[Paper Review] Skorokhod Embeddings via Stochastic Flows on the Space of Measures

Ronen Eldan|arXiv (Cornell University)|Mar 13, 2013

Stochastic processes and financial applications12 references3 citations

TL;DR

This paper introduces a novel stochastic flow-based construction of Skorokhod embeddings using measure-valued diffusions on Gaussian measures, enabling explicit moment and tail bounds for stopping times. The method yields a Markovian structure on conditional laws and provides tight sub-exponential tail estimates for log-concave target measures, with the stopping time's expectation equal to the variance of the target measure.

ABSTRACT

We present a new construction of a Skorohod embedding, namely, given a probability measure mu with zero expectation and finite variance, we construct an integrable stopping time T adapted to a filtration F_t, such that W_t has the law mu, where W_t is a standard Wiener process adapted to the same filtration. We find several sufficient conditions for the stopping time T to be bounded or to have a sub-exponential tail. In particular, our embedding seems rather natural for the case that mu is a log-concave measure and the tail behaviour of $T$ admits some tight bounds in that case. Our embedding admits the property that the stochastic measure-valued process {mu_t} (0

Motivation & Objective

• To develop a new construction of Skorokhod embeddings using stochastic flows on the space of Gaussian measures.
• To establish moment and tail bounds for the stopping time $ T_{\mu} $ under additional assumptions on the target measure $ \mu $.
• To show that the conditional law process $ \{\mu_t\} $, where $ \mu_t $ is the law of $ W_T $ given $ \mathcal{F}_t $, forms a Markov process.
• To unify existing embeddings (e.g., Azéma-Yor, Bass) under a general framework based on kernel-generated stochastic flows on measures.
• To provide a formula for analyzing $ T_{\mu} $'s behavior, particularly for log-concave measures, with sharp quantitative bounds.

Proposed method

• Construct a stochastic flow on the space of Gaussian measures via a system of SDEs driven by Brownian motion.
• Define the stopping time $ T_{\mu} $ as the first time a certain measure-valued process $ \mu_t $, derived from the flow, reaches a target measure $ \mu $.
• Use the inverse function theorem to define a function $ c_{\mu}(a,b) $ that maps the current position $ W_t $ and variance parameter $ b_t $ to a sufficient statistic for the target measure.
• Apply the Picard-Lindelöf theorem to establish existence and uniqueness of the flow parameters $ b_t $, $ c_t $ satisfying the SDE system.
• Apply Itô's formula to derive the SDE dynamics of $ c_t $, showing that the process satisfies the required embedding condition.
• Leverage the Markov property of the conditional measure process $ \{\mu_t\} $ to analyze the stopping time and derive tail bounds.

Experimental results

Research questions

• RQ1 Can a new Skorokhod embedding be constructed via stochastic flows on the space of Gaussian measures, with explicit control over the stopping time distribution?
• RQ2 What conditions ensure that the stopping time $ T_{\mu} $ has sub-exponential tails, particularly when $ \mu $ is log-concave?
• RQ3 Does the process $ \{\mu_t\} $, defined as the conditional law of $ W_T $ given $ \mathcal{F}_t $, form a Markov process with time-homogeneous transition kernels?
• RQ4 How does the proposed construction relate to classical embeddings such as those by Azéma-Yor and Bass?
• RQ5 What quantitative bounds can be derived for $ \mathbb{E}[T_{\mu}] $, $ \mathbb{P}(T_{\mu} > t) $, and higher moments under log-concavity?

Key findings

• The stopping time $ T_{\mu} $ satisfies $ \mathbb{E}[T_{\mu}] = \text{Var}[\mu] $, matching the second moment of the target measure.
• For log-concave $ \mu $ with zero mean and unit variance, $ \mathbb{P}(T_{\mu} > t) < C e^{-c t} $ for universal constants $ c, C > 0 $, indicating sub-exponential tail decay.
• The bound is tight up to constants, as shown by a lower bound $ \mathbb{P}(T_{\mu} > t) \geq C' e^{-c' t} $ for some $ C', c' > 0 $, using a union bound and Doob's inequality.
• The conditional law process $ \{\mu_t\} $ is a Markov process whose transition kernel does not depend on the initial measure $ \mu $, enabling a general framework for embeddings.
• The construction generalizes known solutions, including those by Azéma-Yor (1979) and Bass (1983), by embedding them within a flow-based framework on measure space.
• The method provides an explicit formula for analyzing $ T_{\mu} $, particularly effective for log-concave measures, due to the smoothness and convexity structure of the underlying density.