Quadratic transportation inequalities for SDEs with measurable drift

Smoothness of the density for McKean-Vlasov SDEs with measurable kernel

Consider the McKean-Vlasov SDE $$ dX_t=\langle b(X_t-\cdot),μ_t\rangle dt+dW_t,\quad μ_t=\operatorname{Law}(X_t), $$ where $W$ is the $n$-dimensional Brownian motion and $b:\mathbb{R}^d\to\mathbb{R}^d$ is a measurable function. First assuming $b\in L^\infty$, we prove that the law $μ_t$ of $X_t$ has a density $p_t$ with respect to the Lebesgue measure, which is continuously differentiable with gradient being $γ$-Hölder continuous for each $γ\in(0,1)$. Assume further that $b\in \mathcal{C}_b^1$, we prove that the density $p_t$ is infinitely differentiable. In the regularization by noise perspec…

Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift

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.

Moment estimates and applications for SDEs driven by fractional Brownian motion with irregular drifts

In this paper, high-order moment, even exponential moment, estimates are established for the Hölder norm of solutions to stochastic differential equations driven by fractional Brownian motion whose drifts are measurable and have linear growth. As applications, we first study the weak uniqueness of solutions to fractional stochastic differential equations. Moreover, combining our estimates and the Fourier transform, we establish the existence of density of solutions to equations with irregular drifts.

Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations

In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in \cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non dens…

Euler scheme for density dependent stochastic differential equations

In this paper we show the existence and uniqueness for a class of density dependent SDEs with bounded measurable drift, where the existence part is based on Euler's approximation for density dependent SDEs and the uniqueness is based on the associated nonlinear Fokker-Planck equation. As an application, we obtain the well-posedness of a nonlinear Fokker-Planck equation.

Strong existence and uniqueness of solutions of SDEs with time dependent Kato class coefficients

Consider stochastic differential equations (SDEs) in $\Rd$: $dX_t=dW_t+b(t,X_t)\d t$, where $W$ is a Brownian motion, $b(\cdot, \cdot)$ is a measurable vector field. It is known that if $|b|^2(\cdot, \cdot)=|b|^2(\cdot)$ belongs to the Kato class $\K_{d,2}$, then there is a weak solution to the SDE. In this article we show that if $|b|^2$ belongs to the Kato class $\K_{d,\a}$ for some $\a \in (0,2)$ ($\a$ can be arbitrarily close to $2$), then there exists a unique strong solution to the stochastic differential equations, extending the results in the existing literature as demonstrated by exam…

On the Existence of Solutions of a Class of SDEs with Discontinuous Drift and Singular Diffusion

The classical result by Itô on the existence of strong solutions of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the case where the drift is only measurable and bounded. These generalizations are based on techniques presented by Zvonkin and Veretennikov, which rely on the uniform ellipticity of the diffusion coefficient. In this paper we study the case of degenerate ellipticity and give sufficient conditions for the existence of a solution. The conditions on the diffusion coefficient are more general than previous results and we gain fundamental insig…

On Weak Solutions of SDEs with Singular Time-Dependent Drift and Driven by Stable Processes

Let $d \ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dS_{t}+b(s+t, X_{t})dt, \quad X_{0}=x, \] where $(s,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$ is the initial starting point, $b: \mathbb{R}_+ \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is measurable, and $S=(S_{t})_{t \ge 0}$ is a $d$-dimensional $α$-stable process with index $α\in (1,2)$. We show that if the $α$-stable process $S$ is non-degenerate and $b \in L_{loc}^{\infty}(\mathbb{R}_{+};L^{\infty}(\mathbb{R}^{d}))+ L_{loc}^{q}(\mathbb{R}_{+};L^{p}(\mathbb{R}^{d}))$ for…

有界且可测测试函数下随机微分方程的高阶弱近似

High order weak approximation of Stochastic Differential Equations for bounded and measurable test functions

提出一种针对有界可测测试函数的随机微分方程高阶弱近似方法，通过在随机时间网格上评估数值格式并利用正则化性质，实现任意速率的收敛。

Strong solutions of some one-dimensional SDEs with random and unbounded drifts

In this paper, we are interested in the following one dimensional forward stochastic differential equation (SDE) \[ d X_{t}=b(t,X_{t},ω)d t +σd B_{t},\quad 0\leq t\leq T,\quad X_{0}=\,x\in \mathbb{R}, \] where the driving noise $B_{t}$ is a $d$-dimensional Brownian motion. The drift coefficient $b:[0,T] \timesΩ\times \mathbb{R}\longrightarrow \mathbb{R}$ is Borel measurable and can be decomposed into a deterministic and a random part, i.e., $b(t,x,ω) = b_1(t,x) + b_2(t,x,ω)$. Assuming that $b_1$ is of spacial linear growth and $b_2$ satisfies some integrability conditions, we obtain the existe…

Optimal pointwise approximation of SDEs based on brownian motion at discrete points

We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a measurable way) on a finite number of sequential observations of the driving Brownian motion. The resulting lower error bounds hold in particular for all methods that are implementable on a computer and use a random number generator to simulate the driving Brownian motion at finitely many points. Our analysis shows that approximation at a single point is strongly con…

On the Robust superhedging of measurable claims

The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by [9,11]. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics.

