In this paper we first deal with the problem of optimal control for zero-sum stochastic differential games. We give a necessary and sufficient maximum principle for that problem with partial information. Then we use the result to solve a problem in finance. Finally, we extend our approach to general stochastic games (nonzero-sum), and obtain an equilibrium point of such game.

We consider the problem of risk indifference pricing on an incomplete market, namely on a jump diffusion market where the controller has limited access to market information. We use the maximum principle for stochastic differential games to derive a formula for the risk indifference price p seller risk G, E of a European-type claim G.If P t p-constant for all t, then the portfolio strategies π t , S 0 t is called self-financing. A contingent claim with expiration date T is a nonnegative F T -measurable random variable G 2 Journal of Applied Mathematics and Stochastic Analysis that represents the time T payoff from seller to buyer. Suppose that for a contingent claim G there exists a self-financing strategy such that X π x where Q is any martingale measure equivalent to P on the probability space Ω, F t , P .In an incomplete market, an exact replication of a contingent claim is not always possible. One of the approaches to solve the replicating problems in an incomplete market is the utility indifference pricing. See, for example, Grasselli and Hurd 1 for the case of stochastic volatility model, Hodges and Neuberger 2 for the financial model with constraints, and Takino 3 for model with incomplete information. The utility indifference price p of a claim G is the initial payment that makes the seller of the contract utility indifferent to the two following alternatives: either selling the contract with initial payment p and with the obligation to pay out G at time T or not selling the contract and hence receiving no initial payment.Recently, several papers discuss risk measure pricing rather than utility pricing in incomplete markets. Some papers related to risk measure pricing are the following: Xu 4 propose risk measure pricing and hedging in incomplete markets; Barrieu and El Karoui 5 study a minimization problem for risk measures subject to dynamic hedging; Klöppel and Schweizer 6 study the indifference pricing of a payoff with a minus dynamic convex risk measure. See also the references in these papers.In our paper, we study a pricing formula based on the risk indifference principle in a jump-diffusion market. The same problem was studied by Øksendal and Sulem 7 with the restriction to Markov controls. So the problem is solved by using the Hamilton-Jacobi-Bellman equation. In our paper, the control process is required to be adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. This makes the control problem non-Markovian. Within the non-Markovian setting, the dynamic programming cannot be used. Here we use the maximum principle approach to find the solution for our problem.The paper is organized as follows. In Section 2, we will implement the option pricing method in an incomplete market. In Section 3, we present our problem in a jump-diffusion market. In Section 4, we use a maximum principle for a stochastic differential game to find the relation between the optimal controls of the stochastic differential game and of a corresponding stochastic control problem. Using th...

In this paper, we initiate a study on optimal control problem for stochastic differential games under generalized expectation via backward stochastic differential equations and partial information. We first prove a sufficient maximum principle for zero-sum stochastic differential game problem. And then extend our approach to general stochastic differential games (nonzero-sum games), and obtain an equilibrium point of such game. Finally we give some examples of applications.

as well as many others who helped me with their advice and inspiration. I am particularly thankful to my primary advisor, Frank Proske, for his constant guidance and instruction. It was due to Prof. Proske that I got an opportunity to write my Ph.D. in Oslo, for which I am very grateful. I would also like to express special thanks to Paul C. Kettler for his selfless help with LaTeX files. Finally, I would like to thank my family and friends for their interest and constant support of my efforts.