# Optimal Time-Consistent Investment Strategy for a Random Household Expenditure with Default Risk under Relative Performance

Authors: Wenjin Guan; Wei Yuan; Sheng Li
Journal: Complexity
 (2021)

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## Abstract

Considering the mind of rivalry between families, each family focuses not only on its own wealth but also on other families, especially neighbors. In this paper, we investigate the non-zero-sum mean-variance game between two families with a random household expenditure under the default risk and relative performance. Applying the stochastic control theory within the framework of the game theory, the extended Hamilton–Jacobi–Bellman equation equations are derived. By solving this equation, we obtain the Nash equilibrium strategies of the two families and the corresponding equilibrium value functions. We also provide a numerical example to analyze the effects of relevant parameters on Nash equilibrium strategy and on the utility loss due to the mind of rivalry.


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## Body

## 1. Introduction

With the strengthening of household residents’ financial awareness and the increasing abundance of financial products, families are increasingly focusing on the issue of optimal asset portfolio selection. Optimal investment portfolios have been widely studied in recent years. Hipp and Plum [1], David Promislow and Young [2], Milevsky et al. [3], Chen et al. [4], and Bayraktar and Zhang [5] consider the optimal investment for minimizing the ruin probability. Wang [6], Bo and Capponi [7], Delong [8], Yuan and Lai [9], and Li et al. [10] obtain the optimal investment strategy under maximizing the expected utility of terminal wealth.And then the mean-variance criteria are another objective of great interest to insurers. In contrast to maximizing expected utility and minimizing ruin probability, mean-variance criteria allow insurers to minimize risk at an acceptable return. This approach takes not only the risk but also the returns into account. Due to its rationality and practicality, mean-variance has become a popular decision criterion in financial theory. The study of the mean-variance can be traced back to Markowitz [11]. Since then, there has been a certain amount of research on the optimization problem and its application in finance and insurance. For example, Li and Ng [12] introduce an embedding technique to transform the mean-variance problem into a stochastic linear quadratic control problem in a discrete-time model, which was extended to the corresponding continuous-time model by Zhou and Li [13]. The application of the mean-variance criterion to optimal reinsurance or/and investment can be found in Bäuerle [14], Bai and Zhang [15], and Bi and Guo [16]; however, they are all in the optimal strategy based on initial information, which is time-inconsistent. It is well known that the dynamic mean-variance criterion lacks iterated-expectation property, which leads to time-inconsistent in the sense that the Bellman optimality principle does not hold anymore. To this end, Björk et al. [17] solve the time-inconsistent problem by reconstructing stochastic control theory in the framework of game theory and obtain a time-consistent equilibrium strategy which is not only optimal at the current time but also optimal in the future. In recent years, there has been an increased interest in finding a time-consistent equilibrium strategy for the mean-variance portfolio problem, such as Zeng and Li [18], Li et al. [19], and Zhang and Liang [20].Further, high-yield corporate bonds are becoming increasingly attractive to investors in today’s financial markets. As a result, the optimal portfolio problem with defaulted securities has become an important area of research. Bielecki and Jang [21] consider the portfolio problem of investing wealth in credit risk assets, bank accounts, and stocks to maximize exponential utility maximization. Li et al. [22] investigate the DC pension management problem with a default bond under the CEV model and obtain an optimal time-consistent investment strategy. Zhao et al. [23] and Zhang and Chen [24] derive the optimal time-consistent investment and reinsurance strategies under default risk.Similar to Yuan and Lai [9] and Li et al. [10], we study households with stochastic expenditure optimal investment strategies. Unlike them, we consider that households can invest in corporate bonds and study the optimal time-consistent strategy under the mean-variance. Further, relative performance concerns are prevalent, such as pension managers, insurers, and other firms, who are concerned not only with their absolute wealth but also with the gap between their wealth and that of their competitors. And considering that there is a mind of rivalry among families, and they not only focus on their own wealth but also on the wealth of other families, especially their neighbors; thus, we construct a non-zero-sum mean-variance game between two families under relative performance (see Espinosa and Touzi [25]). Applying the stochastic control theory in the game theory framework, we obtain the Nash equilibrium strategies of the two households and the corresponding equilibrium value functions, and in addition, we analyze the utility loss caused by the mind of rivalry.The remainder of this paper is organized as follows. Section2 formulates the mean-variance game between two families for a random household expenditure with default risk. In Section 3, by solving the extended HJB equation, we derive the time-consistent Nash equilibrium strategy and the corresponding equilibrium value functions. In Section 4, we provide a numerical example to simulate the influence of relevant parameters on the equilibrium strategy and give an economic explanation. Section 5 concludes this paper.


