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Exponential Utility Maximization with Delay in a Continuous Time Gaussian Framework

Yan Dolinsky

TL;DR

This paper addresses exponential utility maximization in continuous time under delayed information within a Gaussian one-asset market. It develops a purely probabilistic approach based on Radon-Nikodym derivatives for Gaussian measures to handle the non-Markovian control problem induced by information delay. The authors prove existence and uniqueness of a pair of Volterra-type kernels $(\kappa,g)$ that determine the optimal trading strategy $\hat{\gamma}_t=a(t)+\int_{0}^t \kappa(t,s)\, dX_s$ and derive an explicit expression for the maximal (negative exponential) utility value, including a delay-penalty term. They also provide a Computation Example with a Gauss-Markov process to illustrate how the delay affects the value and the optimal strategy. The results extend prior discrete-time/constant-delay analyses to a continuous-time setting with general delay, offering a tractable framework for non-Markovian utility optimization under Gaussian dynamics.

Abstract

In this work we study the continuous time exponential utility maximization problem in the framework of an investor who is informed about the price changes with a delay. This leads to a non-Markovian stochastic control problem. In the case where the risky asset is given by a Gaussian process (with some additional properties) we establish a solution for the optimal control and the corresponding value. Our approach is purely probabilistic and is based on the theory for Radon-Nikodym derivatives of Gaussian measures developed by Shepp [6] and Hitsuda [5].

Exponential Utility Maximization with Delay in a Continuous Time Gaussian Framework

TL;DR

This paper addresses exponential utility maximization in continuous time under delayed information within a Gaussian one-asset market. It develops a purely probabilistic approach based on Radon-Nikodym derivatives for Gaussian measures to handle the non-Markovian control problem induced by information delay. The authors prove existence and uniqueness of a pair of Volterra-type kernels that determine the optimal trading strategy and derive an explicit expression for the maximal (negative exponential) utility value, including a delay-penalty term. They also provide a Computation Example with a Gauss-Markov process to illustrate how the delay affects the value and the optimal strategy. The results extend prior discrete-time/constant-delay analyses to a continuous-time setting with general delay, offering a tractable framework for non-Markovian utility optimization under Gaussian dynamics.

Abstract

In this work we study the continuous time exponential utility maximization problem in the framework of an investor who is informed about the price changes with a delay. This leads to a non-Markovian stochastic control problem. In the case where the risky asset is given by a Gaussian process (with some additional properties) we establish a solution for the optimal control and the corresponding value. Our approach is purely probabilistic and is based on the theory for Radon-Nikodym derivatives of Gaussian measures developed by Shepp [6] and Hitsuda [5].
Paper Structure (3 sections, 2 theorems, 29 equations)

This paper contains 3 sections, 2 theorems, 29 equations.

Table of Contents

  1. Introduction and the Main Result
  2. Computational Example
  3. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

There exists a unique pair $(\kappa,g)\in\mathcal{V}\times\mathcal{S}$ such that $\kappa(t,s)=0$ for $t<\tau^{-1}(s)$, $g(t,s)=0$ for $t\geq \tau^{-1}(s)$ and The maximizer $\hat{\gamma}=(\hat{\gamma}_t)_{0\leq t \leq T}$ for the optimization problem (2) is unique and is given by the linear form The corresponding value is given by where $\tilde{g}(s,u):=g(s,u)-\int_{0}^u g(s,v)g(u,v)dv$, $0\leq

Theorems & Definitions (3)

  • Theorem 1
  • Remark 2
  • Lemma 3