Uniqueness In Law Research Articles

A stochastic maximum principle for partially observed general mean-field control problems with only weak solution

In this paper we focus on a general type of mean-field stochastic control problem with partial observation, in which the coefficients depend in a non-linear way not only on the state process Xt and its control ut but also on the conditional law E[Xt|FtY] of the state process conditioned with respect to the past of observation process Y. We first deduce the well-posedness of the controlled system by showing weak existence and uniqueness in law. Neither supposing convexity of the control state space nor differentiability of the coefficients with respect to the control variable, we study Peng’s stochastic maximum principle for our control problem. The novelty and the difficulty of our work stem from the fact that, given an admissible control u, the solution of the associated control problem is only a weak one. This has as consequence that also the probability measure in the solution Pu=LTuQ depends on u and has a density LTu with respect to a reference measure Q. So characterizing an optimal control leads to the differentiation of non-linear functions f(Pu∘{EPu[Xt|FtY]}−1) with respect to (LTu,Xt). This has as consequence for the study of Peng’s maximum principle that we get a new type of first and second order variational equations and adjoint backward stochastic differential equations, all with new mean-field terms and with coefficients which are not Lipschitz. For their estimates and for those for the Taylor expansion new techniques have had to be introduced and rather technical results have had to be established. The necessary optimality condition we get extends Peng’s one with new, non-trivial terms.

Mutually interacting superprocesses with migration

AbstractA system of mutually interacting superprocesses with migration is constructed as the limit of a sequence of branching particle systems arising from population models. The uniqueness in law of the superprocesses is established using the pathwise uniqueness of a system of stochastic partial differential equations, which is satisfied by the corresponding system of distribution function-valued processes.

Spatial birth-and-death processes with a finite number of particles

The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on ${\mathbb{R}^{\mathrm{d}}}$. Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over ${\mathbb{R}^{\mathrm{d}}}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator. The pathwise behavior of an aggregation model is also studied. The probability of extinction and the growth rate of the number of particles under condition of nonextinction are estimated.

An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

We show uniqueness in law for the critical SPDE dXt=AXtdt+(−A)1/2F(X(t))dt+dWt,X0=x∈H, where A:dom(A)⊂H→H is a negative definite self-adjoint operator on a separable Hilbert space H having A−1 of trace class and W is a cylindrical Wiener process on H. Here, F:H→H can be continuous with, at most, linear growth (some functions F which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn–Hilliard-type equations which have interesting applications. To get weak uniqueness, we use an infinite dimensional localization principle and also establish a new optimal regularity result for the Kolmogorov equation λu−Lu=f associated to the SPDE when F=0 (λ>0, f:H→R Borel and bounded). In particular, we prove that the first derivative Du(x) belongs to dom((−A)1/2), for any x∈H, and supx∈H|(−A)1/2Du(x)|H=‖(−A)1/2Du‖0≤C‖f‖0.

Weak solutions and invariant measures of stochastic Oldroyd-B type model driven by jump noise

In this work, we consider sub-critical and critical models for viscoelastic flows driven by pure jump Lévy noise. Due to the elastic property, the noise in the equation for the stress tensor is considered in the Marcus canonical form. We investigate existence of a weak martingale solution for stochastic Oldroyd-B models, with full dissipation in whole of Rd,d=2,3. The key ingredients of the proof are classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Pathwise uniqueness, existence of a strong solution and uniqueness in law for the two-dimensional model are also shown. We also prove, in a Poincaré domain in two-dimensions, existence of an invariant measure using bw-Feller property of the associated Markov semigroup.

Uniqueness for nonlinear Fokker–Planck equations and weak uniqueness for McKean–Vlasov SDEs

One proves the uniqueness of distributional solutions to nonlinear Fokker–Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean–Vlasov stochastic differential equation (SDE).

Existence and regularity of infinitesimally invariant measures, transition functions and time-homogeneous Itô-SDEs

We show existence of an infinitesimally invariant measure $m$ for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some local integrability order. Subsequently, we derive regularity properties of the corresponding semigroup which is defined in $L^s(\mathbb{R}^d,m)$, $s\in [1,\infty]$, including the classical strong Feller property and classical irreducibility. This leads to a transition function of a Hunt process that is explicitly identified as a solution to an SDE. Further properties of this Hunt process, like non-explosion, moment inequalities, recurrence and transience, as well as ergodicity, including invariance and uniqueness of $m$, and uniqueness in law, can then be studied using the derived analytical tools and tools from generalized Dirichlet form theory.

Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients

We show uniqueness in law for a general class of stochastic differential equations in R d , d ≥ 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. Points of degeneracy have a d-dimensional Lebesgue–Borel measure zero. Weak existence is obtained for a more general, but not necessarily locally bounded drift coefficient.

Uniqueness in Law for Stable-Like Processes of Variable Order

Let $$d\ge 1$$ . Consider a stable-like operator of variable order $$\begin{aligned} {\mathcal {A}}f(x)&=\int _{{\mathbb {R}}^{d}\backslash \{0\}} \left[ f(x+h)-f(x)-\mathbf {1}_{\{|h|\le 1\}}h\cdot \nabla f(x)\right] n(x,h)|h|^{-d-\alpha (x)}\mathrm{d}h, \end{aligned}$$ where $$0<\inf _{x}\alpha (x)\le \sup _{x}\alpha (x)<2$$ and n(x, h) satisfies $$\begin{aligned} n(x,h)=n(x,-h),\quad 0<\kappa _{1}\le n(x,h)\le \kappa _{2}, \quad \forall x,h\in {\mathbb {R}}^{d}, \end{aligned}$$ with $$\kappa _{1}$$ and $$\kappa _{2}$$ being some positive constants. Under some further mild conditions on the functions n(x, h) and $$\alpha (x)$$ , we show the uniqueness of solutions to the martingale problem for $${\mathcal {A}}$$ .

