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SUMMARY:Lie group analysis of an optimization problem for a portfolio with
an illiquid asset
DTSTART;VALUE=DATE-TIME:20180828T123000Z
DTEND;VALUE=DATE-TIME:20180828T130000Z
DTSTAMP;VALUE=DATE-TIME:20220122T233507Z
UID:indico-contribution-3153@indico.math.cnrs.fr
DESCRIPTION:Speakers: Ljudmila A. Bordag (University of Applied Sciences Z
ittau/Gorlitz)\nWorking in the Merton's optimal consumption framework wit
h continuous time we consider an optimization problem for a portfolio with
an illiquid\, a risky and a risk-free asset. Our goal in this paper is to
carry out a complete Lie group analysis of PDEs describing value function
and investment and consumption strategies for a portfolio with an illiqui
d asset that is sold in an exogenous random moment of time $T$ with a pres
cribed liquidation time distribution. The problem of such type leads to th
ree dimensional nonlinear Hamilton-Jacobi-Bellman (HJB) equations. Such eq
uations are not only tedious for analytical methods but are also quite cha
llenging form a numeric point of view. To reduce the three-dimensional pro
blem to a two-dimensional one or even to an ODE one usually uses some subs
titutions\, yet the methods used to find such substitutions are rarely dis
cussed by the authors.\n\nWe use two types of utility functions: general H
ARA type utility and logarithmic utility. We carry out the Lie group analy
sis of the both three dimensional PDEs and are able to obtain the admitted
symmetry algebras. Then we prove that the algebraic structure of the PDE
with logarithmic utility can be seen as a limit of the algebraic structure
of the PDE with HARA-utility as $\\gamma \\to 0$. Moreover\, this relatio
n does not depend on the form of the survival function $\\overline{\\Phi}
(t)$ of the random liquidation time $T$.\nWe find the admitted Lie algebra
for a broad class of liquidation time distributions in cases of HARA and
log utility functions and formulate corresponding theorems for all these c
ases.\n\nWe use found Lie algebras to obtain reductions of the studied equ
ations. Several of similar substitutions were used in other papers before
whereas others are new to our knowledge. This method gives us the possibil
ity to provide a complete set of non-equivalent substitutions and reduced
equations.\n\nWe also show that if and only if the liquidation time defin
ed by a survival function $\\overline{\\Phi} (t)$ is distributed exponent
ially\, then for both types of the utility functions we get an additional
symmetry. We prove that both Lie algebras admit this extension\, i.e. we o
btain the four dimensional $L^{HARA}_4$ and $L^{LOG}_4$ correspondingly fo
r the case of exponentially distributed liquidation time.\nWe list reduced
equations and corresponding optimal policies that tend to the classical M
erton policies as illiquidity becomes small.\n\n This research was support
ed by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Ag
reement Number 304617 (FP7 Marie Curie Action\, Project Multi-ITN STRIKE -
Novel Methods in Computational Finance)\n\nhttps://indico.math.cnrs.fr/ev
ent/3123/contributions/3153/
LOCATION:Angers - France
URL:https://indico.math.cnrs.fr/event/3123/contributions/3153/
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