Abstract
In this article, we consider stochastic differential game where the state process is governed by a controlled Itô–Lévy process and the information available to the controllers is possibly less than the general information. All the system coefficients and the objective performance functional are assumed to be random. We use Malliavin calculus to derive a maximum principle for the optimal control of such problem. The results are applied to solve a worst-case scenario portfolio problem in finance.
Received2/18/2011; Accepted 5/23/2012; Final 5/29/2012
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References
An, T.T.K., Øksendal, B.: A maximum principle for stochastic differential games with partial information. J. Optim. Theor. Appl. 139, 463–483 (2008)
Bensoussan, A.: Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (1992)
Benth, F.E., Di Nunno, G., Løkka, A., Øksendal, B., Proske, F.: Explicit representation of the minimal variance portfolio in markets driven by Lévy processes. Math. Financ. 13, 55–72 (2003)
Baghery, F., Øksendal, B.: A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25, 705–717 (2007)
Di Nunno, G., Meyer-Brandis, T., Øksendal, B., Proske, F.: Malliavin calculus and anticipative Itô formulae for Lévy processes. Inf. Dim. Anal. Quant. Probab. 8, 235–258 (2005)
Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Lévy Processes and Applications to Finance. Springer, Berlin (2009)
Karatzas, I., Ocone, D.: A generalized Clark representation formula, with application to optimal portfolios. Stoch. Stoch. Rep. 34, 187–220 (1991)
Karatzas, I., Xue, X.: A note on utility maximization under partial observations. Math. Financ. 1, 57–70 (1991)
Lakner, P.: Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77–97 (1998)
Meyer-Brandis, T., Øksendal, B., Zhou, X.Y.: A mean-field stochastic maximum principle via Malliavin calculus. Stoch. 84, 643–666 (2012)
Nualart, D.: Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)
Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin (2007)
Øksendal, B., Sulem, A.: A game theoretic approach to martingale measures in incomplete markets. Surv. Appl. Ind. Math. 15, 18–24 (2008)
Pham, H., Quenez M.-C.: Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab. 11, 210–238 (2001)
Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)
Acknowledgements
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ ERC grant agreement no [228087].
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Kieu, A.T.T., Øksendal, B., Okur, Y.Y. (2013). A Malliavin Calculus Approach to General Stochastic Differential Games with Partial Information. In: Viens, F., Feng, J., Hu, Y., Nualart , E. (eds) Malliavin Calculus and Stochastic Analysis. Springer Proceedings in Mathematics & Statistics, vol 34. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5906-4_22
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