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Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market


We apply dynamic programming principle to discuss two optimal investment problems by using zero-sum and nonzero-sum stochastic game approaches in a continuous-time Markov regime-switching environment within the frame work of behavioral finance. We represent different states of an economy and, consequently, investors’ floating levels of psychological reactions by a D-state Markov chain. The first application is a zero-sum game between an investor and the market, and the second one formulates a nonzero-sum stochastic differential portfolio game as the sensitivity of two investors’ terminal gains. We derive regime-switching Hamilton–Jacobi–Bellman–Isaacs equations and obtain explicit optimal portfolio strategies with Feynman–Kac representations of value functions. We illustrate our results in a two-state special case and observe the impact of regime switches by comparative results.

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Correspondence to E. Savku.

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Authors’ contributions

This work extends a part of Emel Savku’s PhD thesis and the second author is the advisor of the thesis. All authors contributed to the study conception, design and the second author motivated about neuroscience. The first draft of the manuscript was written by Emel Savku as the principal author and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript

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Let us show that the nonzero-sum game application illustrated in Sect. 4.1 satisfies the conditions of the verification theorem, Theorem 2.

Please remember that \(\mu _{m}(t):=\mu _{m}(t,\alpha (t))=\left\langle \mu _{m},\alpha (t)\right\rangle =(\mu _{m}^{1},\mu _{m}^{2},\ldots ,\mu _{m}^{D})^{T}\in \mathbb {R}^{D}\), \(\sigma _{m}(t):=\sigma _{m}(t,\alpha (t))=\left\langle \sigma _{m},\alpha (t)\right\rangle =(\sigma _{m}^{1},\sigma _{m}^{2},\ldots ,\sigma _{m}^{D})^{T}\in \mathbb {R}^{D}\), and \(\eta _{m}(t,z):= \eta _{m}(t,\alpha (t),z)=\left\langle \eta _{m},\alpha (t)\right\rangle =(z\eta _{m}^{1},z\eta _{m}^{2},\ldots ,z\eta _{m}^{D})^{T}\in \mathbb {R}^{D}\) denote the appreciation rate, the volatility rate, and the jump size of the mth risky asset for \(m=1,2\), \(t\in [0,T]\).

Hence, we are working on a family of geometric Lévy processes, where we assume

$$\begin{aligned} \nu (\left\{ 0\right\} )=0 \qquad \hbox {and} \qquad \int _{\mathbb {R}}(1\wedge |z|^{2})\nu (dz)<\infty . \end{aligned}$$

Infinitely many jumps can occur around the origin (i.e., small jumps); however, the mass away from the origin is bounded; i.e., only a finite number of big jumps can occur [for more details see (Papantaleon 2000)]. Then, \(\pi _{1}^{*}\) and \(\pi _{2}^{*}\), given by Eqs. (15) and (16) correspondingly, are constant functions at each state, \(e_{i}, \ i=1,2,\ldots ,D\). Therefore, \(h(t,\alpha (t))\), \(t\in [0,T]\) is a piecewise continuous function, then \(h(t,\alpha (t))\), \(t\in [0,T]\) is uniformly bounded. Consequently,

$$\begin{aligned} k_{m}(t,e_{i})=\gamma _{m}E\biggl [\exp \biggl (\int _{t}^{T}h(s,\alpha (s))ds\biggr )|\alpha (t)=e_{i}\biggr ], \qquad \hbox {for} \quad m=1,2,\ e_{i}\in S, \end{aligned}$$

are well defined. By dominated convergence theorem, \(k_{m}(t,e_{i})\) is continuously differentiable with respect to t for each \(e_{i}\in S\) and \(m=1,2\).

Then, we can easily see that the value functions

$$\begin{aligned} V_{m}(t,x_{1},x_{1},e_{i})=k_{m}(t,e_{i})x_{1}x_{2}, \quad m=1,2, \end{aligned}$$

have continuous partial derivatives with respect to \(x_{1}\) and \(x_{2}\) in any order. Therefore, the value functions are in \(C^{1,2}(G)\cap C(\bar{G})\) for each \(e_{i}\in S\) and for each investor as required to satisfy Theorem 2.

The conditions (i)–(ii) are direct consequences of first order conditions, which have been applied to obtain \(k_{m}(t,\alpha (t))\), \(m=1,2, \ t\in [0,T]\).

