The Annals of Applied Probability

On dynamic deviation measures and continuous-time portfolio optimization

Martijn Pistorius and Mitja Stadje

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Abstract

In this paper, we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency, we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of stochastic differential equations. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively $m$-stable dual sets. Using this notion of dynamic deviation measure, we formulate a dynamic mean-deviation portfolio optimization problem in a jump-diffusion setting and identify a subgame-perfect Nash equilibrium strategy that is linear as function of wealth by deriving and solving an associated extended HJB equation.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3342-3384.

Dates
Received: May 2016
Revised: January 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328703

Digital Object Identifier
doi:10.1214/17-AAP1282

Mathematical Reviews number (MathSciNet)
MR3737927

Zentralblatt MATH identifier
1382.60089

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 90C46: Optimality conditions, duality [See also 49N15] 91A10: Noncooperative games 91B70: Stochastic models 93E99: None of the above, but in this section

Keywords
Deviation measure time-consistency portfolio optimization extended HJB equation

Citation

Pistorius, Martijn; Stadje, Mitja. On dynamic deviation measures and continuous-time portfolio optimization. Ann. Appl. Probab. 27 (2017), no. 6, 3342--3384. doi:10.1214/17-AAP1282. https://projecteuclid.org/euclid.aoap/1513328703


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