Optimal Investment Policy for Insurers under the Constant Elasticity of Variance Model with a Correlated Random Risk Process

Abstract

This paper investigates the optimal portfolio choice problem for a large insurer with negative exponential utility over terminal wealth under the constant elasticity of variance (CEV) model. The surplus process is assumed to follow a diffusion approximation model with the Brownian motion in which is correlated with that driving the price of the risky asset. We first derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation and then obtain explicit solutions to the value function as well as the optimal control by applying a variable change technique and the Feynman–Kac formula. Finally, we discuss the economic implications of the optimal policy.

The Lie symmetry approach on (1+2)-dimensional financial models

“…We note that this special case of the class (1) admits the additional Lie symmetry 4 . In several models the condition C(y) = − 1 2 A 2 (y) occurs [3,[8][9][10],…”

“…That is, in addition to the Lie symmetries admitted by the general class (1), the class (7) admits the infinite dimensional symmetry Most of the models that appear in the literature are members of the general class (1) and they are subject to a terminal condition of the form [3][4][5][6] u(T , x, y) = x γ , (10) where γ ∈ (0, 1). We find those Lie symmetries admitted by (1) which leave invariant the terminal condition (10). We consider the linear combination of the three finite Lie symmetries given in (3),…”

“…where a i are real constants. Applying this general symmetry on the terminal condition (10) written as two equations…”

“…Equation ( 18) is subject to the terminal condition (10). We substitute C(y 13) to find the reduced partial differential equation…”

“…Such problems are, for example, the investment where an investor can invest in a saving account, stocks and bonds and tries to have the maximum utility from terminal wealth [3][4][5] and the spot price evolution of commodities like gold and coffee or energies like oil and gas [6,7]. Amongst a number of publications which deal with such models, some recent work can be found in [8][9][10][11]. Examples of such equations which were studied from the point of view of Lie symmetry analysis can be found in [9,[11][12][13].…”