(iii) By above explanations, \(\left| k_{m}(t,\alpha (t))\right| \le \gamma _{m}\exp ((T-t)L)\),   \(L>0\). Then, by dominated convergence theorem,

$$\begin{aligned}&\lim _{t\rightarrow T^{-}}\phi _{m}(t,y^{\pi _{1},\pi _{2}},\alpha (t))=\lim _{t\rightarrow T^{-}}k_{m}(t,\alpha (t))x_{1}x_{2} \\&\quad =\lim _{t\rightarrow T^{-}}\gamma _{m}E\biggl [\exp \biggl (\int _{t}^{T}h(s,\alpha (s))ds\biggr )|\alpha (t)=e_{i}\biggr ]x_{1}x_{2} \\&\quad =\gamma _{m}x_{1}x_{2}, \end{aligned}$$

for all \((\pi _{1},\pi _{2})\in \varTheta _{1}\times \varTheta _{2}, \ m=1,2.\)

(iv) If we show that the family \((\phi _{m}(t,Y^{\pi _{1},\pi _{2}}(t),\alpha (t)))_{t\in [0,T]}\) is in \(\mathbb {L}^{2}\) for all \((\pi _{1},\pi _{2})\in \varTheta _{1}\times \varTheta _{2}, \ m=1,2\), then uniform integrability condition will be satisfied.

Our wealth processes are as follows:

$$\begin{aligned} dX_{m}(t)&=X_{m}(t-)\biggl (\pi _{m}(t)\mu _{m}(t,\alpha (t-))+(1-\pi _{m}(t))r(t,\alpha (t-))\biggr )dt+X_{m}(t-) \nonumber \\&\quad \times \pi _{m}(t)\biggl (\sigma _{m}(t,\alpha (t-))dW(t)+\int _{\mathbb {R}_{0}}\eta _{m}(t,\alpha (t-),z)\tilde{N}(dt,dz)\biggr ),\ t\in [0,T], \\&\quad X_{m}(0)=x_{m}>0. \end{aligned}$$

By Itô’s differentiation rule on \(f(s,X_{m}(s),\alpha (s))=\ln (X_{m}(s))\), \(m=1,2, \ s\in [0,T]\) (see Theorem 4.1 by Zhang et al. (2012)), we obtain:

$$\begin{aligned} X_{m}(t)&=x_{m}\exp \biggl \{\int _{0}^{t} \biggl ((\pi _{m}(s)\mu _{m}(s,\alpha (s-))+(1-\pi _{m}) r(s,\alpha (s-)))-\frac{1}{2}\pi _{m}^{2}(s)\\&\quad \times \sigma _{m}(s,\alpha (s-))\biggr )ds +\int _{0}^{t}\int _{\mathbb {R}_{0}} \biggl (\ln (1+\pi _{m}(s)\eta _{m}(s,\alpha (s-),z))\\&\quad -\pi _{m}(s)\eta _{m}(s,\alpha (s-),z)\biggr ) \nu (dz)ds+\int _{0}^{t}\pi _{m}(s)\sigma _{m}(s,\alpha (s-))dW(s)\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}} \ln (1+\pi _{m}(s)\eta _{m}(s,\alpha (s-),z))\tilde{N}(ds,dz)\biggr \}. \end{aligned}$$


$$\begin{aligned}&X_{1}(t)X_{2}(t)=x_{1}x_{2}\exp \biggl \{ \int _{0}^{t}\biggl (\pi _{1}(s)\mu _{1}(s,\alpha (s-))+\pi _{2}(s)\mu _{2}(s,\alpha (s-)) \\&\quad +(2-\pi _{1}(s)-\pi _{2}(s))r(s,\alpha (s-))-\frac{1}{2}(\pi _{1}^{2}(s)\sigma _{1}^{2}(s,\alpha (s-))\\&\quad +\pi _{2}^{2}(s)\sigma _{2}^{2}(s,\alpha (s-)))\biggl )ds+\int _{0}^{t}\biggl (\pi _{1}(s)\sigma _{1}(s,\alpha (s-))+\pi _{2}(s)\sigma _{2}(s,\alpha (s-))\biggr )dW(s)\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl (\ln (1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))+\ln (1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )\tilde{N}(ds,dz)\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl ( (\ln (1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))+\ln (1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z)))\\&\quad -\pi _{1}(s)\eta _{1}(s,\alpha (s-),z)-\pi _{2}(s)\eta _{2}(s,\alpha (s-),z)\biggr )\nu (dz)ds \biggr \}. \end{aligned}$$