## 2. Problem Formulation

In this paper, we consider a complete probability spaceΩ,ℱ,ℙ equipped with a right continuous and ℙ-complete filtration ℱtt∈0,T, where ℱtt∈0,T represents the information of the market available up to time t. All stochastic processes introduced below are defined in the filtered probability space Ω,ℱ,ℱtt∈0,T,ℙ.Suppose that the financial market consists of three tradable assets: a risk-free asset, a risky asset, and a defaultable bond. The price of a risk-free asset satisfies the following process:(1)dRt=rRtdt,R0=R0,where r>0 is the risk-free interest rate. The price St of the stock is given by(2)dSt=St−μdt+σdW1t+d∑i=1NtYi,S0=S0,where μ>0 and σ>0 are the expected return rate and the instantaneous volatility, respectively. Ntt∈0,T, denoting the number of the price jumps that occur in the risky asset during time interval 0,t, is a homogeneous Poisson process with an intensity λ; Yl is the lth jump amplitude of the risky asset price; Yl,l=1,2,… are i.i.d random variables with distribution function Fy, finite first-order moment EYl=μY, and second-order moment EYl2=σY2. Besides, we suppose that ℙYl≥−1for alll≥1=1 to ensure that the risky asset price remains positive. And we assume that μ+λμY>r.Different from the price dynamics of the risk-free asset and risky asset which are defined under the real-world probability measureℙ directly, the price process of the defaultable bond is firstly given by the martingale probability measure ℚ and then turned into the price dynamics under the probability measure ℙ. Following Bielecki and Jang [21], the definition of the default process is given by Zt=1τ≤t, where τ is a nonnegative random variable representing the default time of the company issue the bond, 1 is an indicator which takes value one if the default occurs, and zero otherwise. That is, the default process Zt is nondecreasing right continuous and has a jump at the random time τ. Furthermore, according to Driessen [26] and Duffie and Singleton [27], Zt is supposed as the default process with a constant intensity kℙ under the probability measure ℙ. Then, the martingale default process Kℙt under probability measure ℙ is given by Kℙt:=Zt−∫0t1−Zu−kℙdu, where its differential form is dKℙt=dZt−kℙ1−Zt−dt.Then, the price process of the defaultable bond under the martingale probability measureℚ is considered. Similar to Bielecki and Jang [21], the defaultable bond is assumed as a zero-coupon bond with the one-unit face value and maturity T¯. Further, we assume that in the event that a default occurs, the insurer recovers a fraction of the market value of the defaultable bond just prior to default, and then, the postdefault value of the defaultable bond is zero. The loss rate is denoted by ζ; then, 1−ζ is the default recovery rate. In addition, we denote by kℚ the constant intensity of the default process under the martingale probability measure ℚ, then ϑ=kℚζ is the risk-neutral credit spread, and