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Global Maximum Principle for Partially Observed Risk-Sensitive Progressive Optimal Control of FBSDE with Poisson Jumps

Jingtao Lin, Jingtao Shi

Abstract

This paper is concerned with one kind of partially observed progressive optimal control problems of coupled forward-backward stochastic systems driven by both Brownian motion and Poisson random measure with risk-sensitive criteria. The control domain is not necessarily convex, and the control variable can enters into all the coefficients. The observation equation also has correlated noises with the state equation. Under the Poisson jump setting, the original problem is equivalent to a complete information stochastic recursive optimal control problem of a forward-backward system with quadratic-exponential generator. In order to establish the first- and second-order variations, some new techniques are introduced to overcome difficulties caused by the quadratic-exponential feature. A new global stochastic maximum principle is deduced. As an application, a risk-sensitive optimal investment problem with factor model is studied. Moreover, the risk-sensitive stochastic filtering problem is also studied, which involves both Brownian and Poissonian correlated noises. A modified Zakai equation is obtained.

Global Maximum Principle for Partially Observed Risk-Sensitive Progressive Optimal Control of FBSDE with Poisson Jumps

Abstract

This paper is concerned with one kind of partially observed progressive optimal control problems of coupled forward-backward stochastic systems driven by both Brownian motion and Poisson random measure with risk-sensitive criteria. The control domain is not necessarily convex, and the control variable can enters into all the coefficients. The observation equation also has correlated noises with the state equation. Under the Poisson jump setting, the original problem is equivalent to a complete information stochastic recursive optimal control problem of a forward-backward system with quadratic-exponential generator. In order to establish the first- and second-order variations, some new techniques are introduced to overcome difficulties caused by the quadratic-exponential feature. A new global stochastic maximum principle is deduced. As an application, a risk-sensitive optimal investment problem with factor model is studied. Moreover, the risk-sensitive stochastic filtering problem is also studied, which involves both Brownian and Poissonian correlated noises. A modified Zakai equation is obtained.

Paper Structure

This paper contains 13 sections, 16 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and preliminaries
  3. Equivalent stochastic recursive optimal control problem
  4. $L^p$ solution of FBSDEPs
  5. Well-posedness of quadratic-exponential BSDEPs
  6. Main results
  7. Spike variation
  8. Stochastic Lipschitz BSDEPs
  9. First- and second-order variational equations
  10. Adjoint equations and optimal condition
  11. Applications to an optimal investment model
  12. Risk-sensitive filtering and modified Zakai equation
  13. Concluding remarks

Key Result

Lemma 2.1

If $g$ is a positive $E_i$-progressive measurable process that $\bar{\mathbb{E}}\int_0^T\int_{E_i}gN_i(de,dt)<\infty,i=1,2$, then we have the following results: where $x^p$ is the dual predictable projection of $x$.

Theorems & Definitions (27)

  • Definition 2.1
  • Lemma 2.1
  • Remark 2.1
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.2
  • Remark 2.2
  • ...and 17 more