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Risk minimizing portfolios and HJBI equations for stochastic differential games. (English) Zbl 1145.93054
Summary: We consider the problem to find a market portfolio that minimizes the convex risk measure of the terminal wealth in a jump diffusion market. We formulate the problem as a two player (zero-sum) stochastic differential game. To help us find a solution, we prove a theorem giving the Hamilton-Jacobi-Bellman-Isaacs (HJBI) conditions for a general zero-sum stochastic differential game in a jump diffusion setting. We then use the theorem to study particular risk minimization problems. Finally, we extend our approach to cover general stochastic differential games (not necessarily zero-sum), and we obtain similar HJBI equations for the Nash equilibria of such games.

93E20Optimal stochastic control (systems)
60G51Processes with independent increments; Lévy processes
60H10Stochastic ordinary differential equations
60J75Jump processes
91A15Stochastic games
91A23Differential games (game theory)
91B28Finance etc. (MSC2000)
49L20Dynamic programming method (infinite-dimensional problems)
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