Optimal investment under behavioural criteria in incomplete diffusion market models

TL;DR

This paper extends the theory of optimal investment under behavioral preferences, specifically cumulative prospect theory, to incomplete diffusion market models with economic factors influencing returns.

Contribution

It introduces existence results for optimal strategies in incomplete diffusion models considering CPT preferences, including cases with correlated economic factors.

Findings

Existence of optimal strategies under mild conditions.

Applicable to models with independent economic factors.

Extended to certain models with correlated factors.

Abstract

The most commonly accepted model for investors' preferences is expected utility theory. More recently, other theories have emerged and pose new challenges to mathematics. The present paper treats preferences of cumulative prospect theory (CPT), where an "S-shaped" utility function is considered (i.e. convex up to a certain point and concave from there on). Also, distorted probability measures are applied for calculating the utility of a given position with respect to a (possibly random) benchmark $G$. Such problems have heretofore been solved essentially for complete continuous-time market models only. In the present paper we make a step forward and consider incomplete models of a diffusion type where the return of the investment in consideration depends on some economic factors. Our main result asserts, under mild assumptions, the existence of an optimal strategy when the driving noise of the economic factors is independent of that of the investment and the rate of return is non-negative. We are also able to accommodate models of a specific type where the factor may have non-zero correlation with the investment.