Nonzero-sum risk-sensitive stochastic differential games with discounted costs

Abstract

We study nonzero-sum stochastic differential games with risk-sensitive discounted cost criteria. Under fairly general conditions on drift term and diffusion coefficients, we establish a Nash equilibrium in Markov strategies for the discounted cost criterion. We achieve our results by studying relevant systems of coupled HJB equations.

Keywords:

• Risk-sensitive criterion
• controlled diffusions
• coupled HJB equations
• Nash equilibrium
• eventually stationary strategy

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 91A15
• Secondary: 91A23

Acknowledgment

We wish to thank an anonymous referee for pointing out some flaws in an earlier version of this article.

Notes

1 Note that from [25, p. 234], the p.d.e. $$\frac{\partial \widehat{\psi }}{\partial t}=\underset{v\in V}{\inf }[〈\tilde{b}(t,x,v),\nabla \phi 〉+\tilde{r}(t,x,v)\phi +\tilde{g}(t,x,v\left)\right]+\frac{1}{2}\text{trace}\left(a\right(t,x\left){\nabla }^2\overline{\phi }\right)+\widehat{\mu }\overline{\phi }+\widehat{\mu }\widehat{\psi }+\frac{1}{2}\text{trace}\left(a\right(t,x\left){\nabla }^2\widehat{\psi }\right),\widehat{\psi }(0,x)=0$$ has a unique solution in $W^{1,2,p,\lambda }\left(\right(0,\hspace{0.5em}\Theta )\times {\mathbb{R}}^d)$ for $\widehat{\mu }\ge {\mu }_0,$ for some ${\mu }_0>0$ and hence $\psi =\widehat{\psi }+\overline{\phi }$ is a unique solution to (2.11).

Additional information

Funding

The research of Mrinal K. Ghosh is partially supported by UGC Centre for Advanced Study. The research of Somnath Pradhan was supported in part by DST-SERB, India, grant EMR/2016/004810. The research of Suresh Kumar was supported partially by SERB, India, grant MTR/2017/000416.