In this paper we show that a path-wise solution to the following integral equation $$ Y_t = \int_0^t f(Y_t) dX_t \qquad Y_0=a \in \R^d $$ exists under the assumption that X_t is a Lévy process of finite p-variation for some $p \geq1$ and that f is an $α$-Lipschitz function for some alpha>p. There are two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric the other forward. The geometric solution is obtained by adding fictitious time and solving an associated integral equation. The forward solution is derived from the geometric solution by correcting the solution's jump behaviour. Lévy processes, generally, have unbounded variation. So we must use a pathwise integral different from the Lebesgue-Stieltjes integral. When X has finite p-variation almost surely for p<2 we use Young's integral. This is defined whenever f and g have finite p and q-variation for 1/p+1/q>1 (and they have no common discontinuities). When p>2 we use the integral of Lyons. In order to use this integral we construct the Lévy area of the Lévy process and show that it has finite (p/2)-variation almost surely.
We provide a support theorem for the law of the solution to an SDE with jump noise. This theorem applies to general SDEs with jumps and is illustrated by examples of SDEs with quite degenerate jump noises where the theorem leads to an informative description of the support.
We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results th…
By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived.
We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to $α$-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the $L^1$-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which is derived f…
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 describe a class of explicit invariant measures for both finite and infinite dimensional Stochastic Differential Equations (SDE) driven by Lévy noise. We first discuss in details the finite dimensional case with a linear, resp. non linear, drift. In particular, we exhibit a class of such SDEs for which the invariant measures are given in explicit form, coherently in all dimensions. We then indicate how to relate them to invariant measures for SDEs on separable Hilbert spaces.
The paper studies the rate of convergence of the weak Euler approximation for solutions to SDEs driven by Levy processes, with Hoelder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. The equation considered has a non-degenerate main part driven by a spherically-symmetric stable process.
Consider the following stochastic differential equation (SDE) $$dX_t = b(t,X_{t-}) \, dt+ dL_t, \quad X_0 = x,$$ driven by a $d$-dimensional Lévy process $(L_t)_{t \geq 0}$. We establish conditions on the Lévy process and the drift coefficient $b$ such that the Euler--Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of $b$ and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples…
In this paper, an integration by parts formula was derived for jump processes on Hilbert spaces. Using this formula, we investigated derivative formula and exponential ergodicity for nonlinear SPDEs driven by purely jump processes.
We describe an Euler scheme to approximate solutions of Lévy driven Stochastic Differential Equations (SDE) where the grid points are random and given by the arrival times of a Poisson process. This result extends a previous work of the authors in Ferreiro-Castilla et al. (2012). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and proof that the approximation converges in mean square error with rate $\mathcal{O}(n^{-1/2})$. The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the S…
We study the rate of convergence of some recursive procedures based on some "exact" or "approximate" Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by exact and approximate Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.
We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive Lévy noise.
In this paper we study the convergence of solutions for (possibly degenerate) stochastic differential equations driven by Lévy processes, when the coefficients converge in some appropriate sense. First, we prove, by means of a superposition principle, a limit theorem of stochastic differential equations driven by Lévy processes. Then we apply the result to a type of nonlinear filtering problems and obtain the convergence of the nonlinear filterings.
Let $(L_t)_{t \geq 0}$ be a $k$-dimensional Lévy process and $σ: \mathbb{R}^d \to \mathbb{R}^{d \times k}$ a continuous function such that the Lévy-driven stochastic differential equation (SDE) $$dX_t = σ(X_{t-}) \, dL_t, \qquad X_0 \sim μ$$ has a unique weak solution. We show that the solution is a Feller process whose domain of the generator contains the smooth functions with compact support if, and only if, the Lévy measure $ν$ of the driving Lévy process $(L_t)_{t \geq 0}$ satisfies $$ν(\{y \in \mathbb{R}^k; |σ(x)y+x|<r\}) \xrightarrow[]{|x| \to \infty} 0.$$
For solutions $X=(X_t)_{t\in[0,T]}$ of Lévy-driven Marcus stochastic differential equations we study the Wong--Zakai type time discrete approximations $\bar X=(\bar X_{kh})_{0\leq k\leq T/h}$, $h>0$, and establish the first order convergence $|E f(X_T)-E f(X^h_T)|\leq C h$ for $f\in C_b^4$.
We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations
The paper studies the rate of convergence of the weak Euler approximation for solutions to possibly completely degenerate SDEs driven by Levy processes, with Hoelder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. In addition, the rate robustness to the approximation of the increments of the driving process is studied and approximate Euler scheme considered. A rate of convergence is derived for an approximate jump-adapted scheme as well.
In this paper, a class of reflected generalized backward doubly stochastic differential equations (reflected GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of reflected stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.
