Solvable stochastic differential games in rank one compact symmetric spaces¹

1 This paper is dedicated to Alexander S. Poznyak on the occasion of his seventieth birthday.

ABSTRACT

Some nonlinear stochastic differential games are formulated in the family of complex and quaternion projective spaces that are among the rank one compact symmetric spaces that consist of the spheres, the projective spaces over $\mathbb{R},\mathbb{C}$, and $\mathbb{H}$ and one arising from an exceptional Lie algebra called the Cayley plane. The payoff functionals for the differential games are obtained from some eigenfunctions of the radial part of the Laplacians for these Riemannian manifolds and these payoffs induce symmetries for the game problems that reduce the required analysis to a radial direction in these manifolds. These projective spaces are given a natural Riemannian metric from the Killing forms of the compact Lie groups for these spaces that are the special unitary groups, SU(n), and the symplectic groups, Sp(n). Explicit optimal control strategies are obtained for these differential games and the explicit payoffs are given. A countable family of distinct solvable stochastic differential games can be obtained for each of these compact symmetric spaces.

KEYWORDS:

• Games in compact rank one symmetric spaces
• solvable stochastic differential games
• stochastic systems

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This paper is dedicated to Alexander S. Poznyak on the occasion of his seventieth birthday.

Additional information

Funding

Research supported by NSF [grant number DMS 1411412]; AFOSR [grant number FA9550-17-1-0073]; ARO [grant number W911NF-14-10390]; Simons Fellowship for the first author.