[Paper Review] Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift

Stéphane Menozzi, A. Pesce+1 more|arXiv (Cornell University)|Jun 12, 2020

Stochastic processes and financial applications25 references4 citations

TL;DR

This paper establishes two-sided Gaussian bounds and pointwise gradient estimates for the transition density and its derivatives up to order two of non-degenerate Itô stochastic differential equations (SDEs) with unbounded measurable drift and Hölder continuous diffusion coefficients. Using a parametrix method combined with a flow-based regularization to handle the unbounded drift, the authors derive estimates that reflect the transport of the initial condition by the drift through an auxiliary flow, extending classical Aronson-type bounds to the unbounded drift regime with explicit dependence on the drift's spatial regularity.

ABSTRACT

We consider non degenerate Brownian SDEs with H{ö}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{ö}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.

Motivation & Objective

• To extend classical Aronson-type two-sided density estimates to non-degenerate SDEs with unbounded measurable drift and Hölder continuous diffusion coefficients.
• To provide pointwise bounds on the first and second spatial derivatives of the transition density under minimal regularity assumptions on the drift.
• To incorporate the effect of the unbounded drift on the density by introducing a deterministic flow that transports the initial condition.
• To establish estimates that reflect the transport of the initial point by the drift, even when the drift is not bounded, using a regularized flow approach.
• To generalize existing results on gradient estimates for SDEs with bounded coefficients to the case of unbounded drift with linear growth and Hölder continuity.

Proposed method

• Applies the parametrix method to construct a series expansion for the transition density of the SDE with unbounded drift.
• Introduces a deterministic flow $\theta_t(x)$ associated with the drift $b$, which captures the transport of the initial condition $x$.
• Uses a regularized version of the flow to control the unboundedness of the drift in the parametrix kernel expansions.
• Employs Hölder continuity of the drift in space and boundedness away from zero of the diffusion coefficient to derive uniform estimates.
• Derives bounds on the density and its derivatives by controlling the iterated kernels in the parametrix expansion using time-space scaling and Gaussian-type estimates.
• Applies interpolation and gradient estimates on the heat kernel to control the Hölder continuity of the density and its derivatives in the spatial variable.

Experimental results

Research questions

• RQ1 Can two-sided Gaussian bounds for the transition density of a non-degenerate SDE be extended to the case of unbounded measurable drift with linear growth?
• RQ2 How do the first and second spatial derivatives of the density behave when the drift is unbounded but Hölder continuous in space?
• RQ3 To what extent does the transport of the initial condition by the drift influence the shape and decay of the transition density?
• RQ4 Can the parametrix method be adapted to handle unbounded drifts by incorporating a regularized flow?
• RQ5 What role does the Hölder continuity of the drift play in obtaining uniform gradient estimates for the density?

Key findings

• The transition density $p(0,x,t,y)$ satisfies two-sided Gaussian bounds of the form $C^{-1}g_{\lambda^{-1}}(t,\theta_t(x)-y) \leq p(0,x,t,y) \leq C g_\lambda(t,\theta_t(x)-y)$, where $\theta_t(x)$ is the flow associated with the drift.
• The first and second spatial derivatives of the density satisfy pointwise estimates $|\nabla^j_x p(0,x,t,y)| \lesssim t^{-j/2} g_\lambda(t,\theta_t(x)-y)$ for $j=1,2$, reflecting the transport by the drift flow.
• The constants $C$ and $\lambda$ depend on the non-degeneracy of the diffusion, the Hölder norm of the drift, the dimension $d$, and the maximal time interval, but not on the initial point $x$.
• The estimates are robust under regularization of the drift, and the flow $\theta_t(x)$ effectively shifts the center of the Gaussian density to account for the unbounded drift.
• The method successfully controls the parametrix expansion even when the drift is unbounded, by using the flow to absorb the drift's effect in the kernel estimates.
• The results generalize classical Aronson bounds to the unbounded drift case, with the key innovation being the explicit incorporation of the drift-induced flow in the Gaussian center.