[Paper Review] Homogenization and asymptotics for small transaction costs: the multidimensional case

Dylan Possamaï, H. Meté Soner+1 more|arXiv (Cornell University)|Dec 26, 2012

Advanced Mathematical Modeling in Engineering31 references4 citations

TL;DR

This paper establishes an asymptotic expansion for the multidimensional infinite-horizon optimal consumption-investment problem with small proportional transaction costs, using viscosity solutions and homogenization techniques. It proves the existence of a corrector solution to a singular ergodic control problem and derives the first-order correction term, which is characterized via a system of corrector equations without explicit solutions in higher dimensions.

ABSTRACT

In the context of the multi-dimensional infinite horizon optimal consumption-investment problem with proportional transaction costs, we provide the first order expansion in small transact costs. Similar to the one-dimensional derivation in our accompanying paper [42], the asymptotic expansion is expressed in terms of a singular ergodic control problem, and our arguments are based on the theory of viscosity solutions, and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available anymore. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.

Motivation & Objective

• To extend the one-dimensional asymptotic analysis of small transaction costs to the multidimensional setting.
• To characterize the first-order correction term in the expansion of the value function under small transaction costs.
• To establish the existence and regularity of a corrector solution to a singular ergodic control problem in high dimensions.
• To develop a homogenization framework where the fast variable emerges from the limit problem, not from initial oscillations.
• To provide numerical evidence for the structure of optimal trading strategies in the multidimensional case.

Proposed method

• Applies the viscosity solution approach of Evans to homogenization, combined with relaxed limits from Barles and Perthame.
• Uses a change of variables that introduces a fast variable dependent on the limit problem's optimal strategy, rather than predefining oscillatory terms.
• Derives a system of corrector equations linked to a singular ergodic control problem, analogous to the monotone follower problem.
• Employs recent results by Hynd (2013, 2014) to establish existence and $C^{1,1}$ regularity of the solution to the eigenvalue problem.
• Applies comparison principles and inf-convolution techniques to analyze the free boundary and second-order regularity.
• Uses uniform $C^{1,1}$ estimates and compactness arguments to pass to the limit and construct the corrector in the asymptotic regime.

Experimental results

Research questions

• RQ1 How can the asymptotic expansion for the multidimensional optimal consumption-investment problem with small transaction costs be rigorously derived?
• RQ2 What is the structure of the first-order correction term in the presence of multiple risky assets and proportional transaction costs?
• RQ3 Can a corrector solution be constructed for the singular ergodic control problem that arises in the multidimensional setting?
• RQ4 How does the homogenization framework adapt when the fast variable is not pre-specified but emerges from the limit problem?
• RQ5 What are the regularity properties of the value function and its second-order derivatives in the asymptotic regime?

Key findings

• The first-order correction term in the asymptotic expansion is characterized by a singular ergodic control problem that arises from the homogenization process.
• A $C^{1,1}$ unique solution to the corrector equation exists and satisfies a precise growth condition, even without explicit solutions in higher dimensions.
• The second derivative of the corrector function is bounded almost everywhere and vanishes almost everywhere in the complement of the effective region.
• The value function's second-order regularity is preserved uniformly in the small-cost limit, enabling convergence in $C^1_{ ext{loc}}$.
• Numerical results illustrate that the first-order optimal controls exhibit a piecewise linear structure, consistent with the theoretical framework.
• The homogenization framework successfully handles non-periodic, non-oscillatory fast variables that emerge from the limit problem's structure.