[Paper Review] Stochastic Utilities With a Given Optimal Portfolio : Approach by Stochastic Flows

Nicole El Karoui, Mohamed Mrad|arXiv (Cornell University)|Apr 29, 2010

Stochastic processes and financial applications31 references19 citations

TL;DR

This paper introduces a novel stochastic flows-based method to construct consistent stochastic utilities whose optimal portfolio is pre-specified, leveraging duality and stochastic change of variables. The key contribution is a general, minimal-regularity framework that constructs all such utilities via composition of stochastic flows of homeomorphisms, ensuring the given benchmark portfolio remains optimal under convex test-process constraints.

ABSTRACT

The paper generalizes the construction by stochastic flows of consistent utility processes introduced by M. Mrad and N. El Karoui in (2010). The utilities random fields are defined from a general class of processes denoted by $\GX$. Making minimal assumptions and convex constraints on test-processes, we construct by composing two stochastic flows of homeomorphisms, all the consistent stochastic utilities whose the optimal-benchmark process is given, strictly increasing in its initial condition. Proofs are essentially based on stochastic change of variables techniques.

Motivation & Objective

• To generalize the construction of consistent stochastic utilities with a given optimal benchmark portfolio using stochastic flows.
• To establish a framework that ensures the optimal portfolio remains consistent across time without a fixed investment horizon.
• To minimize regularity assumptions while maintaining the validity of utility processes under convex test-process constraints.
• To provide a duality-based method that avoids reliance on classical duality in utility maximization.
• To demonstrate the stability of consistent utilities under change of numeraire and time-shifted formulations.

Proposed method

• The method constructs consistent stochastic utilities by composing two stochastic flows of homeomorphisms: one derived from the inverse of the optimal wealth process $X^*$, and another from the dual optimal process $Y^*$.
• It uses the duality identity $U_x(t, X^*_t(x)) = Y^*_t(U_x(0, ilde{X}_t(x)))$ to express the marginal utility $U_x$ in terms of the dual process and the reverse flow $\tilde{X}$.
• The utility $U(t,x)$ is recovered via integration: $U(t,x) = \int_0^x Y^*_t(\tau, U_x(\tau, \mathcal{X}_\tau(t,z))) \, dz$, where $\mathcal{X}_\tau$ is the reverse flow at time $\tau$.
• Stochastic change of variables techniques are central to proving that the constructed utility process yields the given $X^*$ as optimal portfolio.
• The approach ensures that $U(t, X^*_t(x))$ is a martingale, a key requirement for optimality in consistent utility theory.
• The method is extended to intermediate times via a time-shifted formulation, showing that the utility construction is valid for any stopping time $\tau$.

Experimental results

Research questions

• RQ1 How can consistent stochastic utilities be systematically constructed when the optimal portfolio is given a priori?
• RQ2 What is the role of stochastic flows of homeomorphisms in generating consistent utility processes with a fixed optimal strategy?
• RQ3 Can the duality relationship between the primal and dual processes be leveraged to define the utility field without assuming a terminal time horizon?
• RQ4 Under what minimal assumptions does the constructed utility process remain consistent and yield the prescribed optimal portfolio?
• RQ5 How does the method ensure stability under change of numeraire and time-shifted initial conditions?

Key findings

• All consistent stochastic utilities that generate a given optimal portfolio $X^*$ are constructed via composition of stochastic flows of homeomorphisms, under minimal regularity and convexity assumptions.
• The marginal utility $U_x(t,x)$ is expressed as $Y^*_t(\tau, U_x(\tau, \mathcal{X}_\tau(t,x)))$, where $\mathcal{X}_\tau$ is the reverse flow of $X^*$, enabling recursive construction.
• The utility process $U(t, X^*_t(x))$ is proven to be a martingale, confirming the optimality of $X^*$ under the constructed utility.
• The method generalizes prior results by allowing richer classes of test-processes as long as they are convex, without requiring additional smoothness.
• The construction is stable under change of numeraire and extends naturally to any stopping time $\tau$, not just $t=0$.
• The proofs rely solely on stochastic change of variables and duality, avoiding complex computations and reducing technical overhead.