## Abstract

In this paper, we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward SDEs and forward-backward SDEs. The controlled process (X^{ν}, Y^{ν}) takes values in ℝ^{d} × ℝ and a given initial data for X^{ν}(0). Then the control problem is to find the minimal initial data for Y^{ν} so that it reaches a stochastic target at a specified terminal time T. The main application is from financial mathematics, in which the process X^{ν} is related to stock price, Y^{ν} is the wealth process, and ν is the portfolio. We introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation. The boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense. This provides a unique characterization of the value function which is the minimal initial data for Y^{ν}.

Original language | English (US) |
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Pages (from-to) | 404-424 |

Number of pages | 21 |

Journal | SIAM Journal on Control and Optimization |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - 2003 |

## Keywords

- Discontinuous viscosity solutions
- Dynamic programming
- Forward-backward SDEs
- Stochastic control

## ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics