[Paper Review] Stochastic equations with singular drift driven by fractional Brownian motion

Oleg Butkovsky, Khoa Lê+1 more|arXiv (Cornell University)|Feb 23, 2023

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes weak existence and strong well-posedness for stochastic differential equations with singular drift driven by fractional Brownian motion (fBm) using a novel stochastic sewing lemma. It proves optimal conditions: $\frac{d}{p} < \frac{1}{H} - 1$ for $L_p$ drifts and $H < \frac{1}{d+1}$ for Radon measure drifts, with a counter-example confirming optimality, extending Krylov-Röckner's classical result to the non-Markovian fBm setting.

ABSTRACT

We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$, $d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$ we show weak existence of solutions to this equation under the condition $$ \frac{d}p&lt;\frac1H-1, $$ which is an extension of the Krylov-Röckner condition (2005) to the fractional case. We construct a counter-example showing optimality of this condition. If $b$ is a Radon measure, particularly the delta measure, we prove weak existence of solutions to this equation under the optimal condition $H&lt;\frac1{d+1}$. We also show strong well-posedness of solutions to this equation under certain conditions. To establish these results, we utilize the stochastic sewing technique and develop a new version of the stochastic sewing lemma.

Motivation & Objective

• To extend the Krylov-Röckner condition for SDEs with Brownian motion to the case of fractional Brownian motion with Hurst parameter $H \in (0,1)$.
• To establish weak existence of solutions when the drift $b$ is in $L_p(\mathbb{R}^d)$ or a finite signed Radon measure, under optimal conditions.
• To prove optimality of the derived conditions via a counter-example construction.
• To develop a new version of the stochastic sewing lemma with relaxed moment assumptions, enabling analysis of non-semimartingale noise.
• To establish strong well-posedness in the one-dimensional case under suitable conditions.

Proposed method

• The authors introduce a new Rosenthal-type stochastic sewing lemma that relaxes assumptions on high-order moments of increment processes, enabling analysis of irregular fBm paths.
• They apply the stochastic sewing technique to control the convergence of approximating solutions to the SDE with singular drift.
• For $L_p$ drifts, the method relies on moment estimates and interpolation to derive the condition $\frac{d}{p} < \frac{1}{H} - 1$.
• For Radon measure drifts, including the Dirac delta, the method uses local time regularity and approximation via mollifiers to define the integral $\int b(X_t)dt$.
• A counter-example is constructed using a singular drift in $L_p$ with $\frac{d}{p} \geq \frac{1}{H} - 1$ to show non-existence of solutions, proving optimality.
• The proof of strong well-posedness in $d=1$ uses pathwise uniqueness and the new sewing lemma to control the drift's irregularity.

Experimental results

Research questions

• RQ1 What is the optimal condition on the drift $b$ and Hurst parameter $H$ for weak existence of solutions to SDEs driven by fractional Brownian motion?
• RQ2 Can the Krylov-Röckner condition for Brownian motion be extended to the fractional Brownian motion case, and if so, in what form?
• RQ3 Is the derived condition $\frac{d}{p} < \frac{1}{H} - 1$ for $L_p$ drifts optimal, and can this be proven via counter-example?
• RQ4 What is the optimal condition for weak existence when $b$ is a Radon measure, particularly the Dirac delta?
• RQ5 Can strong well-posedness be established in the one-dimensional case under the new framework?

Key findings

• Weak existence of solutions is established for $b \in L_p(\mathbb{R}^d)$ under the condition $\frac{d}{p} < \frac{1}{H} - 1$, which is shown to be optimal via a counter-example.
• For $b$ a finite signed Radon measure, weak existence holds under the optimal condition $H < \frac{1}{d+1}$, including for the Dirac delta measure.
• The paper constructs a counter-example showing that if $\frac{d}{p} \geq \frac{1}{H} - 1$, then no solution exists for some $b \in L_p$, proving the condition is sharp.
• A new stochastic sewing lemma is developed with relaxed moment assumptions, enabling analysis of non-semimartingale noise such as fBm.
• Strong well-posedness is proven for $d=1$ under certain conditions, extending pathwise uniqueness to the fBm setting.
• As a byproduct, the authors give a direct proof of the classical result that $W^H$ has a jointly continuous local time if $Hd < 1$, and extend it to $H(d+1) < 1$ for the sum of fBm and bounded variation processes.