受控粗糙随机微分方程、路径随机控制与动态规划原理

Controlled rough SDEs, pathwise stochastic control and dynamic programming principles

本文研究粗糙随机微分方程的最优控制，建立路径动态规划原理与HJB稳定性，无需布朗统计，统一不同控制类别。

Antithetic multilevel particle system sampling method for McKean-Vlasov SDEs

Let $μ\in \mathcal{P}_2(\mathbb R^d)$, where $\mathcal{P}_2(\mathbb R^d)$ denotes the space of square integrable probability measures, and consider a Borel-measurable function $Φ:\mathcal P_2(\mathbb R^d)\rightarrow \mathbb R $. IIn this paper we develop Antithetic Monte Carlo estimator (A-MLMC) for $Φ(μ)$, which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws $μ^N = \frac1N \sum_{i=1}^{N}δ_{X_i}$, where a) $(X_i)_{i=1}^N$ is a sequence of i.i.d samples from $μ$ or b) $(X_i)_{i=1}^N$ is a system of interacting particles (diffusions)…

Criteria for the Finiteness of the Strong $p$-Variation for Lévy-type Processes

Using generalized Blumenthal--Getoor indices, we obtain criteria for the finiteness of the $p$-variation of Lévy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations (SDEs), certain Feller processes and solutions of Lévy driven SDEs. The class of processes is wider than in earlier contributions and using fine continuity we are able to handle general measurable subsets of $R^d$ as state spaces. Furthermore, in contrast to previous contributions on the subject, we introduce a local index in order to complement the upper index. …

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also pr…

Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms \[\mathbb{R}^d\ni x\quad\longmapsto\quadϕ_{s,t}(x)\in \mathbb{R}^d,\qquad s,t\in\mathbb{R}\] for a stochastic differential equation (SDE) of the form \[dX_t=b(t,X_t)\,dt+dB_t,\qquad s,t\in\mathbb{R},X_s=x\in\mathbb{R}^d.\] The above SDE is driven by a bounded measurable drift coefficient $b:\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}^d$ and a $d$-dimensional Brownian motion $B$. More specifically, we show that the stochastic flow $ϕ_{s,t}(\cdot)$ of the SDE lives in the space $L^2(Ω;W^{1,p}(\mathbb{…

On the stochastic Magnus expansion and its application to SPDEs

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time τ and provide a novel asymptotic estimate of the cumulative distribution function of t. As an application, we propose a new method for the numerical solution of stochastic partial differential equation…

关于退化伊藤随机微分方程在局部有界漂移下的点wisewell-posedness结果

Pointwise well-posedness results for degenerate Itô-SDEs with locally bounded drifts

本文基于前人研究，探讨了退化伊藤随机微分方程在局部有界漂移下的点wisewell-posedness问题，通过椭圆正则性结果和额外正则性假设，提供欧几里得空间中每个起始点的点wise分析，主要贡献是弱well-posedness，即弱存在性和唯一性。

Finitely additive mass transportation

Some classical mass transportation problems are investigated in a finitely additive setting. Let $Ω=\prod_{i=1}^nΩ_i$ and $\mathcal{A}=\otimes_{i=1}^n\mathcal{A}_i$, where $(Ω_i,\mathcal{A}_i,μ_i)$ is a ($σ$-additive) probability space for $i=1,\ldots,n$. Let $c:Ω\rightarrow [0,\infty]$ be an $\mathcal{A}$-measurable cost function. Let $M$ be the collection of finitely additive probabilities on $\mathcal{A}$ with marginals $μ_1,\ldots,μ_n$. If couplings are meant as elements of $M$, most classical results of mass transportation theory, including duality and attainability of the Kantorovich inf…

Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction

We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a multidimensional Lévy-driven stochastic differential equation and $f$ is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Lévy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions $f$ that are Lipschitz continuous with respect to the supremum nor…

带$L^{\infty}$向量场的双曲型随机偏微分方程解的光滑性

Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields

针对有界可测向量场下拟线性双曲型SPDE确定性形式不适定的问题，结合马利亚万微积分与推广的变分技术，首次证明其存在唯一马利亚万可微强解，还得到解对初值的索伯列夫可微性，推进了平面SDE领域研究。

具有非Lipschitz漂移的耦合前向-后向SDE的密度分析及其应用

Density Analysis for coupled forward-backward SDEs with non-Lipschitz drifts and Applications

本文研究了具有非Lipschitz漂移的耦合二次前向-后向SDEs解的密度存在性，提出基于弱解耦场和Itô-Tanaka技巧的框架，并应用于利率模型与不可交易资产的对冲问题。

具有可测系数和$L_d$-漂移的随机最优控制问题

Stochastic optimal control problems with measurable coefficients and $L_d$-drift

研究具有可测系数和$L_d$-漂移的受控SDEs，利用强Markov选择原理建立无限和有限时间区间下值函数的粘性解正则性及非线性半群性质，涵盖缺乏唯一性的框架。

具有 Hölder 漂移项的 SDE 的未调整 Langevin 算法

Unadjusted Langevin Algorithms for SDEs with Hoelder Drift

本文研究具有 Hölder 连续漂移项的 SDE，在耗散条件下分析未调整 Langevin/Euler-Maruyama 逼近，给出其同时逼近真实过程与唯一不变测度的显式 Wasserstein 收敛速率。

具有有界变分漂移的系统随机最优控制：最大原理方法

Stochastic Optimal Control for Systems with Drifts of Bounded Variation: A Maximum Principle Approach

本文研究非线性系统在随机微分方程中具有不规则漂移的控制问题，通过最大原理建立理论框架，并应用到保险盈余模型中，提出最优走廊型资本调整策略。