Equilibrium for a Time-Inconsistent Stochastic Linear–Quadratic Control System with Jumps and Its Application to the Mean-Variance Problem

Abstract

This paper studies a kind of time-inconsistent linear–quadratic control problem in a more general framework with stochastic coefficients and random jumps. The time inconsistency comes from the dependence of the terminal cost on the current state as well as the presence of a quadratic term of the expected terminal state in the objective functional. Instead of finding a global optimal control, we look for a time-consistent locally optimal equilibrium solution within the class of open-loop controls. A general sufficient and necessary condition for equilibrium controls via a flow of forward–backward stochastic differential equations is derived. This paper further develops a new methodology to cope with the mathematical difficulties arising from the presence of stochastic coefficients and random jumps. As an application, we study a mean-variance portfolio selection problem in a jump-diffusion financial market; an explicit equilibrium investment strategy in a deterministic coefficients case is obtained and proved to be unique.

Appendix

In this appendix, we provide an essential estimate assisting the proof of Proposition 3.2. To ease the explosion of the results, we only consider the case for $n=1$, and the extension to the multidimensional case is straightforward. If not specified, we will denote by C some positive constants that may differ from line to line in the following estimates.

Lemma A.1

For each $t\in [0,T]$, let $(\varPsi (s))_{s\in [t,T]}$ be a progressively measurable process, such that, for any $k\ge 1$

$$\begin{aligned} {\mathbb {E}}_t\bigg [\sup \limits _{s\in [t,T]}\big |\varPsi (s)\big |^{k}\bigg ]\le C. \end{aligned}$$

(43)

Then, there exists a function $\rho : \varOmega \times ]0,\infty [\rightarrow ]0,\infty [$ with $\rho (\varepsilon )\downarrow 0$ as $\varepsilon \downarrow 0,\ a.s.$, such that

$$\begin{aligned} \int _t^T\big |{\mathbb {E}}_t[\varPsi (s)Y^\varepsilon (s)] \big |^2\mathrm{d}s\le \varepsilon \rho (\varepsilon ). \end{aligned}$$

(44)

Proof

Define an auxiliary process $(\varLambda (s))_{s\in [t,T]}$ with $\varLambda (t)=1$ by:

$$\begin{aligned} \varLambda (s)= & {} \exp \bigg \{\int _t^s\bigg [-A(r)+\frac{1}{2}\sum _{i=1}^dC_i^2(r)\\&\quad +\sum _{j=1}^m\int _{{\mathbb {R}}_0}\Big (E_j(r,e)+\ln \frac{1}{1+E_j(r,e)}\Big )\nu _j(de)\bigg ]\mathrm{d}r\\&\quad -\sum _{i=1}^d\int _t^sC_i(r)\mathrm{d}W_i(r) +\sum _{j=1}^m\int _t^s\int _{{\mathbb {R}}_0}\ln \frac{1}{1+E_j(r,e)} {\widetilde{N}}_j(dr,de)\bigg \}. \end{aligned}$$

Since $A, C_i, E_j$ are uniformly bounded, for any $k\ge 1$, there exists a positive constant C, such that

$$\begin{aligned} {\mathbb {E}}_t\bigg [\sup \limits _{s\in [t,T]}\big (\big |\varLambda (s) \big |^{k}+\big |\varGamma (s)\big |^{k}\big )\bigg ]\le C, \end{aligned}$$

(45)

where $\varGamma (s)=\varLambda (s)^{-1}$. Furthermore, in view of (43), we can easily obtain

$$\begin{aligned} {\mathbb {E}}_t\bigg [\sup \limits _{s\in [t,T]}\big |\varPsi (s)\varGamma (s)\big |^{k}\bigg ]\le C. \end{aligned}$$

(46)

Following the martingale representation theorem (see, e.g., [20, Lemma 2.3]), for every $s\in [t,T]$, there exists a unique pair $(\xi (\cdot ;s),\beta (\cdot ,\cdot ;s))\in L^2_{{\mathcal {F}},p}(t,s;{\mathbb {R}}^d) \times F^2_{p}(t,s;{\mathbb {R}}^m)$ such that

$$\begin{aligned} \varPsi (s)\varGamma (s)&={\mathbb {E}}_t[\varPsi (s)\varGamma (s)] +\sum _{i=1}^d\int _t^s\xi _i(r;s)\mathrm{d}W_i(r)\nonumber \\&\quad +\sum _{j=1}^m\int _t^s\int _{{\mathbb {R}}_0}\beta _j(r,e;s){\widetilde{N}}_j(dr,de), \end{aligned}$$

(47)

where $\xi (\cdot ;s)=(\xi _1(\cdot ;s),\ldots ,\xi _d(\cdot ;s))$ and $\beta (\cdot ,\cdot ;s)=(\beta _1(\cdot ,\cdot ;s),\ldots ,\beta _m(\cdot ,\cdot ;s))$.

