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Stochastic maximum principle for problems with delay with general dependence on the past

Giuseppina Guatteri, Federica Masiero

Abstract

We prove a stochastic maximum principle for a control problem where the state equation is delayed both in the state and in the control, and also the final cost functional may depend on the past trajectories. The adjoint equations turn out to be a new form of linear anticipated backward stochastic differential equations (ABSDEs in the following), and we prove a direct formula to solve these equations.

Stochastic maximum principle for problems with delay with general dependence on the past

Abstract

We prove a stochastic maximum principle for a control problem where the state equation is delayed both in the state and in the control, and also the final cost functional may depend on the past trajectories. The adjoint equations turn out to be a new form of linear anticipated backward stochastic differential equations (ABSDEs in the following), and we prove a direct formula to solve these equations.

Paper Structure

This paper contains 6 sections, 7 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. A new form of anticipated backward stochastic differential equations
  4. The controlled problem and the stochastic maximum principle
  5. Delay equations arising in advertising models
  6. An optimal portfolio problem with execution delay

Key Result

Lemma 2.2

Assume Hypothesis hyp:BSDEtrascinata holds true, then the BSDE (BSDEtrascinata) admits a unique adapted solution, that is a pair of processes $(p,q)\in \mathcal{L}^2_{\mathcal{F}}(\Omega\times [0,T], \mathbb R^{n})\times \mathcal{L}^2_{\mathcal{F}}(\Omega\times [0,T], \mathbb R^{n\times m})$, satis

Theorems & Definitions (13)

  • Lemma 2.2
  • proof
  • Remark 1
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 3.2
  • ...and 3 more