying on the price of the risky asset at time T. The total discounted contingent claim for N insured individuals (contracts) and the corresponding obligation of the insurance company is represented by the present value.

$$H=\frac{\mathrm{P}\left(S_T\right)}{S_T^0}\sum _{i=1}^{N}11_{\{T_i>T\}}=\frac{\mathrm{P}\left(S_T\right)}{S_T^0}\left(N-\sum _{i=1}^{N}11_{\{T_i\le T\}}\right)=\frac{\mathrm{P}\left(S_T\right)}{S_T^0}\left(N-N_T^d\right)$$

(43)

The non-discounted insurance claim $H{S}_T^0$ in the form of (43) depends on $S_T$ and $N_T^d$, where $N-N_T^d$ represents the number of survivors at the end of the insurance period. The Föllmer–Schweizer decomposition of the claim H is provided for $0\le t\le T$:

$$V_{t,T}\left(\varphi \right)=V_{0,T}\left(\varphi \right)+{\int }_0^t(N-N_{s-}^d){ }_{T-s}{p}_{x+s}\phi (s,T)d{\tilde{S}}_s+{\varrho }^H(t,T)$$

where

$$\begin{array}{ccc}\hfill {\varrho }^H(t,T)& =& {\int }_0^t{(N-N_{s-}^d)}_{T-s}{p}_{x+s}{\psi }^{\left(1\right)}(s,T)d{W}_s\hfill \\ & & +{\int }_0^t{\int }_{\mathbb{R}}(N-N_{s-}^d){ }_{T-s}{p}_{x+s}{\psi }^{\left(2\right)}(s,z,T)\tilde{\aleph }(ds,dz)\hfill \\ & & -{\int }_0^t\frac{1}{S_s^0}F(s,S_{s-},T){ }_{T-s}{p}_{x+s}d{\tilde{N}}_s^d.\hfill \end{array}$$

The optimal strategy is characterized, $\forall 0\le t\le T$, by:

• ${\phi }^{★}(t,T)=(N-N_{t-}^d){ }_{T-t}{p}_{x+t}\phi (t,T)$, in the risky assets;
• $(N-N_{t-}^d){ }_{T-t}{p}_{x+t}\frac{1}{S_t^0}F(t,S_{t-},T)-{\phi }^{★}(t,T){\tilde{S}}_t$, in the riskless assets.

The associated risk process reads:

$$\begin{array}{ccc}\hfill R^{{\varphi }^{★}}(t,T)& =& {\int }_t^T\{{ }_{T-s}{p}_{x+s}^2{\mathbb{E}}_{\mathbb{P}}\left[{(N-N_{s-}^d)}^2\right]\hfill \\ & & \times {\mathbb{E}}_{\mathbb{P}}\left[{\left({\psi }^{\left(1\right)}(s,T)\right)}^2+{\int }_{\mathbb{R}}{\left({\psi }^{\left(2\right)}(s,z,T)\right)}^2\nu \left(dz\right)|{\mathcal{F}}_t\right]\}ds\hfill \\ & & +(N-N_{t-}^d){ }_{T-t}{p}_{x+t}{\int }_t^T\left\{{ }_{T-s}{p}_{x+s}{\mu }_{x+s}\mathbb{E}\left[{\left(S_s^0\right)}^{-2}{F}^2(s,S_{s-},T)|{\mathcal{F}}_t\right]\right\}ds.\hfill \end{array}$$

6. Conclusions

In this work, we addressed the challenge of the actuarial valuation and hedging of life insurance liabilities in the presence of mortality risk, employing the local risk-minimizing hedging approach. Our framework assumes that the lifetimes of policyholders in an insurance portfolio are modeled by a point process with stochastic intensity determined by a diffusion process. The share price process is considered as a regime-switching Lévy process with a non-zero regime-switching drift, where the parameters are contingent on the economic state.

Utilizing the Föllmer–Schweizer decomposition, we derived the main hedging result for a contingent payment process under the original probability measure. In most cases, the equivalence between the Föllmer–Schweizer decomposition and the local risk-minimizing hedging strategy still holds. We explored specific scenarios, including the local risk-minimizing strategies for a stream of liability payments and a unit-linked contract.

We believe that our findings have practical implications in the framework of risk management and hedging for insurance companies. However, due to the length of the present paper practical examples and numerical applications will be incorporated in future works.