Stochastic Cucker-Smale flocking dynamics of jump-type

We present a stochastic version of the Cucker-Smale flocking dynamics based on a markovian $N$-particle system of pair interactions with unbounded and, in general, non-Lipschitz continuous interaction potential. We establish the infinite particle limit $N \to \infty$ and identify the limit as a solution with a nonlinear martingale problem describing the law of a weak solution to a Vlasov-McKean stochastic equation with jumps. Moreover, we estimate the total variation and Wasserstein distance for the time-marginals from which uniqueness in the class of solutions having some finite exponential moments is deduced. Based on the uniqueness for the time-marginals we prove uniqueness in law for the Vlasov-McKean equation, i.e. we establish propagation of chaos.

On uniqueness of solutions to martingale problems — counterexamples and sufficient criteria

The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process in the sense of [4, 5, 11] is not necessarily determined by its generator if it has jumps. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.

Uniqueness of the nonlinear Schrödinger equation driven by jump processes

In a recent paper by the first two authors, existence of martingale solutions to a stochastic nonlinear Schrödinger equation driven by a Lévy noise was proved. In this paper, we prove pathwise uniqueness, uniqueness in law and existence of strong solutions to this problem using an abstract uniqueness result of Kurtz.

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

On étudie certaines EDS à sauts et les équations de Fokker–Planck (ou Kolmogorov progressives) correspondantes, qui sont des EDP non-locales. On suppose seulement que les coefficients sont mesurables et à croissance au plus linéaire. On montre que pour toute solution faible $(f_{t})_{t\in[0,T]}$ de l’EDP, il existe une solution faible à l’EDS, dont les lois marginales sont données par $(f_{t})_{t\in[0,T]}$. On en déduit que pour toute donnée initiale, l’existence pour l’EDP est équivalente à l’existence faible pour l’EDS, et que l’unicité en loi pour l’EDS implique l’unicité pour l’EDP. Nous étendons ainsi des idées de Figalli (J. Funct. Anal. 254 (2008) 109–153) concernant des EDS continues et des EDP locales.

Weak solutions for stochastic differential equations with additive fractional noise

In this paper, by using a transformation formula for fractional Brownian motion (fBm), we prove the existence of weak solutions to stochastic differential equations driven by an additive fBm with Hurst parameter [Formula: see text] under the linear growth condition. Furthermore, we also consider the uniqueness in law and the pathwise uniqueness of the weak solution.

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.

Invariance principle for non-homogeneous random walks

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in ${\mathbb R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X} $ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq 2$. To characterize $\mathcal{X} $, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in ${\mathbb R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X} $ and thus develop the excursion theory of $\mathcal{X} $ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X} $ in ${\mathbb R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X} $ is time-reversible. If so, the excursions of $\mathcal{X} $ in ${\mathbb R}^d$ generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions.

Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise

Existence theory of optimal relaxed control problem for a class of stochastic hereditary evolution equations driven by Lévy noise has been studied. We formulate the problem in the martingale sense of Stroock and Varadhan to establish existence of optimal controls. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod theorem for nonmetric spaces, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. As application of the abstract theory, Oldroyd and Jeffreys fluids have been studied and existence of optimal relaxed control is established. Existence and uniqueness of a strong solution and uniqueness in law for the two-dimensional Oldroyd and Jeffreys fluids are also shown.

Martingale problems for some degenerate Kolmogorov equations

We obtain Calderón–Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous nonlinear coefficients. We then derive the well-posedness of the martingale problem associated with related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.

Existence, uniqueness and the strong Markov property of solutions to Kimura diffusions with singular drift

Motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics, we study existence, uniqueness, and the strong Markov property of weak solutions to a class of degenerate stochastic differential equations. The stochastic differential equations considered in our article admit solutions supported in the set [ 0 , ∞ ) n × R m [0,\infty )^n\times \mathbb {R}^m , and they are degenerate in the sense that the diffusion matrix is not strictly elliptic, as the smallest eigenvalue converges to zero at a rate proportional to the distance to the boundary of the domain, and the drift coefficients are allowed to have power-type singularities in a neighborhood of the boundary of the domain. Under suitable regularity assumptions on the coefficients, we establish existence of solutions that satisfy the strong Markov property, and uniqueness in law in the class of Markov processes.

Weak solutions of mean-field stochastic differential equations

ABSTRACTIn this article, we study weak solutions of mean-field stochastic differential equations (SDEs), also known as McKean–Vlasov equations, whose drift , and diffusion coefficient depend not only on the state process Xs but also on its law. We suppose that b and σ are bounded and continuous in the state as well as the probability law; the continuity with respect to the probability law is understood in the sense of the 2-Wasserstein metric. Using the approach through a local martingale problem, we prove the existence and the uniqueness in law of the weak solution of mean-field SDEs. The uniqueness in law is obtained if the associated Cauchy problem possesses for all initial condition a classical solution. However, unlike the classical case, the Cauchy problem is a mean-field PDE as recently studied by Buckdahn et al. [arXiv:1407.1215, 2014]. In our approach, we also extend the Itô formula associated with mean-field problems given by Buckdahn et al. to a more general case of coefficients.