Constrained LQ Problem with a Random Jump and Application to Portfolio Selection
This paper deals with a constrained stochastic linear-quadratic (LQ for short) optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the Itô-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations (BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.
KeywordsBackward stochastic Riccati equation Default time Mean-variance problem
2000 MR Subject Classification60H15 35R60 93E20
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The author would like to thank his advisor, Prof. Shanjian Tang from Fudan University, for the helpful comments and discussions. The author would also thank the referees of this paper for helpful comments.