# Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation

## TL;DR

This paper introduces a Riccati transformation method to solve a complex Hamilton-Jacobi-Bellman equation in stochastic portfolio optimization, enabling more accurate computation of optimal strategies considering intertemporal utility.

## Contribution

The paper presents a novel Riccati method for transforming nonlinear HJB equations into a quasi-linear form, facilitating numerical solutions in portfolio optimization.

## Key findings

- The numerical scheme achieves second-order convergence.
- Optimal strategies are significantly affected by intertemporal utility considerations.
- Application to German DAX 30 data demonstrates practical relevance.

## Abstract

In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) equation. We propose the so-called Riccati method for transformation of the fully nonlinear HJB equation into a quasi-linear parabolic equation with non-local terms involving the intertemporal utility function. As a numerical method we propose a semi-implicit scheme in time based on a finite volume approximation in the spatial variable. By analyzing an explicit traveling wave solution we show that the numerical method is of the second experimental order of convergence. As a practical application we compute optimal strategies for a portfolio investment problem motivated by market financial data of German DAX 30 Index and show the effect of considering intertemporal utility on optimal portfolio selection.