then, we derive the following price process of the defaultable bond under the probability measure ℚ:(3)pt,T¯=1τ>te−r+ϑT¯−t+1τ≤t1−ζe−t+ϑT¯−tert−τ,where pt,T¯ is a fictitious bond rather than a bond actually traded, and T<T¯. That is, as Bielecki and Jang [21] and Driessen [26] mentioned, the fictitious bond allows us to take into consideration the jump risk premium in the expected return of the defaultable bond. Applying Itô formula to the above expression, we derive(4)dpt,T¯=rpt,T¯dt−ζe−r+ϑT¯−tdKℚt,where Kℚt is a martingale default process under the martingale probability measure ℚ.Next, we need to transform the price process of defaultable bonds under the martingale probability measureℚ into the real-world probability measure ℙ. Based on the above notations, by Girsanov theorem (see Kusuoka [28]), there exists a predictable process Δ such that the evolutions of the price process for the defaultable bond under real-world probability measure ℙ as follows:(5)dpt,T¯=pt−,T¯rdt+1−Ztϑ1−Δdt−1−Zt−ζdKℙt,where Δ=kℚ/kℙ and Kℙt=Zt−kℚ∫0t1−Zu−Δdu is a ℱ-martingale under probability measure ℙ. Similar to Duffie and Singleton [27], the probability of default under the real-world probability measure ℙ is lower than that under the martingale probability measure ℚ, that is, 1/Δ>1. According to Yu [29], equation (5) consists of two components: the first component is the return of a default-free bond. The second is the difference between the risk-neutral credit spread and the real-world credit spread provided that the default has not occurred by time t.In addition, we assume that household expenditures include household consumption expenditures, transaction costs, and other expenditures. Thus, the family expenditure has stochastic property. We denote the family expenditure at the timet by wt which is characterized by an arithmetic Brownian motion:(6)dwt=αdt+βdW2t,w0=w0>0,where α>0 represents the average expenditure. βdW2t denotes expenditure subject to random factors, in which β>0 and W2t are positive constants and standard Brownian motion, respectively. And the correlation coefficient of W1t and W2t is denoted by ρ, i.e., EW1tW2t=ρt. To simplify the calculation later, we assume that the expenditures are the same between the two households. In practice, the expenditures of the two households may be different, but it does not bring essential changes to the solution of the optimization problem but only increases the variables and parameters in the optimal solution.LetXit denote the wealth at the time t∈0,T for family i, and pit and qit denote the amount at time t for family i invested in the risky asset and defaultable bond, respectively; then, the amount for family i invested in risk-free assets is Xit−pit−qit. Define πit=pit,qit as the strategy of family i at time t; thus, the dynamics of the wealth of family i under strategy πi satisfies the following stochastic differential equation (SDE):(7)dXiπit=rXiπit+pitμ−r+qit1−Ztϑ1−Δdt+pitσdW1t+∫−1+∞pityNdt,dy−qit1−Zt−ζdKℙ−wtdt,X0=X0.Definition 1.
