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Random Operators and Stochastic Equations

Editor-in-Chief: Girko, Vyacheslav

Managing Editor: Molchanov, S.

Editorial Board: Accardi, L. / Albeverio, Sergio / Carmona, R. / Casati, G. / Christopeit, N. / Domanski, C. / Drygas, Hilmar / Gupta, A.K. / Ibragimov, I. / Kirsch, Werner / Klein, A. / Kondratyev, Yuri / Kurotschka, V. / Leonenko, N. / Loubaton, Philippe / Orsingher, E. / Pastur, L. / Rodrigues, Waldyr A. / Shiryaev, Albert / Turbin, A.F. / Veretennikov, Alexandre

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Volume 26, Issue 4

Issues

An HJB approach to a general continuous-time mean-variance stochastic control problem

Georgios AivaliotisORCID iD: http://orcid.org/0000-0003-4018-4820 / A. Yu. Veretennikov
  • School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom; and National Research University Higher School of Economics, Russian Federation & Institute of Information Transmission Problems, Moscow
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Published Online: 2018-11-14 | DOI: https://doi.org/10.1515/rose-2018-0020

Abstract

A general continuous mean-variance problem is considered for a diffusion controlled process where the reward functional has an integral and a terminal-time component. The problem is transformed into a superposition of a static and a dynamic optimization problem. The value function of the latter can be considered as the solution to a degenerate HJB equation either in the viscosity or in the Sobolev sense (after a regularization) under suitable assumptions and with implications with regards to the optimality of strategies. There is a useful interplay between the two approaches – viscosity and Sobolev.

Keywords: Mean-variance; stochastic control; Hamilton–Jacobi–Bellman; Sobolev solutions; viscosity solutions

MSC 2010: 93E20; 60H10.

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About the article


Received: 2018-02-07

Revised: 2018-10-03

Accepted: 2018-10-16

Published Online: 2018-11-14

Published in Print: 2018-12-01


Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: EP/N013980/1

Funding Source: Alan Turing Institute

Award identifier / Grant number: EP/N510129/1

The first author has been partially supported by EPSRC grant EP/N013980/1 and the Alan Turing Institute EP/N510129/1. For the second author this study has been funded by the Russian Academic Excellence Project ’5-100’.


Citation Information: Random Operators and Stochastic Equations, Volume 26, Issue 4, Pages 225–234, ISSN (Online) 1569-397X, ISSN (Print) 0926-6364, DOI: https://doi.org/10.1515/rose-2018-0020.

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