Xue Dong He, Zhaoli Jiang
In this paper, we prove a sufficient and necessary condition for the transition probability distribution of a general, time-inhomogeneous linear SDE to possess a density function and study the differentiability of the density function and the transition quantile function of the SDE. Moreover, we completely characterize the support of the marginal distribution of this SDE.
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This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the corresponding Kolmogorov equation whose coefficient is a distribution.
In this paper, we aim to study a stochastic process from a macro point of view, and thus periodic solution of a stochastic process in distributional sense is introduced. We first give the definition and then establish the existence of periodic solution on bounded domain. Lastly, for the case that probability density function exists, we obtain the existence periodic solutions of the probability density function corresponding to the stochastic process by using the technique of deterministic partial differential equations.
We consider the stochastic differential equation $$ dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $α$-stable Lévy processes, $α\in (1, 2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--Hölder space $\mathcal{C}^β$ for $β>1/2-α/2$.
Mean field stochastic differential equations with a diffusion coefficient with irregular distributional dependence
研究扩散系数不连续依赖分布的平均场随机微分方程，通过Lamperti型变换转化为确定性系数方程，证明存在唯一性并讨论失效情形。
We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^β_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $β-1/q\ge-1$, $β>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=δ_0$ which corresponds to the …
We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward pr…
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a H{ö}lder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes th…
We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition and blow-up times. We prove a global existence and uniqueness result assuming further properties on the non-li…
We review the formulation of the stochastic Burgers equation as a martingale problem. One way of understanding the difficulty in making sense of the equation is to note that it is a stochastic PDE with distributional drift, so we first review how to construct finite-dimensional diffusions with distributional drift. We then present the uniqueness result for the stationary martingale problem of [Gubinelli, Perkowski 2016], but we mainly emphasize the heuristic derivation and also we include a (very simple) extension of [Gubinelli, Perkowski 2016] to a non-stationary regime.
In this paper we present a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a rate of convergence in a suitable $L^1$-norm and we implement the scheme numerically. To the best of our knowledge this is the first paper to study (and implement) numerical solutions of SDEs whose drift lives in a space of distributions. As a byproduct we also obtain an estimate of the convergence rate for a numerical scheme applied to SDEs with drift…
We consider stochastic gradient estimation using only black-box function evaluations, where the function argument lies within a probability simplex. This problem is motivated from gradient-descent optimization procedures in multiple applications in distributionally robust analysis and inverse model calibration involving decision variables that are probability distributions. We are especially interested in obtaining gradient estimators where one or few sample observations or simulation runs apply simultaneously to all directions. Conventional zeroth-order gradient schemes such as simultaneous p…
Euler--Maruyama scheme for $α$-stable SDE with distributional drift
本文研究了由对称非退化α-稳定过程驱动的随机微分方程，分析了欧拉方案在有界漂移情况下的量化估计及负阶贝索空间漂移下的弱收敛速率。
Sensitivity Analysis of Distributionally Robust BSDEs and RBSDEs
研究在漂移参考测度不确定性下，分布鲁棒最优控制和最优停止问题中出现的倒向随机微分方程和反射倒向随机微分方程的敏感性，并给出显式公式。
Hess-Schrader-Uhlenbrock inequality for the heat semigroup on differential forms over Dirichlet spaces tamed by distributional curvature lower bounds
本文在驯化Dirichlet空间框架下，建立了1-形式的Hess-Schrader-Uhlenbrock不等式，推广了Braun在附加条件下的结果。
Distributionally-Robust Optimization with Noisy Data for Discrete Uncertainties Using Total Variation Distance
本文研究从含噪声数据推断分布的随机规划，提出基于总变差距离构建模糊集的分布鲁棒优化方法，证明其渐近一致性并转化为可处理的凸优化问题，适用于大规模随机问题。
Numerical approximation of the stochastic heat equation with a distributional reaction term
该研究针对带分布反应项的随机热方程，采用带软化漂移项的驯服欧拉有限差分格式进行数值逼近，基于Besov正则性得到$L^m(\Omega)$空间的收敛速率，并运用随机缝合技术推导离散Ornstein-Uhlenbeck过程的新正则性质。
Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts
该研究针对带迭代对数校正的临界分布漂移随机微分方程，通过光滑逼近构造弱解，结合多种估计证明律的唯一性，还得到确定初值解的时间边缘分布密度的双侧高斯估计。
We study the fundamental task of estimating the median of an underlying distribution from a finite number of samples, under pure differential privacy constraints. We focus on distributions satisfying the minimal assumption that they have a positive density at a small neighborhood around the median. In particular, the distribution is allowed to output unbounded values and is not required to have finite moments. We compute the exact, up-to-constant terms, statistical rate of estimation for the median by providing nearly-tight upper and lower bounds. Furthermore, we design a polynomial-time diffe…
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