(admissible strategy). For any fixedt∈0,T, a strategy πi=pit,qitt∈0,T is said to be admissible if(i)
πi is ℱ-progressively measurable(ii)
E∫0Tpiu2+qiu2du<∞(iii)
The SDE (7) has a unique strong solution
For any initial statet,xi,w,z∈0,T×ℝ×ℝ×0,1, the corresponding set of all admissible strategies for family i is denoted by Πi. When default has occurred, i.e., τ≤t, we assume that pt−,T¯=0 and fix p2t=0 afterward. And we let z denote the initial default state, where z=1 and z=0 correspond to the postdefault cases τ>t and the predefault case τ≤t, respectively.
Due to the mind of rivalry, each family is concerned not only with their own terminal wealth but also with the differences between them and other families, especially neighbors. Hence, we formulate this optimization problem as a non-zero-sum stochastic differential game among two households. Following Espinosa and Touzi [25], we define the relative performance X^iπit for family i as follows:(8)X^iπi,πjt=1−κiXiπit+κiXiπit−Xjπjt,i≠j∈1,2,where κi∈0,1 captures the intensity of the family i’s relative concern and measures his sensitivity to the performance of his competitors. Together with equations (7) and (8), we derive(9)dX^iπi,πjt=rX^iπi,πjt+μ−rpit−κipjt+1−Ztϑ1−Δqit−κiqjt−1−κiwtdt+pit−κipjtσdW1t+∫−1+∞pit−κipjtNdt,dy−qit−κiqjt1−ZtζdKℙt.
We assume that each family has mean-variance criteria and define the following value function for familyi:(10)Jiπi∗,πj∗t,x^i,w,z=supπi∈ΠiEt,x^i,w,zX^iπi,πj∗T−γi2Vart,x^i,w,zX^iπi,πj∗T,where Et,x^i,w,z⋅ and Et,x^i,w,z⋅ are the conditional expectation and variance given X^iπi,πjt=x^i, wt=w, and Zt=z for t,x^i,w,z∈0,T×ℝ×ℝ×0,1, and γi>0 is the risk-aversion coefficient of family i.Problem 1.
In the classical non-zero-sum stochastic differential game, we find a Nash equilibriumπi,πj∈Πi×Πj such that(11)Jiπi∗,πj∗t,x^i,w,z≥Jiπi,πj∗t,x^i,w,z,i≠j∈1,2.
Because there exists a nonlinear function of the expectation of terminal wealth in the objective functional (10), the optimization problem is time-inconsistent. In fact, time consistency cannot be ignored for a rational decision-maker who aims to seek an equilibrium strategy that is optimal at a time and still can be optimal as time goes forward into the future. Hence, we shall define the following time-consistent equilibrium strategy according to Björk et al. [17].Definition 2.
Consider an admissible strategyπi∗ for family i, which can be informally viewed as a candidate equilibrium strategy. And choose arbitrarily a fixed π^i∈Πi, a real number ε>0, and a given initial point t,x^i,w,z∈0,T×ℝ×ℝ×0,1; the strategy πiε for family i is defined as follows:(12)πiε=π^iu,x^i,w,z,u,x^i,w,z∈t,t+ε×ℝ×ℝ×0,1,πi∗u,x^i,w,z,u,x^i,w,z∈t+ε,T×ℝ×ℝ×0,1.
If(13)limε↓0infJiπi∗,πj∗t,x^i,w,z−Jiπiε,πj∗t,x^i,w,zε≥0,we call that πi∗ is an equilibrium strategy of family i and the equilibrium value function of family i is given by Jiπi∗,πj∗t,x^i,w,z.


