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2021 Issue №1

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Portfolio optimization in the case of an exponential utility function and in the presence of an illiquid asset

Pages
73-102

Abstract

We study an optimization problem for a portfolio with a risk-free, a li­quid, and an illiquid risky asset. The illiquid risky asset is sold in an exogenous random moment with a prescribed liquidation time distribution. We assume that the investor chooses an exponential utility function. Study of op­timization problems with three assets including an illiquid asset leads to three-dimensional nonlinear Hamilton — Jacobi — Bellman (HJB) equations.

It is well known that the exponential utility function is connected with the HARA utility function through a limiting procedure if the parameter of the HARA utility function is going to infinity. We show that the optimization problem with the exponential utility function is not connected to the optimization problem with the HARA utility by the limiting procedure and we obtain essentially different results. We provide the Lie group analysis of the corresponding HJB equation.

For the main three-dimensional PDE with the exponential utility function, we obtain the complete set of the nonequivalent Lie group invariant reductions to two-dimensional PDEs according to an optimal system of subalgebras of the admitted Lie algebra. We prove that in just one case the invariant reduction is consistent with the boundary condition. This reduction represents a significant simplification of the original problem.

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