[Paper Review] Yet another notion of irregularity through small ball estimates

Marco Romito, Leonardo Tolomeo|arXiv (Cornell University)|Jul 6, 2022

Stochastic processes and financial applications4 citations

TL;DR

This paper introduces a new notion of path irregularity—Small Ball Estimate (SBE)—based on controlling the size of small balls via the occupation measure. It establishes that SBE implies Besov regularity of the occupation measure, extending prior results on regularization by noise to general Besov spaces, and shows that SBE holds for stochastic processes under simple local non-determinism conditions, including fractional Brownian motion with $ H < 1/d $. The framework enables pathwise solutions to SDEs with singular drifts in Besov spaces.

ABSTRACT

We introduce a new notion of irregularity of paths, in terms of control of growth of the size of small balls by means of the occupation measure of the path. This notion ensures Besov regularity of the occupation measure and thus extends the analysis of Catellier and Gubinelli (2016) to general Besov spaces. On stochastic processes this notion is granted by suitable properties of local non-determinism.

Motivation & Objective

• To develop a new notion of irregularity for paths that generalizes existing frameworks like $(\rho,\gamma)$-irregularity.
• To establish that SBE regularity implies Besov regularity of the occupation measure, extending results from Fourier-Lebesgue to Besov spaces.
• To show that SBE is invariant under bi-Lipschitz reparametrization and stable under regular perturbations.
• To prove that SBE holds for stochastic processes under simple local non-determinism conditions, particularly for fractional Brownian motion.
• To enable pathwise analysis of SDEs with singular drifts by leveraging SBE-based regularity of the occupation measure.

Proposed method

• Define the SBE condition as a control on the measure of small balls via the occupation measure of a path.
• Prove that SBE implies regularity of the occupation measure in Besov spaces, using maximal function estimates and interpolation techniques.
• Establish invariance of SBE under bi-Lipschitz reparametrization by controlling measure distortion under such maps.
• Demonstrate stability of SBE under $C^1$ perturbations using norm estimates and measure comparison.
• Use Malliavin calculus and moment estimates to derive bounds on the density of increments for stochastic processes.
• Apply these bounds to show SBE regularity for fractional Brownian motion and solutions of SDEs driven by it.

Experimental results

Research questions

• RQ1 Can a new notion of irregularity be defined that ensures Besov regularity of the occupation measure, beyond the Fourier-Lebesgue framework of [CG16]?
• RQ2 Is the SBE condition invariant under bi-Lipschitz reparametrization and stable under regular perturbations?
• RQ3 Under what conditions on the law of increments does a stochastic process satisfy the SBE condition?
• RQ4 How does SBE regularity enable pathwise solution of SDEs with singular drifts in Besov spaces?
• RQ5 What is the time regularity of the occupation measure of solutions to SDEs driven by fractional Brownian motion?

Key findings

• The SBE condition is equivalent to Besov regularity of the occupation measure, allowing extension of regularization by noise results to general Besov spaces.
• SBE is invariant under bi-Lipschitz reparametrization and stable under $C^1$ perturbations, ensuring robustness.
• For fractional Brownian motion with Hurst index $H < 1/d$, the occupation measure is in $C^{p\text{-var}}([a,T];\textsc{S\kern-1.19995pt\raise-1.29167pt\hbox{B}\kern-1.19995pt\hbox{E}}}^{\alpha,2}_{2})$ for $\alpha < \frac{1}{2H} - \frac{d}{2}$ and $p > 1 - \frac{1}{2}H(d + 2\alpha)$.
• For $d \leq 3$ and $H \in (\frac{1}{4},1)$ with $H < \frac{1}{d}$, the occupation measure of solutions to SDEs driven by fBm lies in $V^p([a,T];\textsc{S\kern-1.19995pt\raise-1.29167pt\hbox{B}\kern-1.19995pt\hbox{E}}}^{\alpha,2}_{2})$ for $p > 1 - \frac{1}{2}H(d + 2\alpha)$.
• The SBE condition is granted by local non-determinism, and for Gaussian processes, it reduces to a simple condition on the variance of increments.
• The framework enables pathwise existence and uniqueness for SDEs with drifts in Besov spaces, as shown in Corollary 7.1 for $f \in C^{r_2-\text{var}}(B^{\alpha_2}_{p_2,q_2})$ under suitable integrability and regularity conditions.