## 3. Nash Equilibrium Strategies

In this section, we derive the Nash equilibrium strategy between two families. According to Definition2, the equilibrium strategy above is time-consistent. We aim to seek an equilibrium strategy π1∗,π2∗ and the corresponding to equilibrium value functions. To give the extended HJB system and verification theorem conveniently, we define a variational operator. Let C1,2,20,T×ℝ×ℝ×0,1 denote a space of any function φt,x^i,w,z which φt,x^i,w,z itself and its derivatives ∂φ/∂t, ∂φ/∂x^i, ∂2φ/∂x^i2, ∂φ/∂w, ∂2φ/∂w2, and ∂2φ/∂x^i∂w are continuous on 0,T×ℝ×ℝ×0,1. For any function φt,x^i,w,z∈C1,2,20,T×ℝ×ℝ×0,1, the variational operator is defined as follows:(14)ℒπi,πjφt,x^i,w,z=∂φ∂t+rx^i+μ−rpit−κipjt−1−κiw+1−zϑqit−κiqjt∂φ∂x^i+α∂φ∂w+12pit−κipjt2σ2∂2φ∂x^i2+12β2∂2φ∂w2+ρβσpit−κipjt∂2φ∂x^i∂w+λEφt,x^i+pit−κipjty,w,z−φt,x^i,w,z+kℙ1−zφt,x^i−qit−κiqjtζ,w,1−φt,x^i,w,0.Before finding the equilibrium strategy, the following theorem gives the verifications for the extended HJB equation corresponding to Problem1.Theorem 1.
(verification theorem). For Problem1, we assume that there exist two real-valued functions Vit,x^i,w,z, git,x^i,w,z∈C1,2,20,T×ℝ×ℝ×0,1 satisfying the following extended HJB equations:(15)supπi∈Πiℒπi,πj∗Vit,x^i,w,z−γi2ℒπi,πj∗gi2t,x^i,w,z+γigit,x^i,w,zℒπi,πj∗git,x^i,w,z=0,ℒπi∗,πj∗git,x^i,w,z=0,πi∗=argsupπi∈Πiℒπi,πj∗Vit,x^i,w,z−γi2ℒπi,πj∗gi2t,x^i,w,z+γigit,x^i,w,zℒπi,πj∗git,x^i,w,z,ViT,x^i,w,z=x^i,giT,x^i,w,z=x^i,then Et,x^i,w,zX^iπi∗,πj∗T=git,x^i,w,z, Jiπi∗,πj∗t,x^i,w,z=Vit,x^i,w,z, and πi∗ is an equilibrium strategy of family i.Proof.
The proof of the verification theorem can be adapted from Theorem 5.2 in [17], and thus the proof is placed in Appendix A.Theorem 2.
For Problem1, the optimal time-consistent Nash equilibrium strategies for family i≠j∈1,2 are given by(16)pi∗t=e−rT−t1−κiκjrμ−r+λμY+1−κiγiρβσerT−t−1γirσ2+λσY2+κirμ−r+λμY+1−κjγjρβσerT−t−1γjrσ2+λσY2,(17)qi∗t=e−rT−t1−κiκjϑγiζ2kℙ−1γiζ+cit−c¯itζ+κiϑγjζ2kℙ−1γjζ+cjt−c¯jtζ1τ>t,and the equilibrium value function of family i is(18)Vit,x^i,w,z=erT−tx^i+Bitw+Cit,z=1,erT−tx^i+B¯itw+C¯it,z=0.Moreover, the expectation and variance of the terminal value corresponding to the equilibrium strategy for familyi are as follows:(19)Et,x^i,w,z=erT−tx^i+bitw+cit,z=1,erT−tx^i+b¯itw+c¯it,z=0,Vart,x^i,w,z=2γiCit−cit,z=1,2γiC¯it−c¯it,z=0,where Bit, bit, cit, Cit, B¯it, b¯it, c¯it, C¯it, cjt, and c¯jt are given by (B.18)–(B.20), (B.40)–(B.42), (B.45), and (B.47), respectively.Proof.
See Appendix B.
Ifκ1=κ2=0, then the model degenerates to the case where the competition factor is not considered; i.e., there is no psychological widespread between the two families; we have the following corollary.Corollary 1.
