Appl Math Optim (2015) 72:469–491
Robust Consumption-Investment Problem on Inﬁnite
Published online: 17 January 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract In our paper we consider an inﬁnite horizon consumption-investment prob-
lem under a model misspeciﬁcation in a general stochastic factor model. We formulate
the problem as a stochastic game and ﬁnally characterize the saddle point and the value
function of that game using an ODE of semilinear type, for which we provide a proof
of an existence and uniqueness theorem for its solution. Such equation is interested
on its own right, since it generalizes many other equations arising in various inﬁnite
horizon optimization problems.
Keywords Robust optimization· Stochastic differential games· Model uncertainty·
Optimal consumption · Portfolio optimization
Mathematics Subject Classiﬁcation 91G80 · 91G10 · 91A15 · 91A25 · 49N90 ·
A major weakness of a portfolio optimization is a huge sensitivity to estimation errors
and a model misspeciﬁcation. The concern about a model uncertainty should lead the
investor to design a strategy which is robust to model imperfections. In this paper
a max–min robust version of the classical Merton optimal investment-consumption
model is presented. We consider a ﬁnancial market consisting of a stock and a bond.
A stock and a bond dynamics are assumed to be stochastic differential equations.
In addition coefﬁcients of our model are affected by a non-tradable but observable
D. Zawisza (
Institute of Mathematics, Faculty of Mathematics and Computer Science,
Jagiellonian University in Krakow, Łojasiewicza 6, 30-348 Kraków, Poland