We study an optimization problem for a portfolio with a risk-free, a liquid risky , and an illiquid asset which is sold in an exogenous random moment of time with a prescribed liquidation time distribution. Problems of such type lead to three dimensional nonlinear partial differential equations (PDEs) on the value function. We study the optimization problem with a utility function of a CARA type, i.e. with negative and positive exponential utility functions (EXPn and EXPp) . It is well known that both the LOG and the EXPn utility functions are connected with the HARA utility function by means of a limiting procedure : in the first case the parameter of a HARA utility function is going to zero and in the second case to infinity. In our previous papers devoted to the optimization problem with a HARA and LOG utility functions we proved that also the corresponding analytical and Lie algebraic structures are connected with the same limiting procedure. In this paper we show that the case of EXPn utility function differs from the case of the HARA utility and is not connected to the HARA case by the limiting procedure . We carry out the Lie group analysis of the PDEs for the cases EXPn and EXPp utility functions and proved that they are connected by a one-to-one analytical substitution and are identical from the economical, analytical or Lie algebraic point of view. The complete set of nonequivalent group invariant reductions to two dimensional PDEs is provided for the three dimensional PDE with the EXPn utility function in accordance with an optimal system of sub algebras of the admitted Lie algebra. We prove that in  one case the invariant reduction is consistent with the boundary condition. We can use the reduced two dimensional PDE to study the properties of the optimal solution and the investment - consumption strategies .
We study an optimization problem for a portfolio with a risk-free, a liquid , and an illiquid risky asset. The illiquid risky asset is sold in an exogenous random moment with a prescribed liquidation time distribution. The investor prefers a negative or a positive exponential utility function. We prove that both cases are connected by a one-to-one analytical substitution and are identical from the economic, analytical, or Lie algebraic points of view . It is well known that the exponential utility function is connected with the HARA utility function through a limiting procedure if the parameter of the HARA utility function is going to infinity. We show that the optimization problem with the exponential utility function is not connected to the HARA case by the limiting procedure and we obtain essentially different results. For the main three dimensional PDE with the exponential utility function we obtain the complete set of the nonequivalent Lie group invariant reductions to two dimensional PDEs according to an optimal system of subalgebras of the admitted Lie algebra. We prove that in just one case the invariant reduction is consistent with the boundary condition. This reduction represents a significant simplification of the original problem .
D
risky
R
asset which
risky asset. The illiquid risky asset
D
of time
R
Problems of such type lead to three dimensional nonlinear partial differential equations (PDEs) on the value function. We study the optimization problem with a utility function of a CARA type, i.e. with negative and
The investor prefers a negative or a
R
functions (EXPn and EXPp)
function. We prove that both cases are connected by a one-to-one analytical substitution and are identical from the economic, analytical, or Lie algebraic points of view
R
both the LOG and the EXPn utility functions are
the exponential utility function is
R
by means of
through
R
: in the first case the parameter of a
if the parameter of the
R
zero and in the second case to infinity. In our previous papers devoted to
infinity. We show that
R
a HARA and LOG utility functions we proved that also the corresponding analytical and Lie algebraic structures are connected with the same limiting procedure. In this paper we show that the case of EXPn utility function differs from the case of the HARA utility and
the exponential utility function
R
. We carry out the Lie group analysis of the PDEs for the cases EXPn and EXPp utility functions and proved that they are connected by a one-to-one analytical substitution and are identical from the economical, analytical or Lie algebraic point of view. The
and we obtain essentially different results. For the main three dimensional PDE with the exponential utility function we obtain the
R
nonequivalent
the nonequivalent Lie
R
is provided for the three dimensional PDE with the EXPn utility function in accordance with
according to
R
sub algebras
subalgebras
A
just
R
We can use the reduced two dimensional PDE to study the properties of the optimal solution and the investment - consumption strategies
This reduction represents a significant simplification of the original problem