Whenκ1=κ2=0, the optimal time-consistent Nash equilibrium strategies for family i≠j∈1,2 are as follows:(20)p˜i∗t=e−rT−trμ−r+λμY+γiρβσerT−t−1γirσ2+λσY2,(21)q˜i∗t=e−rT−tϑγiζ2kℙ−1γiζ+ci0t−c¯i0tζ1τ>t,and the equilibrium value function of family i is(22)V˜it,xi,w,z=erT−txi+Bi0tw+Ci0t,z=1,erT−txi+B¯i0tw+C¯i0t,z=0.
Moreover, the expectation and variance of the terminal value corresponding to the equilibrium strategy for familyi are as follows:(23)E˜t,xi,w,z=erT−txi+bi0tw+ci0t,z=0,erT−txi+b¯i0tw+c¯i0t,z=1,Var˜t,xi,w,z=2γiCi0t−ci0t,z=1,2γiC¯i0t−c¯i0t,z=0,where(24)Bi0t=bi0t=B¯i0t=b¯i0t=−1rerT−t−1,(25)ci0t=−α−μ−r+λμYρβσσ2+λσY21r2erT−t−1−α−μ−r+λμYρβσσ2+λσY21rt−T−μ−r+λμY2γiσ2+λσY2t−T,(26)Ci0t=α−μ−r+λμYρβσσ2+λσY2−1r2erT−t−1rt−T+1r2+γiβ221−ρ2σ2σ2+λσY2−12r3e2rT−t+2r3erT−t+1r2t−T+12r3−2r3−μ−r+λμY22γiσ2+λσY2t−T,(27)c¯i0t=eϑt/ζ∫tTe−ϑu/ζα−μ−r+λμYρβσσ2+λσY2b¯i0u+μ−r+λμY2γiσ2+λσY2+kℙγi−2ϑγiζ+ϑ2γiζ2kℙdu,(28)C¯i0t=ekℙt∫tTe−kℙuαB¯i0u−γiβ221−ρ2σ2σ2+λσY2b¯i02u−μ−r+λμYρβσσ2+λσY2b¯i0u+kℙCi0u−kℙ−ϑζci0u−c¯i0u+μ−r+λμY22γiσ2+λσY2+kℙ2γi−ϑγiζ+ϑ22γiζ2kℙdu.Remark 1.
(i)
From Theorem2, we can find that the amounts invested in risky assets pi∗ and p˜i∗ are influenced by the diffusion parameter β of random expenditures but are independent of the drift parameter α of random expenditures. This is the same as Yuan and Lai [9]. By equations (B.16) and (B.37), we have(29)c¯i′t−ci′t−ϑζc¯it−cit+kℙγi−2ϑγiζ+ϑ2γiζ2kℙ=0,c¯iT−ciT=0.Solving the above equation and noting thatϑ=kℚζ and 1/Δ=kℚ/kℙ, we derive(30)c¯it−cit=ζkℙ−ϑ21−e−ϑT−t/ζγiζϑkℙ=Δ−121−e−ϑT−t/ζγiΔ.Symmetrically, we have(31)c¯jt−cjt=ζkℙ−ϑ21−e−ϑT−t/ζγjζϑkℙ=Δ−121−e−ϑT−t/ζγjΔ.Noting equations (17) and (21), the amount invested in defaultable bonds is easily found to be independent of random expenditures.(ii)
From equations (18), (B.18), (B.40), (22), and (24), we have(32)∂Vit,x^i,w,z∂w=−1−κirerT−t−1<0,∂V˜it,xi,w,z∂w=−1rerT−t−1<0,that is, expenditure brings about loss of utility.(iii)
In fact, similar to the Cournot duopoly game in classical economics, there is utility loss in the psychology of mutual comparison between the two family homes we consider. Combining Theorem2 and Corollary 1, we define the utility loss as follows:(33)ULit≔Vit,x^i,w,z−V˜it,xi,w,z=erT−tκixj−κirerT−t−1w+α−μ−r+λμYρβσσ2+λσy2−κirerT−t−κirt−T+κir2+γiβ21−ρ2σ2σ2+λσY2κi2−2κi2r3e2rT−t+4κi−2κi2r3erT−t+2κi−κi2r2t−T+2κi−κi22r3+2κi2−4κir3,z=1,erT−tκixj−κirerT−t−1w+ekℙt∫tTe−kℙuαB¯i0u−B¯iu−γiβ221−ρ2σ2σ2+λσY2b¯i02u−b¯i2u−μ−r+λμYρβσσ2+λσY2b¯i0u−b¯iu+kℙCi0u−Ciu−kℙ−ϑζci0u−c¯i0u−ciu−c¯iudu,z=0.Notice that the utility lossULit≡0 when κi=0.