Therefore, we can state:

$$\begin{aligned}&E[X_{1}^{2}(t)X_{2}^{2}(t)]=x_{1}x_{2}E\biggl [\exp \biggl \{ \biggl (\int _{0}^{t}2\pi _{1}(s)\mu _{1}(s,\alpha (s-))+2\pi _{2}(s)\mu _{2}(s,\alpha (s-))\\&\quad +2(2-\pi _{1}(s)-\pi _{2}(s))r(s,\alpha (s-))-\pi _{1}^{2}(s)\sigma _{1}^{2}(s,\alpha (s-))-\pi _{2}^{2}(s)\sigma _{2}^{2}(s,\alpha (s-))\biggr )ds\\&\quad +\int _{0}^{t}2\biggl (\pi _{1}(s)\sigma _{1}(s,\alpha (s-))+\pi _{2}(s)\sigma _{2}(s,\alpha (s-))\biggr )dW(s) \\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2}\tilde{N}(ds,dz) \\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl (\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2} \\&\quad -2\pi _{1}(s)\eta _{1}(s,\alpha (s-),z)-2\pi _{2}(s)\eta _{2}(s,\alpha (s-),z)\biggr )\nu (dz)ds\biggr \}\biggr ]. \end{aligned}$$

Since \(\pi _{m}\) and \(\sigma _{m}\), for \(m=1,2\), are piecewise constant, then we can find a \(K>0\) such that

$$\begin{aligned}&E\biggl [\exp \biggl (\int _{0}^{t}2(\pi _{1}(s)\sigma _{1}(s,\alpha (s-))+\pi _{2}(s)\sigma _{2}(s,\alpha (s-)))dW(s)\biggr )\biggr ] \\&\quad \le E\biggl [\exp \biggl (\int _{0}^{t}2K\biggr )dW(s)\biggr )\biggr ]=\exp (2K^{2}t)< \infty . \end{aligned}$$

Moreover, if we apply Example 1.6 in Øksendal and Sulem (2007) state by state, we reach

$$\begin{aligned}&E\biggl [\exp \biggl (\int _{0}^{t}\int _{\mathbb {R}_{0}}\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2}\tilde{N}(ds,dz)\biggr )\biggr ] \\&\quad =\exp \biggl (\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl \{2(1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))-1\\&\quad -\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2}\biggr \}\nu (dz)ds\biggr ). \end{aligned}$$

Hence, by Eq. (22) and by the construction of coefficients,

$$\begin{aligned}&E[k_{m}^{2}(t,\alpha (t))X_{1}^{2}(t)X_{2}^{2}(t)]\le \gamma ^{2}_{m}\exp (2(T-t)L))E[X_{1}^{2}(t)X_{2}^{2}(t)] \\&\quad \le \gamma ^{2}_{m}\exp (2(T-t)L+2K^{2}t)x_{1}x_{2}E\biggl [\exp \biggl \{\biggl (\int _{0}^{t}2\pi _{1}(s)\mu _{1}(s,\alpha (s-))+2\pi _{2}(s)\mu _{2}(s,\alpha (s-))\\&\quad +2(2-\pi _{1}(s)-\pi _{2}(s))r(s,\alpha (s-))-\pi _{1}^{2}(s)\sigma _{1}^{2}(s,\alpha (s-))-\pi _{2}^{2}(s)\sigma _{2}^{2}(s,\alpha (s-))\biggr )ds\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}(1+\pi _{1}(s)\pi _{2}(s)\eta _{1}(s,\alpha (s-),z)\eta _{2}(s,\alpha (s-),z))\nu (dz)ds\biggr ]< \infty , \quad m=1,2, \end{aligned}$$


$$\begin{aligned}&E[k_{m}^{2}(t,e_{i})X_{1}^{2}(t)X_{2}^{2}(t)]\le \gamma ^{2}_{m}\exp (2(T-t)L))E[X_{1}^{2}(t)X_{2}^{2}(t)] \\&\quad \le \gamma ^{2}_{m}\exp (2(T-t)L+2K^{2}t)x_{1}x_{2}E\biggl [\exp \biggl \{\biggl (\int _{0}^{t}2\pi _{1}(s)\mu ^{i}_{1}+2\pi _{2}(s)\mu ^{i}_{2}\\&\quad +2(2-\pi _{1}(s)-\pi _{2}(s))r^{i}-\pi _{1}^{2}(s)(\sigma ^{i}_{1})^{2}-\pi _{2}^{2}(s)(\sigma ^{i}_{2})^{2}\biggr )ds\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}(1+\pi _{1}(s)\pi _{2}(s)\eta ^{i}_{1}\eta ^{i}_{2}z^{2})\nu (dz)ds\biggr ]< \infty , \quad m=1,2, \quad i=1,2,\ldots ,D. \end{aligned}$$

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Savku, E., Weber, GW. Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market. Ann Oper Res (2020).

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  • Control
  • Stochastic processes
  • Behavioral finance
  • Game theory
  • Dynamic programming