Stochastic target problems, dynamic programming, and viscosity solutions

H. Mete Soner, Nizar Touzi

Research output: Contribution to journalArticle

56 Scopus citations

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 (Xν, Yν) takes values in ℝd × ℝ and a given initial data for Xν(0). Then the control problem is to find the minimal initial data for Yν so that it reaches a stochastic target at a specified terminal time T. The main application is from financial mathematics, in which the process Xν is related to stock price, Yν 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 Yν.

Original languageEnglish (US)
Pages (from-to)404-424
Number of pages21
JournalSIAM Journal on Control and Optimization
Volume41
Issue number2
DOIs
StatePublished - Jan 1 2003
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Applied Mathematics

Keywords

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

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