## 4. Sensitivity Analysis

In this section, we provide a numerical example to illustrate the optimal time-consistent investment strategy with a random household expenditure under the relative performance, in which we analyze the effect of important parameters on results and give an economic explanation. According to the model settings inSection 2, unless otherwise stated, we select the parameters in Table 1 throughout this section. Next, we vary the value of one parameter by fixing other parameters to analyze the effect of the optimal time-consistent strategy. Due to symmetry, the effects of the parameters have the same trend for family 1 and family 2; thus, we only analyze the effects of the parameters on the optimal results for family 1.Table 1
Model parameters.
rμσμYλκ1κ2γ1γ2ϑζkℙαβρT0.030.120.20.20.50.20.40.50.90.010.50.0050.10.6−0.53Figures1–4 depict the effects of parameters κ1, γ1, β, ρ, and α on family 1’s optimal time-consistent investment strategy for risky assets, respectively. From Figure 1, it can be observed that the larger the parameter κ1 that portrays the relative performance concern sensitivity, the larger the number of investments in risky assets by family 1. This is because the larger κ1 is, the greater family 1 is concerned about the comparison of the wealth of its neighbors, and if family 1 increases its exposure to risky assets, then its probability of creating more wealth than neighbors at the terminal moment increases. In Figure 2, the number of investments in risky assets decreases as the parameter γ1 increases. γ1 represents the degree of risk aversion, and a larger γ1 indicates that family 1 is more risk-averse, and therefore, family 1 will reduce the number of investments in risky assets. Figure 3 shows that the amounts invested in risky assets decrease with the β for ρ<0, and the amounts invested in risky assets increase with the β for ρ>0. By equation (16), we have ∂p1∗/∂β=e−rT−t1−κ1ρσerT−t−1/1−κ1κ2rσ2+λσY2. ∂p1∗/∂β>0 when ρ>0, which means that p1∗ increase with β, the family 1 is willing to invest more money in the risk asset to get more profits. In contrast, ∂p1∗/∂β<0 when ρ<0, that is to say, p1∗ decrease with β, and the family 1 will put less money in the risky asset. From Figure 4, we can see that the amount of investments in risky assets is not affected by the average expenditure α, which is consistent with equation (16).Figure 1
The effect ofκ1 on p1t.Figure 2
The effect ofγ1 on p1t.Figure 3
The effect ofβ on p10.Figure 4
The effect ofα on p10.Figures5–9 illustrate the effect of parameters κ1, γ1, ϑ, ζ, and kℙ on the amounts invested in the defaultable bonds. Figures 5 and 6 show that the amounts invested in the defaultable bonds increase with κ1 and decrease with γ1, in which the economic explanation is the same as Figures 1 and 2. Figures 7 and 8 show that the amounts invested in the defaultable bond q1t increase with the credit spread ϑ and decrease with the loss rate ζ because the lower credit spread ϑ or higher loss rate ζ can induce a larger potential loss. Figure 9 presents that the greater the default intensity kℙ, the less money will be invested in default bonds. This is because the greater the default intensity kℙ, the higher the default risk, and the insurer will naturally reduce the money of defaultable bonds.Figure 5
The effect ofκ1 on q1t.Figure 6
The effect ofγ1 on q1t.Figure 7
The effect ofϑ on q1t.Figure 8
The effect ofζ on q1t.Figure 9
The effect ofkℙ on q1t.Figure10 depicts the effect of the average expenditure α on the equilibrium value function for family 1 and it can be found that the equilibrium value function decreases monotonically with α, which can be interpreted as a utility loss due to the expenditure. Also, we can find that the predefault equilibrium value function is higher than the postdefault equilibrium value function, which can be interpreted as a loss of utility due to the default event. In order to portray the effect of the intensity of family 1’s relative concerns parameter κ1 on utility loss in Figure 11, we set x2=w=1. From Figure 11, it can be seen that the utility loss of family 1 increases with κ1; i.e., the higher the degree of mind of rivalry, the greater the utility loss. And it can be noted that when κ1=0, the utility loss is equal to 0, which is consistent with (iii) in Remark 1.Figure 10
The effect ofα on V10.Figure 11
The effect ofκ1 on UL10.


## 5. Conclusion

This paper investigates the stochastic differential game between two families for a random household expenditure with default risk under relative performance. There is a climbing mentality among families, especially neighbors, who will not only focus on their own wealth but also on that of their neighbors; thus, we assume that two families constitute a non-zero-sum game and that they face the same financial market consisting of a risk-free asset, a risky asset which is characterized by jump-diffusion model. And each family’s random expenditure is described by Brownian motion with drift. In the mean-variance criteria and Nash equilibrium framework, by applying stochastic control theory within the framework of game theory, we obtain an explicit expression for each family’s optimal time-consistent strategy and the corresponding equilibrium value function. We find that the heavier the climbing psychology, the higher the number of investments in risky assets and defaultable bonds. In addition, we define the utility loss from climbing and find that the utility loss increases with climbing psychology, similar to the static Cournot duopoly game in classical economics.



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*Source: 1274649-2021-10-08.xml*