[Paper Review] Asymptotic Optimal Strategy for Portfolio Optimization in a Slowly Varying Stochastic Environment

Jean‐Pierre Fouque, Ruimeng Hu|arXiv (Cornell University)|Mar 11, 2016

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper develops a rigorous asymptotic approximation for portfolio optimization in a slowly varying stochastic environment, where asset returns and volatility depend on a slow-moving factor. It proves that a zeroth-order suboptimal strategy derived heuristically is asymptotically optimal under mild regularity conditions, with convergence rates depending on the perturbation scaling.

ABSTRACT

In this paper, we study the portfolio optimization problem with general utility functions and when the return and volatility of underlying asset are slowly varying. An asymptotic optimal strategy is provided within a specific class of admissible controls under this problem setup. Specifically, we first establish a rigorous first order approximation of the value function associated to a fixed zeroth order suboptimal trading strategy, which is given by the heuristic argument in [J.-P. Fouque, R. Sircar and T. Zariphopoulou, {\it Mathematical Finance}, 2016]. Then, we show that this zeroth order suboptimal strategy is asymptotically optimal in a specific family of admissible trading strategies. Finally, we show that our assumptions are satisfied by a particular fully solvable model.

Motivation & Objective

• To establish a rigorous first-order approximation of the value function for portfolio optimization under a slowly varying stochastic environment.
• To prove that a zeroth-order suboptimal strategy—previously derived heuristically—is asymptotically optimal within a specific class of admissible controls.
• To validate the asymptotic framework by showing that the assumptions hold in a fully solvable model.
• To analyze the convergence rate of the approximation under different scaling regimes of the slow factor.

Proposed method

• Uses regular perturbation theory to expand the value function as $ V^\delta = v^{(0)} + \sqrt{\delta}v^{(1)} + \delta v^{(2)} + \cdots $, where $ \delta $ measures the slowness of the factor.
• Derives the leading-order value function $ v^{(0)} $ and first-order correction $ v^{(1)} $ via asymptotic equations (2.16) and (2.18) under the slow factor dynamics.
• Defines a zeroth-order suboptimal strategy $ \pi^{(0)} = -\frac{\lambda(z)}{\sigma(z)} \frac{v^{(0)}_x}{v^{(0)}_{xx}} $, based on the leading-order value function.
• Establishes asymptotic optimality by proving that the expected utility under $ \pi^{(0)} $ converges to the true optimal value at rate $ O(\delta^{\alpha}) $ for $ \alpha > 0 $, under boundedness assumptions on key moments.
• Introduces a family of perturbed strategies $ \pi = \widetilde{\pi}^0 + \delta^\alpha \widetilde{\pi}^1 $ and derives sufficient conditions (Assumption C.1) for uniform boundedness of moments involving $ v^{(0)}, v^{(1)} $, and their derivatives.
• Applies the framework to a fully solvable model where the assumptions are verified, confirming the validity of the asymptotic approximation.

Experimental results

Research questions

• RQ1 Can the heuristic zeroth-order strategy derived in Fouque et al. (2016) be rigorously proven to be asymptotically optimal in a slowly varying stochastic environment?
• RQ2 What are the necessary conditions on the coefficients and moments to ensure the asymptotic optimality of the zeroth-order strategy?
• RQ3 How does the convergence rate of the approximation depend on the scaling parameter $ \delta $ and the perturbation order $ \alpha $?
• RQ4 In what class of admissible strategies is the zeroth-order strategy asymptotically optimal?
• RQ5 Does the asymptotic framework hold for a fully solvable model with explicit solutions?

Key findings

• The zeroth-order suboptimal strategy $ \pi^{(0)} $ is asymptotically optimal in the class of strategies $ \widetilde{\pi}^0 + \delta^\alpha \widetilde{\pi}^1 $, with convergence rate $ O(\delta^\alpha) $ for $ \alpha > 1/4 $.
• For $ 0 < \alpha < 1/4 $, the asymptotic optimality holds under modified moment bounds involving $ \widetilde{v}^{2\alpha} $, the coefficient of $ \delta^{2\alpha} $ in the expansion of the value function.
• In the critical case $ \alpha = 1/4 $, the convergence rate is $ O(\delta^{1/4}) $, and the first-order correction $ \widetilde{v}^{(1)} $ satisfies a linear PDE (4.10).
• The assumptions in Assumption C.1 ensure uniform boundedness of key moment terms, which is essential for proving the asymptotic optimality of the strategy.
• The framework is validated on a fully solvable model where the conditions are satisfied, confirming the theoretical convergence rates.
• The value function approximation $ \widetilde{V}^\delta $, built from $ v^{(0)} $ and $ v^{(1)} $, achieves accuracy $ O(\delta^{1/2}) $ under the stated boundedness conditions.