Following from (46), the Burkholder–Davis–Gundy inequality, and Doob’s maximal inequality, we get for $k>1$

$$\begin{aligned}&{\mathbb {E}}\bigg [\bigg (\sum _{i=1}^d\int _t^s|\xi _i(r;s)|^2\mathrm{d}r +\sum _{j=1}^m\int _t^s\int _{{\mathbb {R}}_0}|\beta _j(r,e;s) |^2\nu _j(de)\mathrm{d}r\bigg )^{\frac{k}{2}}\bigg ]\nonumber \\&\quad \le C{\mathbb {E}}\bigg [\sup _{u\in [t,s]}\bigg |\sum _{i=1}^d\int _t^u\xi _i(r;s)\mathrm{d}W_i(r) +\sum _{j=1}^m\int _t^u\int _{{\mathbb {R}}_0}\beta _j(r,e;s) {\widetilde{N}}_j(dr,de)\bigg |^k\bigg ] \nonumber \\&\quad \le C\bigg (\frac{k}{k-1}\bigg )^k{\mathbb {E}} \bigg [\bigg |\sum _{i=1}^d\int _t^s\xi _i(r;s)\mathrm{d}W_i(r) +\sum _{j=1}^m\int _t^s\int _{{\mathbb {R}}_0}\beta _j(r,e;s) {\widetilde{N}}_j(dr,de)\bigg |^k\bigg ]\nonumber \\&\quad \le C{\mathbb {E}}\Big [\big |\varPsi (s) \varGamma (s)-{\mathbb {E}}_t\big [\varPsi (s)\varGamma (s)\big ]\big |^k\Big ] \le C\Big ({\mathbb {E}}\Big [\big |\varPsi (s)\varGamma (s)\big |^k\Big ] +{\mathbb {E}}\Big [\big |{\mathbb {E}}_t\big [\varPsi (s) \varGamma (s)\big ]\big |^k\Big ]\Big )\nonumber \\&\quad \le C{\mathbb {E}}\bigg [\big |\varPsi (s)\varGamma (s)\big |^k\bigg ] \le C{\mathbb {E}}\bigg [\sup \limits _{s\in [t,T]} \big |\varPsi (s)\varGamma (s)\big |^{k}\bigg ]\le C. \end{aligned}$$

(48)

For $k=1$, by using Hölder’s inequality, we can easily get

$$\begin{aligned}&{\mathbb {E}}\bigg [\bigg (\sum _{i=1}^d\int _t^s|\xi _i(r;s)|^2\mathrm{d}r +\sum _{j=1}^m\int _t^s\int _{{\mathbb {R}}_0}|\beta _j(r,e;s)| ^2\nu _j(de)\mathrm{d}r\bigg )^{\frac{1}{2}}\bigg ]\le C. \end{aligned}$$

(49)

Combining (48) and (49), we have for any $k\ge 1$

$$\begin{aligned}&\sup _{s\in [t,T]} {\mathbb {E}}\bigg [\bigg (\sum _{i=1}^d\int _t^s|\xi _i(r;s)|^2\mathrm{d}r +\sum _{j=1}^m\int _t^s\int _{{\mathbb {R}}_0}|\beta _j(r,e;s)|^2 \nu _j(de)\mathrm{d}r\bigg )^{\frac{k}{2}}\bigg ]\le C. \end{aligned}$$

(50)

Now, using Itô’s formula to $s\mapsto \varLambda (s)Y^\varepsilon (s)$ yields

$$\begin{aligned} Y^\varepsilon (s)= & {} \varGamma (s)\int _t^s\varLambda (r) \bigg \{\Big [B(r)-\sum _{i=1}^dC_i(r)D_i(r) \\&\quad -\sum _{j=1}^m\int _{{\mathbb {R}}_0}\frac{E_j(r,e)F_j(r,e)}{1+E_j(r,e)}\nu _j(de)\Big ]v\mathbf{1}_{[t,t+\varepsilon [}(r)\bigg \}\mathrm{d}r\\&\quad +\sum _{i=1}^d\varGamma (s)\int _t^s\varLambda (r)D_i(r)v\mathbf{1} _{[t,t+\varepsilon [}(r)\mathrm{d}W_i(r)\\&\quad +\sum _{j=1}^m\varGamma (s)\int _t^s \int _{{\mathbb {R}}_0}\varLambda (r-)\frac{F_j(r,e)}{1+E_j(r,e)}v\mathbf{1}_{[t,t+\varepsilon [}(r){\widetilde{N}}_j(dr,de). \end{aligned}$$

