N. V. Krylov, R. Liptser
Convergence of stochastic processes with jumps to diffusion processes is investigated in the case when the limit process has discontinuous coefficients. An example is given in which the diffusion approximation of a queueing model yields a diffusion process with discontinuous diffusion and drift coefficients.
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In this paper we study a stochastic differential equation driven by a fractional Brownian motion with a discontinuous coefficient. We also give an approximation to the solution of the equation. This is a first step to define a fractional version of the skew Brownian motion.
In this paper we study the strong convergence for the Euler-Maruyama approximation of a class of stochastic differential equations whose both drift and diffusion coefficients are possibly discontinuous.
We study one-dimensional stochastic differential equations of form $dX_t = σ(X_t)dY_t$, where $Y$ is a suitable Hölder continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac12$. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients $σ$ for which we assume very mild conditions. In particular, we allow $σ$ to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.
Mean field stochastic differential equations with a discontinuous diffusion coefficient
研究具有不连续扩散系数的平均场随机微分方程，证明在强漂移下存在唯一全局强解，并探讨全局解失效的典型情况。
In this work the existence of solutions of one-dimensional backward dou- bly stochastic differential equations (BDSDEs in short) where the coefficient is left-Lipschitz in y (may be discontinuous) and Lipschitz in z is studied. Also, the associated comparison theorem is obtained.
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the equation.
The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion.
In this paper we study jump-diffusion stochastic differential equations (SDEs) with a discontinuous drift coefficient and a possibly degenerate diffusion coefficient. Such SDEs appear in applications such as optimal control problems in energy markets. We prove existence and uniqueness of strong solutions. In addition we study the strong convergence order of the Euler-Maruyama scheme and recover the optimal rate $1/2$.
This paper deals with generalized backward doubly stochastic differential equations driven by a Lévy process (GBDSDEL, in short). Under left or right continuous and linear growth conditions, we prove the existence of minimal (resp. maximal) solutions.
We establish the large deviation principle for solutions of one-dimensional SDEs with discontinuous coefficients. The main statement is formulated in a form similar to the classical Wentzel--Freidlin theorem, but under the considerably weaker assumption that the coefficients have no discontinuities of the second kind.
We establish the existence and uniqueness for a one-dimensional stochastic differential equation driven by a Brownian motion and a pure jump {\levy} process. It is shown that under fairly general conditions on the coefficients, pathwise uniqueness holds based on the methods of weak uniqueness and local time technique.
The paper considers parabolic equations in non-divergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a solution for nonlinear parabolic or elliptic equations can fail, and there is no a priory estimate for partial derivatives of a solution. In this paper, existence and regularity results are obtained under some Cordes type restrictions on the coefficients. The results are applied to diffusion processes.
We consider a one-dimensional stochastic differential equations (SDE) with irregular coefficients. The purpose of this paper is to estimate the $L^p(Ω)$-difference of SDEs using the norm of the difference of coefficients, where the discontinuous drift coefficient satisfies a one-sided Lipschitz condition and the diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. As an application, we consider the stability problem.
We construct a stochastic flow generated by an SDE with Lévy noise and a drift coefficient being a function of bounded variation on R. It is proved that this flow is non-coalescing and Sobolev differentiable with respect to initial data. The representation for the derivative is given.
In a companion paper, the authors have characterized all deterministic semigroups, and all Markov semigroups, whose trajectories are Carathe'odory solutions to a given ODE x'=f(x), with f possibly discontinuous. The present paper establishes two approximation results. Namely, every deterministic semigroup can be obtained as the pointwise limit of the flows generated by a sequence of ODEs $x'=f_n(x) with smooth right hand sides. Moreover, every Markov semigroup can be obtained as limit of a sequence of diffusion processes with smooth drifts and with diffusion coefficients approaching zero.
We show the existence of Lévy-type stochastic processes in one space dimension with characteristic triplets that are either discontinuous at thresholds, or are stable-like with stability index functions for which the closures of the discontinuity sets are countable. For this purpose, we formulate the problem in terms of a Skorokhod-space martingale problem associated with non-local operators with discontinuous coefficients. These operators are approximated along a sequence of smooth non-local operators giving rise to Feller processes with uniformly controlled symbols. They converge uniformly o…
We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.
