[Paper Review] Muckenhoupt's $(A_p)$ condition and the existence of the optimal martingale measure

Dmitry Kramkov, Kim Weston|RePEc: Research Papers in Economics|Jul 21, 2015

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes that the existence of a martingale measure whose density satisfies the probabilistic Muckenhoupt $(A_p)$ condition for $p = 1/(1-a)$, where $a \in (0,1)$ is a lower bound on relative risk aversion, is both sufficient and sharp for the dual optimizer in utility maximization to be a uniformly integrable martingale. The result ensures the existence of the optimal martingale measure $\widehat{\mathbb{Q}}$, which is crucial for pricing and hedging in incomplete markets.

ABSTRACT

In the problem of optimal investment with utility function defined on $(0,\infty)$, we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt $(A_p)$ condition for the power $p=1/(1-a)$, where $a\in (0,1)$ is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this $(A_p)$ condition is sharp.

Motivation & Objective

• To identify sufficient conditions under which the dual optimizer $\widehat{Y}$ in a utility maximization problem is a uniformly integrable martingale.
• To resolve the longstanding issue in mathematical finance where $\widehat{Y}$ may fail to be a martingale, even in continuous models.
• To show that the Muckenhoupt $(A_p)$ condition with $p = 1/(1-a)$ is sharp for ensuring $\widehat{Y}$ is of class $(\mathbf{D})$, a key step toward uniform integrability.
• To construct a counterexample demonstrating that the $A_p$ condition cannot be weakened for $\widehat{Y}$ to be of class $(\mathbf{D})$ even under power utility and continuous price processes.

Proposed method

• The authors introduce a key condition: the existence of a dual supermartingale $Z$ satisfying the probabilistic Muckenhoupt $(A_p)$ condition for $p = 1/(1-a)$, where $a$ bounds the relative risk aversion of the utility function.
• They prove in Theorem 5.1 that this $A_p$ condition on $Z$ implies $\widehat{Y}$ satisfies $A_{p'}$ for some $p' > 1$, which ensures $\widehat{Y}$ is of class $(\mathbf{D})$.
• The proof leverages connections between the $A_p$ condition and BMO martingales, using known results from harmonic analysis and stochastic analysis.
• A counterexample in Proposition 6.1 constructs a continuous diffusion model with power utility where the $A_p$ condition is violated and $\widehat{Y}$ fails to be of class $(\mathbf{D})$, proving sharpness.
• The construction uses a two-state jump process and a specific stochastic exponential $\mathcal{E}(L)$ to define the dual process $\widehat{Y}$, with explicit expressions for $\mathcal{E}(L)_T$ and $\widehat{Y}_T$.
• The counterexample shows $\mathbb{E}[\widehat{Y}_T] < 1$, proving $\widehat{Y}$ is not a uniformly integrable martingale, thus confirming the sharpness of the $A_p$ condition.

Experimental results

Research questions

• RQ1 Under what conditions is the dual optimizer $\widehat{Y}$ in a utility maximization problem a uniformly integrable martingale?
• RQ2 Is the Muckenhoupt $(A_p)$ condition with $p = 1/(1-a)$ necessary and sufficient for $\widehat{Y}$ to be of class $(\mathbf{D})$?
• RQ3 Can the $A_p$ condition be weakened without losing the uniform integrability of $\widehat{Y}$, even in continuous models with power utility?
• RQ4 Does the existence of an equivalent local martingale measure $\mathbb{Q}$ with density satisfying $A_p$ imply the existence of the optimal martingale measure $\widehat{\mathbb{Q}}$?
• RQ5 Is there a model where the $A_p$ condition fails and $\widehat{Y}$ is not of class $(\mathbf{D})$, even under continuous price processes and power utility?

Key findings

• The dual optimizer $\widehat{Y}$ is a uniformly integrable martingale if there exists a dual supermartingale $Z$ satisfying the Muckenhoupt $(A_p)$ condition with $p = 1/(1-a)$, where $a$ is a lower bound on relative risk aversion.
• This $A_p$ condition on $Z$ implies $\widehat{Y}$ satisfies $A_{p'}$ for some $p' > 1$, which ensures $\widehat{Y}$ is of class $(\mathbf{D})$, a necessary condition for uniform integrability.
• The paper constructs a counterexample with continuous stock prices and power utility where the $A_p$ condition fails and $\mathbb{E}[\widehat{Y}_T] < 1$, proving that $\widehat{Y}$ is not a uniformly integrable martingale.
• The counterexample confirms that the $A_p$ condition with $p = 1/(1-a)$ is sharp: no weaker condition suffices to ensure $\widehat{Y}$ is of class $(\mathbf{D})$.
• The dual minimizer $\widehat{Y}$ is explicitly constructed as $\widehat{Y} = \mathcal{E}(L)Z$, where $Z$ is the density of $\mathbb{Q}$ and $L$ is a stochastic exponential with a jump component.
• In the counterexample, $\mathbb{E}[\widehat{Y}_T] \leq 2^{-\delta} < 1$, which implies $\widehat{Y}$ is not a uniformly integrable martingale, thus demonstrating the sharpness of the $A_p$ condition.