Pathwise Uniqueness Research Articles

Unique strong solutions of Lévy processes driven stochastic differential equations with discontinuous coefficients

ABSTRACTWe study the strong solutions for a class of one-dimensional stochastic differential equations driven by a Brownian motion and a pure jump Lévy process. Under fairly general conditions on the coefficients, we prove the pathwise uniqueness by showing the weak uniqueness and applying a local time technique.

Lattice Birth-and-Death Processes

Lattice birth-and-death Markov dynamics of particle systems with spins from the set of non-negative integers are constructed as unique solutions to certain stochastic equations. Pathwise uniqueness, strong existence, Markov property and joint uniqueness in law are proven, and a martingale characterization of the process is given. Sufficient conditions for the existence of an invariant distribution are formulated in terms of Lyapunov functions. We apply obtained results to discrete analogs of the Bolker--Pacala--Dieckmann--Law model and an aggregation model.

Pathwise Uniqueness of Non-uniformly Elliptic SDEs with Rough Coefficients

In this paper, we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is allowed to vanish on a set of positive measure and is not assumed to be smooth. As opposed to various existing results, our arguments are mainly based on the comparison theorem for local time and the occupation time formula. We apply our pathwise uniqueness results to derive strong existence and other properties of solutions for SDEs with rough coefficients.

Remark on pathwise uniqueness of stochastic differential equations driven by Lévy processes

Consider real-valued processes determined by stochastic differential equations driven by Lévy processes. The jump parts of the driving Lévy process are not always α-stable ones, nor symmetric ones. In the present article, we shall study the pathwise uniqueness of the solutions to the stochastic differential equations under the conditions on the coefficients that the diffusion and the jump terms are Hölder continuous, while the drift one is monotonic. Our approach is based on Gronwall’s inequality.

Periodic Measures of Mean-Field Stochastic Predator-Prey System

This paper discusses the dynamics of the mean-field stochastic predator-prey system. We prove the existence and pathwise uniqueness of the solution for stochastic predator-prey systems in the mean-field limit. Then we show that the solution of the mean-field equation is a periodic measure. Finally, we study the fluctuations of the periodic in distribution processes when the white noise converges to zero.

On the Euler–Maruyama scheme for spectrally one-sided Lévy driven SDEs with Hölder continuous coefficients

We study in this article the strong rate of convergence of the Euler–Maruyama scheme and associated with the jump-type equation introduced in Li and Mytnik (2011). We obtain the strong rate of convergence under similar assumptions for strong existence and pathwise uniqueness. Models of this type can be considered as a generalization of the CIR (Cox–Ingersoll–Ross) process with jumps.

Pathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noise

We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index $\gamma>1-\frac{\eta}{2(\eta+1)}$. Here $\eta\in(0,1)$ is a constant that defines the spatial regularity of the noise.

Martingale solutions for the stochastic nonlinear Schr\xf6dinger equation in the energy space

We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H^1 on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo–Galerkin-method based on the Littlewood–Paley-decomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions.

A fully discrete approximation of the one-dimensional stochastic heat equation

Abstract A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in $L^q(\varOmega )$, for all $q\geqslant 2$, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

On a doubly nonlinear PDE with stochastic perturbation

We consider a doubly nonlinear evolution equation with multiplicative noise and show existence and pathwise uniqueness of a strong solution. Using a semi-implicit time discretization we get approximate solutions with monotonicity arguments. We establish a-priori estimates for the approximate solutions and show tightness of the sequence of image measures induced by the sequence of approximate solutions. As a consequence of the theorems of Prokhorov and Skorokhod we get a.s. convergence of a subsequence on a new probability space which allows to show the existence of martingale solutions. Pathwise uniqueness is obtained by an $$L^1$$ -method. Using this result, we are able to show existence and uniqueness of strong solutions.

Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence, Feller and strong Feller properties, and exponential ergodicity

This paper considers multidimensional jump type stochastic differential equations with super linear and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient local non-Lipschitz conditions for pathwise uniqueness. The non-confluence property for solutions is investigated. Feller and strong Feller properties under local non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Lévy processes and presents a Feynman–Kac formula for Lévy type operators.

On approximations of the Euler-Peano scheme for multivalued stochastic differential equations

ABSTRACTIt is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. Krylov's conditions put forward in On Kolmogorov's equations for finite-dimensional diffusions, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math., 1715, Springer, Berlin, 1999, pp. 1–63). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.

Existence and pathwise uniqueness to an SPDE driven by [formula omitted]-stable colored noise

In this paper we study a stochastic partial differential equation (SPDE) with Hölder continuous coefficient driven by an α-stable colored noise. The pathwise uniqueness is proved by using a backward doubly stochastic differential equation backward (SDE) to take care of the Laplacian. The existence of solution is shown by considering the weak limit of a sequence of SDE system which is obtained by replacing the Laplacian operator in the SPDE by its discrete version. We also study an SDE system driven by Poisson random measures.

Uniqueness for measure-valued equations of nonlinear filtering for stochastic dynamical systems with Lévy noise

Abstract In the paper we study the Zakai and Kushner–Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner–Stratonovich equation is unique in joint law.

Pathwise uniqueness for stochastic evolution equations with Hölder drift and stable Lévy noise

We prove the pathwise uniqueness of solutions to stochastic evolution equations in Hilbert spaces with the $$\alpha $$ -stable Lévy noise and a bounded $$\beta $$ -Hölder continuous drift term. The proof is based on the regularity results of resolvent equations associated to Kolmogorov operators.

Degenerate SDEs with singular drift and applications to Heisenberg groups

By using the ultracontractivity of a reference diffusion semigroup, Krylov's estimate is established for a class of degenerate SDEs with singular drifts, which leads to existence and pathwise uniqueness by means of Zvonkin's transformation. The main result is applied to singular SDEs on generalized Heisenberg groups.

Jump stochastic differential equations with non-Lipschitz and superlinearly growing coefficients

In the paper, we consider the no-explosion condition and pathwise uniqueness for SDEs driven by a Poisson random measure with coefficients that are super-linear and non-Lipschitz. We give a comparison theorem in the one-dimensional case under some additional condition. Moreover, we study the non-contact property and the continuity with respect to the space variable of the stochastic flow. As an application, we will show that there exists a unique strong solution for SDEs with coefficients like .

Projections of spherical Brownian motion

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R} ^{n+\ell }$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R} ^{n+\ell }$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.

On the Pathwise Solutions to the Camassa--Holm Equation with Multiplicative Noise

In this paper we consider the Camassa--Holm (CH) equation with multiplicative noise, which can be obtained when the nonhydrostatic pressure in the deterministic equation is subject to a turbulent velocity field. For the periodic boundary value problem for this SPDE, we establish the local existence and pathwise uniqueness of the pathwise solution in Sobolev spaces $H^s$ with $s>3/2$. For the linear noise case, conditions that lead to the global existence and the blow-up in finite time of the solution, and their associated probabilities, are also acquired. Finally, we study the pathwise dissipative effect of the linear noise on the periodic peakons to the deterministic CH equation.

On the boundary of the support of super-Brownian motion

We study the density $X(t,x)$ of one-dimensional super-Brownian motion and find the asymptotic behaviour of $P(0<X(t,x)\le a)$ as $a\downarrow0$ as well as the Hausdorff dimension of the boundary of the support of $X(t,\cdot)$. The answers are in terms of the leading eigenvalue of the Ornstein–Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.