We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustrate the abstract results, we consider large particle systems with mean-field interaction and common noise.
We derive Wasserstein distance bounds between the probability distributions of a stochastic integral (Itô) process with jumps $(X_t)_{t\in [0,T]}$ and a jump-diffusion process $(X^\ast_t)_{t\in [0,T]}$. Our bounds are expressed using the stochastic characteristics of $(X_t)_{t\in [0,T]}$ and the jump-diffusion coefficients of $(X^\ast_t)_{t\in [0,T]}$ evaluated in $X_t$, and apply in particular to the case of different jump characteristics. Our approach uses stochastic calculus arguments and $L^p$ integrability results for the flow of stochastic differential equations with jumps, without relyi…
Convergence rate to the stationary distribution for continuous-time Markov processes can be studied using Lyapunov functions. Recent work by the author provided explicit rates of convergence in special case of a reflected jump-diffusion on a half-line. These results are proved for total variation distance and its generalizations: measure distances defined by test functions regardless of their continuity. Here we prove similar results for Wasserstein distance, convergence in which is related to convergence for continuou test functions. In some cases, including the reflected Ornstein-Uhlenbeck p…
This work is devoted to the Lipschitz contraction and the long time behavior of certain Markov processes. These processes diffuse and jump. They can represent some natural phenomena like size of cell or data transmission over the Internet. Using a Feynman-Kac semigroup, we prove a bound in Wasserstein metric. This bound is explicit and optimal in the sense of Wasserstein curvature. This notion of curvature is relatively close to the notion of (coarse) Ricci curvature or spectral gap. Several consequences and examples are developed, including an $L^2$ spectral for general Markov processes, expl…
An efficient Wasserstein-distance approach for reconstructing jump-diffusion processes using parameterized neural networks
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Department of Computational Medicine, UCLA（加州大学洛杉矶分校计算医学系）；Nuffield Department of Medicine, University of Oxford（牛津大学纳菲尔德医学院）；Department of Mathematics, UCLA（加州大学洛杉矶分校数学系）
Consider on a manifold the solution $X$ of a stochastic differential equation driven by a Lévy process without Brownian part. Sufficient conditions for the smoothness of the law of $X_t$ are given, with particular emphasis on noncompact manifolds. The result is deduced from the case of affine spaces by means of a localisation technique. The particular cases of Lie groups and homogeneous spaces are discussed.
In his 1972 paper, John Lamperti characterized all positive self-similar Markov processes as time-changes of exponentials of Levy processes. In the past decade the problem of classifying all non-negative self-similar Markov processes that do not necessarily have zero as a trap has been solved gradually via connections to ladder height processes and excursion theory. Motivated by a recent article of Chaumont, Rivero, Panti we classify via jump-type SDEs the symmetric real-valued self-similar Markov processes that only decrease the absolute value by jumps and leave zero continuously. Our constru…
We establish the exponential convergence with respect to the $L^1$-Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equation (SDE) $$d X_t=d Z_t+b(X_t)\,d t,$$ where $(Z_t)_{t\ge0}$ is a pure jump Lévy process whose Lévy measure $ν$ fulfills $$ \inf_{x\in \R^d, |x|\le κ_0} [ν\wedge (δ_x \ast ν)]( \R^d)>0$$ for some constant $κ_0>0$, and the drift term $b$ satisfies that for any $x,y\in \R^d$, $$\langle b(x)-b(y),x-y\rangle\le \begin{cases} Φ_1(|x-y|)|x-y|,& |x-y|\le l_0; -K_2|x-y|^2,& |x-y|> l_0 \end{cases}$$ with some positive constan…
We introduce a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction on the real line as mathematically rigorous description of colloidal motion of fluids. The associated measure-valued process provides a weak solution to a corrected Dean-Kawasaki equation for supercooled liquids without dissipation. Our construction is based on the introduction and analysis of a fundamentally new family of equilibrium measures for the associated dynamics and their Dirichlet forms. We identify the intrinsic metric as the quadrat…
This work is devoted to the study of conservative affine processes on the canonical state space $D = $R_+^m \times \R^n$, where $m + n > 0$. We show that each affine process can be obtained as the pathwise unique strong solution to a stochastic equation driven by Brownian motions and Poisson random measures. Then we study the long-time behavior of affine processes, i.e., we show that under first moment condition on the state-dependent and log-moment conditions on the state-independent jump measures, respectively, each subcritical affine process is exponentially ergodic in a suitably chosen Was…
We give a bare-hands approach to the martingale representation theorem for integer valued random measures, which allows for a wide class of infinite activity jump processes, as well as all processes with well-ordered jumps.
Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves
Institute of Mathematics TU Berlin（柏林工业大学数学研究所）；Faculty of Mathematics Bielefeld University（比勒费尔德大学数学系）；Department of Mathematics and Statistics University of Jyväskylä（于韦斯屈莱大学数学与统计系）
Extending previous work [arXiv:1408.0628] by the first author we present a variant of the Arratia flow, which consists of a collection of coalescing Brownian motions starting from every point of the unit interval. The important new feature of the model is that individual particles carry mass which aggregates upon coalescence and which scales the diffusivity of each particle in an inverse proportional way. In this work we relate the induced measure valued process to the Wasserstein diffusion of [arXiv:0704.0704]. First, we present the process as a martingale solution to a SPDE similar to [arXiv…
After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related with the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metri…
We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean--Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution.