Consider

$$\begin{aligned} \varPsi (s)Y^\varepsilon (s):=L_1(s)+L_2(s)+L_3(s), \quad s\in [t,T], \end{aligned}$$

with

$$\begin{aligned} L_1(s)= & {} \varPsi (s)\varGamma (s)\int _t^s\varLambda (r)\bigg \{\Big [B(r)-\sum _{i=1}^dC_i(r)D_i(r)\\&\quad -\sum _{j=1}^m\int _{{\mathbb {R}}_0}\frac{E_j(r,e) F_j(r,e)}{1+E_j(r,e)}\nu _j(de)\Big ]v\mathbf{1}_{[t,t+\varepsilon [}(r)\bigg \}\mathrm{d}r,\\ L_2(s)= & {} \sum _{i=1}^d\varPsi (s)\varGamma (s) \int _t^s\varLambda (r)D_i(r)v\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}W_i(r),\\ L_3(s)= & {} \sum _{j=1}^m\varPsi (s)\varGamma (s) \int _t^s\int _{{\mathbb {R}}_0}\varLambda (r-)\frac{F_j(r,e)}{1+E_j(r,e)}v\mathbf{1}_{[t,t+\varepsilon [}(r){\widetilde{N}}_j(dr,de). \end{aligned}$$

Actually, by virtue of (45) and (46), the following estimate for ${\mathbb {E}}_t[L_1(s)]$ holds:

$$\begin{aligned} \int _t^T\big |{\mathbb {E}}_t[L_1(s)]\big |^2\mathrm{d}s\le C|v|^2\varepsilon ^2, \quad s\in [t,T]. \end{aligned}$$

(51)

Following the expression of $\varPsi (s)\varGamma (s)$ in (47), we have

$$\begin{aligned} {\mathbb {E}}_t[L_2(s)]&=\sum _{i=1}^d{\mathbb {E}} _t\bigg [\Big (\varPsi (s)\varGamma (s)\Big )\cdot \int _t^s \varLambda (r)D_i(r)v\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}W_i(r)\bigg ]\\&=\sum _{i=1}^d{\mathbb {E}}_t\bigg [\int _t^s\xi _i(r;s)\varLambda (r)D_i(r)v\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\bigg ]\\&\le C\varepsilon ^{\frac{1}{2}}|v| \sum _{i=1}^d{\mathbb {E}}_t\bigg [\sup _{r\in [t,T]}\varLambda (r)\cdot \bigg (\int _t^s\big |\xi _i(r;s)\big |^2\mathbf{1} _{[t,t+\varepsilon [}(r)\mathrm{d}r\bigg )^{\frac{1}{2}}\bigg ]\\&\le C\varepsilon ^{\frac{1}{2}}|v|\sum _{i=1} ^d\bigg ({\mathbb {E}}_t\bigg [\int _t^s\big |\xi _i(r;s)\big |^2\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\bigg ]\bigg )^{\frac{1}{2}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _t^T\big |{\mathbb {E}}_t[L_2(s)]\big |^2\mathrm{d}s&\le C\varepsilon |v|^2\sum _{i=1}^d{\mathbb {E}} _t\bigg [\int _t^T\int _t^s\big |\xi _i(r;s)\big |^2\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\mathrm{d}s\bigg ]. \end{aligned}$$

Setting $\rho _1(\varepsilon ):=C|v|^2\sum \limits _{i=1} ^d{\mathbb {E}}_t\bigg [\int _t^T\int _t^s\big |\xi _i(r;s)\big |^2\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\mathrm{d}s\bigg ]$ and then from (50), we have

$$\begin{aligned} {\mathbb {E}}\bigg [\int _t^T\int _t^s\big |\xi _i(r;s)\big |^2\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\mathrm{d}s\bigg ]&\le C\sup _{s\in [t,T]}{\mathbb {E}} \bigg [\int _t^s\big |\xi _i(r;s)\big |^2\mathrm{d}r\bigg ]<\infty . \end{aligned}$$

