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

  • Soňa Kilianová
  • Daniel ŠevčovičEmail author
Original Paper


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.


Dynamic stochastic portfolio optimization Dynamic utility Hamilton–Jacobi–Bellman equation Riccati transformation Finite volume scheme 

Mathematics Subject Classification

35K55 34E05 70H20 91B70 90C15 91B16 



The authors were supported by VEGA 1/0062/18 and DAAD ENANEFA-2018 Grants.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Appl. Mathematics and Statistics, FMFIComenius University in BratislavaBratislavaSlovakia

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