Mutual Fund Theorem for continuous time markets with random coefficients

TL;DR

This paper extends the Mutual Fund Theorem to continuous time markets with random, observable coefficients, demonstrating that optimal investment strategies can be simplified regardless of utility preferences.

Contribution

It proves a weakened version of the Mutual Fund Theorem in a complex market with stochastic parameters, applicable to a broad class of utility functions.

Findings

Optimal strategies can be approximated by a fixed distribution of risky assets.

The theorem applies to markets with random, observable coefficients.

Expected utility maximization is achieved by strategies with a specific asset distribution.

Abstract

We study the optimal investment problem for a continuous time incomplete market model such that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are assumed to be independent from the driving Brownian motion, and they are supposed to be currently observable. It is shown that some weakened version of Mutual Fund Theorem holds for this market for general class of utilities; more precisely, it is shown that the supremum of expected utilities can be achieved on a sequence of strategies with a certain distribution of risky assets that does not depend on risk preferences described by different utilities.