This paper is devoted to solving a real valued backward stochastic differential equation with jumps where the time horizon may be finite or infinite. Under linear growth generator, we prove existence of a minimal solution. Using a comparison theorem we show existence and uniqueness of solution to such equations when the generator is uniformly continuous and satisfies a weakly monotonic condition.
We study the quantization problem for certain types of jump processes. The probabilities for the number of jumps are assumed to be bounded by Poisson weights. Otherwise, jump positions and increments can be rather generally distributed and correlated. We show in particular that in many cases entropy coding error and quantization error have distinct rates. Finally, we investigate the quantization problem for the special case of $\mathbb{R}^d$-valued compound Poisson processes.
This work concerns a type of coupled McKean-Vlasov stochastic differential equations (MVSDEs in short) with jumps. First, we prove superposition principles for these coupled MVSDEs with jumps and non-local space-distribution dependent Fokker-Planck equations. Since superposition principles are related to the well-posedness of weak solutions for coupled MVSDEs with jumps, then we give some conditions to assure it. After this, we construct space-distribution valued Markov processes associated with these coupled MVSDEs with jumps. Finally, the ergodicity of these coupled MVSDEs with jumps are inv…
We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general Lévy processes in this context.
Given two n-dimensional measures $μ$ and $ν$ on Polish spaces, we propose an optimal transportation's formulation, inspired by classical Kan-torovitch's formulation in the scalar case. In particular, we established a strong duality result and as a consequence, optimality conditions are investigated. Wasserstein's metrics induced by our formulation are also investigated.
In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple reproduction events. We construct families of probability measures on the lookdown space and on an extension of it that allows to include the case with dust. From this construction, we read off the tree-valued $Ξ$-Fleming-Viot processes and deduce path properties. For instance, these processes usually have a.s. càdlàg paths with jumps at the times of large reproduction events. In the case of comin…
We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solution in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution an…
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation.…
This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process $A$ such that $[N,A] = 0$, for any continuous local martingale $N$. Given a function $u:[0,T] \times \mathbb{R} \to \mathbb{R}$, which is of class $C^{0,1}$ (or sometimes less), we provide a chain rule type expansion for $u(t,X_t)$ which stands in applications for a chain Itô type rule.
We establish existence of exponential moments and the validity of the affine transform formula for affine jump-diffusions with a general closed convex state space. This extends known results for affine jump-diffusions with a canonical state space. The key step is to prove the martingale property of an exponential local martingale, using the well-posedness of the associated martingale problem. By analytic extension we obtain the affine transform formula for complex exponentials, in particular for the characteristic function. Our results apply to a wide class of affine processes, including those…
In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the main result: the integral of a set-valued stochastic process with respect to the compensated Poisson measure is a set-valued submartingale but not a martingale unless the integrand degenerates into a single-valued process. Secondly we study the strong solution to the set-valued stochastic integral equation, which includes a set-valued drift, a single-valued…
We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin's Brownian motion on the homeomorphism group. The…
Consider a system $X = ((x_ξ(t)), ξ\in Ω_N)_{t \geq 0}$ of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space $(\CP(\I))^{Ω_N}$, where $\I$ is the type space, ${Ω_N}$ the geographic space is assumed to be a countable group and $\CP$ denotes the probability measures. We establish various duality relations for this process. These dualities are function-valued processes which are driven by a coalescing-branching random walk, that is, an evolving particle system which in addition exhibits certain changes in the fun…
In this paper, we show that the abstract framework developed in Pages & Rey (2017) and inspired by Lamberton & Pages (2002) can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test f…
We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness of generalized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutions for second order integro-differential equation of Fokker-Planck type.
We propose of an improved version of the ubiquitous symmetrization inequality making use of the Wasserstein distance between a measure and its reflection in order to quantify the symmetry of the given measure. An empirical bound on this asymmetric correction term is derived through a bootstrap procedure and shown to give tighter results in practical settings than the original uncorrected inequality. Lastly, a wide range of applications are detailed including testing for data symmetry, constructing nonasymptotic high dimensional confidence sets, bounding the variance of an empirical process, an…
This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685--712]. A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach is based on the entropy techniques developed in the modern theory of empirical processes.
By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe \textit{rigorously} the representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} Φ\circ | ΔX_s |$, where $X : Ω\times \R_+ \longrightarrow \R$ is a stochastic process with regulated trajectories, $t \in \R_+$ and $Φ: \R_+ \longrightarrow \R_+$ is a strictly increasing function which maps 0 to 0 (cf. P…
This letter derives some new exponential bounds for discrete time, real valued, conditionally symmetric martingales with bounded jumps. The new bounds are extended to conditionally symmetric sub/ supermartingales, and they are compared to some existing bounds.
In this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to sup…
This paper is concerned with optimal control of systems driven by G-stochastic differential equations (G-SDEs), with controlled jump term. We study the relaxed problem, in which admissible controls are measurevalued processes and the state variable is governed by an G-SDE driven by a counting measure valued process called relaxed Poisson measure such that the compensator is a product measure. Under some conditions on the coefficients, using the G-chattering lemma, we show that the strict and the relaxed control problems have the same value function. Additionally, we derive a maximum principle …
Jump inequalities are the $r=2$ endpoint of Lépingle's inequality for $r$-variation of martingales. Extending earlier work by Pisier and Xu we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $\mathbb{R}^{d}$ to $\mathbb{Z}^{d}$ for jump estimates for Fourier multipliers.
