Zero-Sum Linear Quadratic Stochastic Differential Game With Jumps

There is no doubt that quadratic linear stochastic control problem is classical,which can be used to fit some nonlinear control problems and applied in other fields,especially in the increasingly prosperous financial market.To be specific,some mean-variance portfolio problems and risk management problems can be transformed into quadratic linear problems.Furthermore,there always exists competitors in real market,there must be some profit conflicts so that some portion of the market will be affected by these competitors.This situation inspires us to think of these problems from a perspective of game theory with stochastic process,especially for Possion jumps,which is in accordance with the law of market.In this paper,we study a zero-sum linear quadratic stochastic differential game problem with Poisson jumps,which diffusion coefficient unequal to zero.We get a closed-loop formation based on related Riccati equation,which expands the results of linear-quadratic problem with jumps in some way.We connect the game problem with another linear-quadratic control problem and talk about the existence of solution of related Hamiltonian system and Riccati equation,which is interesting in itself.The control variable in zero-sum linear quadratic differential game is com-posed of two portion,each of both can be considered as behaviors of one person.Thus,there are two players participating in and affecting this system together.And the state equation is driven by two independent stochastic processes,one is d-dimension standard Brownian motion,the other one is a Possion random martingale measure.The object function is quadratic about the state variable and control variables.Now,we give the two players serial number-1 and 2 respec-tively.Then,we can say that Player 1 want to maximize(or minimize)the object function while Player 2 want to minimize(or maximize)it.Now,the two players are restricted and affected by each other.On account of the particularity of game problem,admissible control,admissible strategy and admissible control strategy pair should be redefined.Like many classical process mode in game theory,we will define the value of Player 1,Player 2 and the game problem respectively.Also,we find a equilibrium point about the system in this paper,and we call the relative control variables of two players a optimal feedback control strategy pair.