Thus, following the dominated convergence theorem for conditional expectations and observing the fact that $\mathbf{1}_{[t,t+\varepsilon [}\rightarrow 0$, we can easily obtain that $\rho _1(\varepsilon )\rightarrow 0$ as $\varepsilon \downarrow 0,\ a.s.$ Hence,

$$\begin{aligned} \int _t^T\big |{\mathbb {E}}_t[L_2(s)]\big |^2\mathrm{d}s&\le \varepsilon \rho _1(\varepsilon ). \end{aligned}$$

(52)

Similarly, we get

$$\begin{aligned} {\mathbb {E}}_t[L_3(s)]&=\sum _{j=1}^m{\mathbb {E}}_t \bigg [\int _t^s\int _{{\mathbb {R}}_0}\beta _j(r,e;s)\varLambda (r) \frac{F_j(r,e)}{1+E_j(r,e)}v\mathbf{1}_{[t,t+\varepsilon [}(r)\nu _j(de)\mathrm{d}r\bigg ]\\&\le C\varepsilon ^{\frac{1}{2}}|v|\sum _{j=1} ^m{\mathbb {E}}_t\bigg [\sup _{r\in [t,T]}\varLambda (r)\cdot \bigg (\int _t^s\int _{{\mathbb {R}}_0}\big |\beta _j(r,e;s) \big |^2\nu _j(de)\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\bigg )^{\frac{1}{2}}\bigg ]\\&\le C\varepsilon ^{\frac{1}{2}}|v|\sum _{j=1}^m \bigg ({\mathbb {E}}_t\bigg [\int _t^s\int _{{\mathbb {R}}_0} \big |\beta _j(r,e;s)\big |^2\nu _j(de)\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\bigg ]\bigg )^{\frac{1}{2}}. \end{aligned}$$

Hence,

$$\begin{aligned} \int _t^T\big |{\mathbb {E}}_t[L_3(s)]\big |^2\mathrm{d}s&\le C\varepsilon |v|^2\sum _{j=1}^m{\mathbb {E}}_t \bigg [\int _t^T\int _t^s\int _{{\mathbb {R}}_0}\big |\beta _j(r,e;s)\big |^2\nu _j(de)\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\mathrm{d}s\bigg ]. \end{aligned}$$

Setting $\rho _2(\varepsilon ):=C\varepsilon |v|^2\sum \limits _{j=1}^m{\mathbb {E}}_t\bigg [\int _t^T\int _t^s \int _{{\mathbb {R}}_0}\big |\beta _j(r,e;s)\big |^2\nu _j(de)\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\mathrm{d}s\bigg ]$ and then from (50) again, we have

$$\begin{aligned}&{\mathbb {E}}\bigg [\int _t^T\int _t^s\int _{{\mathbb {R}}_0} \big |\beta _j(r,e;s)\big |^2\nu _j(de)\mathbf{1}_{[t,t+\varepsilon [}(r)\mathrm{d}r\mathrm{d}s\bigg ]\\&\quad \le C\sup _{s\in [t,T]}{\mathbb {E}} \bigg [\int _t^s\int _{{\mathbb {R}}_0}\big |\beta _j(r,e;s)\big |^2\nu _j(de)\mathrm{d}r\bigg ]<\infty . \end{aligned}$$

Thus, using the dominated convergence theorem once again and observing the fact that $\mathbf{1}_{[t,t+\varepsilon [}\rightarrow 0$, it holds that $\rho _2(\varepsilon )\rightarrow 0$ as $\varepsilon \downarrow 0,\ a.s.$ Hence,

$$\begin{aligned} \int _t^T\big |{\mathbb {E}}_t[L_3(s)]\big |^2\mathrm{d}s&\le \varepsilon \rho _2(\varepsilon ). \end{aligned}$$

(53)

In view of (51)–(53), we have

$$\begin{aligned} \int _t^T\big |{\mathbb {E}}_t[\varPsi (s)Y^\varepsilon (s)]\big |^2\mathrm{d}s&\le C\big (\varepsilon ^2|v|^2+\varepsilon \rho _1 (\varepsilon )+\varepsilon \rho _2(\varepsilon )\big ). \end{aligned}$$

(54)

Now setting $\rho (\varepsilon ):=C(\varepsilon |v|^2+\rho _1(\varepsilon )+\rho _2(\varepsilon ))$ in (54), we obtain estimate (44). $\square$