On mixed fractional SDEs with discontinuous drift coefficient
本文研究由标准与分数布朗运动共同驱动且漂移不连续的混合分数SDE，通过建立适用于导数绝对连续函数的广义Itô法则并证明分布密度存在，得到解的存在唯一性。
We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in Cesaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of "$L^p-$viscosity soluti…
Heavy traffic limit with discontinuous coefficients via a non-standard semimartingale decomposition
本文通过非标准半鞅分解研究了具有不连续系数的重交通极限问题，证明了单服务器队列在重交通下的弱极限由随机微分方程的解给出。
Subsurface flows are commonly modeled by advection-diffusion equations. Insufficient measurements or uncertain material procurement may be accounted for by random coefficients. To represent, for example, transitions in heterogeneous media, the parameters of the equation are spatially discontinuous. Specifically, a scenario with coupled advection- and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. For the numerical approximation of the solution, an adaptive, pathwise discretization scheme based on a Finite Element ap…
We show uniqueness in law for a general class of stochastic differential equations in $\mathbb{R}^d$, $d\ge 2$, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. The points of degeneracy have $d$-dimensional Lebesgue-Borel measure zero. Weak existence is obtained for more general, not necessarily locally bounded drift coefficient.
Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient
研究了具有不连续漂移和退化扩散系数的多维随机微分方程的欧拉-马尔可夫方法的强收敛性，证明了对于任意小的ε>0，收敛阶为1/4-ε，方法通过估计欧拉-马尔可夫方案与另一种数值方法之间的差异来实现。
Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontin…
A higher order approximation method for jump-diffusion SDEs with discontinuous drift coefficient
本文针对不连续漂移跳扩散 SDE 提出首个基于变换的跳跃适应拟 Milstein 高阶格式，并证明其 $L^p$ 收敛阶为 3/4，特殊情形可达 1 阶。
Strong solution of stochastic differential equations with discontinuous and unbounded coefficients
本文研究了漂移系数可能不连续且无界时随机微分方程强解的存在性与唯一性，通过结合 Zvonkin 变换与 Lyapunov 函数方法，并建立局部化偏微分方程联系，同时得出 Krylov 估计与稳定性的局部化结果。
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…
Strong order 1 adaptive approximation of jump-diffusion SDEs with discontinuous drift
本文提出了一种强阶1自适应逼近方案，用于解决具有不连续漂移的跳扩散SDEs问题，通过双适应策略提升收敛性。
In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form $E_x\int_0^τe^{-λs}f(X_s) ds$, $λ\ge0$ for one-dimensional diffusions $X$. We prove a general verification theorem which justifies the modified version of the free boundary problem. In th…
It is well known that a regular diffusion on an interval $I$ without killing inside is uniquely determined by a canonical scale function $s$ and a canonical speed measure $m$. Note that $s$ is a strictly increasing and continuous function and $m$ is a fully supported Radon measure on $I$. In this paper we will associate a general triple $(I,s,m)$, where $s$ is only assumed to be increasing and $m$ is not necessarily fully supported, to certain Markov processes by way of Dirichlet forms. A straightforward generalization of Dirichlet form associated to regular diffusion will be first put forward…
We study a second-order parabolic equation with divergence form elliptic operator, having piecewise constant diffusion coefficients with two points of discontinuity. Such partial differential equations appear in the modelization of diffusion phenomena in medium consisting of three kind of materials. Using probabilistic methods, we present an explicit expression of the fundamental solution under certain conditions. We also derive small-time asymptotic expansion of the PDE's solutions in the general case. The obtained results are directly usable in applications.
The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly modeled by random coefficients. For flows in heterogeneous\textbackslash fractured\textbackslash porous media, the coefficients are modeled as discontinuous random fields, where the interfaces along the stochastic discontinuities represent transitions in the media. More precisely, the random coefficient is given by the sum of a (continuous) Gaussian random field a…
In recent years, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space has begun. In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. For scalar SDEs of that type the best $L_p$-error rate known so far for approximation of the solution at the final time point is $3/4$ in terms of the number of evaluations of the drivin…
In this paper, we consider scalar stochastic differential equations (SDEs) with a superlinearly growing and piecewise continuous drift coefficient. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate 1/2 for all $p\in [1, +\infty)$ is recovered for the tamed Euler scheme. A numerical example is provided to support our conclusion.
In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently it has been shown in [Müller-Gronbach, T., and Yaroslavtseva, L., On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient, arXiv:1809.08423 (2018)] that for scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient the classical Euler-Maruyama…
An adaptive Euler-Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis
本文研究了具有不连续漂移系数和（可能）退化扩散系数的随机微分方程的强近似问题，提出了一种自适应步长策略的显式欧拉-马约拉塔方案，实现了平均计算成本下的强收敛阶为1/2（至对数项），并通过数值例子验证了理论结果。
On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient
本文研究了具有不连续漂移系数的随机微分方程的欧拉-马尔可夫方案的性能，证明了在特定条件下，该方案在$ L_p $-误差率上能达到至少$ 1/2 $，与具有利普希茨系数的SDEs相同。
Convergence of the tamed-Euler-Maruyama method for SDEs with discontinuous and polynomially growing drift
本文针对同时具有多项式增长和不连续漂移系数的SDE，证明了tamed-Euler-Maruyama数值格式具有1/2阶的强收敛性。
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