We propose new jump-adapted weak approximation schemes for stochastic differential equations driven by pure-jump Lévy processes. The idea is to replace the driving Lévy process $Z$ with a finite intensity process which has the same Lévy measure outside a neighborhood of zero and matches a given number of moments of $Z$. By matching 3 moments we construct a scheme which works for all Lévy measures and is superior to the existing approaches both in terms of convergence rates and easiness of implementation. In the case of Lévy processes with stable-like behavior of small jumps, we construct schem…
We consider a class of Lévy-driven stochastic differential equations (SDEs) with McKean-Vlasov (MK-V) interaction in the drift coefficient. It is assumed that the coefficient is bounded, affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration scheme that app…
Recent developments on financial markets have revealed the limits of Brownian motion pricing models when they are applied to actual markets. Lévy processes, that admit jumps over time, have been found more useful for applications. Thus, we suggest a Lévy model based on Forward-Backward Stochastic Differential Equations (FBSDEs) for option pricing in a Lévy-type market. We show the existence and uniqueness of a solution to FBSDEs driven by a Lévy process. This result is important from the mathematical point of view, and also, provides a much more realistic approach to option pricing.
We study a time-inhomogeneous SDE in $\R^d$ driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit `principal part' and a `residual part', subject to certain $L^\infty(dx)\otimes L^1(dy)$ and $L^\infty(dx)\otimes L^\inft…
In this article we show that a finite dimensional stochastic differential equation driven by a Lévy process can be formulated as a stochastic partial differential equation. We prove the existence and uniqueness of strong solutions of such stochastic PDEs. The solutions that we construct have the `translation invariance' property. The special case of this correspondence for diffusion processes was proved in [Rajeev, Translation invariant diffusion in the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013), no.~2, 231--258].
By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX_t=a_θ(X_t)dt + dZ_t with a tempered α-stable process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.
In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with Lévy process (RGBDSDELs, in short) with one continuous barrier. Under uniformly Lipschitz coefficients, we prove existence and uniqueness result by means of the penalization method and the fixed point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs, in short) with a nonlinear Neumann boundary condition.
In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.
It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coeffcients determined by the corresponding SDE. Time-fractional Kolmogorov type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distribut…
In this paper we study the asymptotic properties of the power variations of stochastic processes of the type X=Y+L, where L is an alpha-stable Levy process, and Y a perturbation which satisfies some mild Lipschitz continuity assumptions. We establish local functional limit theorems for the power variation processes of X. In case X is a solution of a stochastic differential equation driven by L, these limit theorems provide estimators of the stability index alpha. They are applicable for instance to model fitting problems for paleo-climatic temperature time series taken from the Greenland ice c…
In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a Lévy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Lévy process with a smooth but unbounded Lévy measure, we develop a stochastic cal…
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…
Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present a general framework based on semigroup expansions for the construction of higher-order discretization schemes and analyze its rate of convergence. We also apply it to approximate general Lévy driven stochastic differential equations.
We stu\dd y a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. Our work is divided in two parts. In the present part I we first define a Hilbert-Banach setting in which we can prove existence and uniqueness of solutions under general assumptions on the drift and the Lévy noise. We then prove a decomposition of the solution process in a stationary component and a component which vanishes asymptotically for large times in the $L^p-$sense, $p\geq1$. The law of the stationary component is identified with the unique invariant proba…
Quantitative Asymptotics for Time-Inhomogeneous Lévy-Driven SDEs with Asymptotically Vanishing Drifts
Center for Applied Mathematics, Tianjin University（天津大学应用数学中心）；School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education) & Fujian Provincial Key Laboratory of Statistics and Artificial Intelligence, Fujian Normal University（福建师范大学数学与统计学院与分析数学及应用教育部重点实验室及福建省统计与人工智能省级重点实验室）
We consider solutions of Lévy-driven stochastic differential equations of the form $\mathrm{d} X_t=σ(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $σ$ is twice continuously differentiable and maximal of linear growth and the driving Lévy process $L=(L_t)_{t\geq0}$ is either vector or matrix-valued. While the almost sure short-time behavior of Lévy processes is well-known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the process $X$. Using methods from stochastic calculus, we derive limiting results for stochastic …
We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by $α$-stable processes which arise as the limiting equations of multiclass queueing models in the Halfin-Whitt regime that have heavy-tailed arrival processes. When the safety staffing parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence to the invariant probability measure in total variation, which is uniform over all stationary Markov controls resulting in a locally Lipschitz continuous drift. We also derive a matching lower bound …
In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$ d X_{t}= σ_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d, $$ where $Z$ is a $d$-dimensioanl nondegenerate $α$-stable-like process with $α\in(0,2)$ (including cylindrical case), and uniform in $t\geq 0$, $x\mapsto σ_t(x): {\bf R}^d\to {\bf R}^d\otimes {\bf R}^d$ is Lipchitz and uniformly elliptic and $x\mapsto b_t (x)$ is $β$-order Hölder continuous with $β\in(1-α/2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $…
We consider an SDE in R^m of the type dX(t)=a(X(t))dt+dU(t) with a Lévy process U and study the problem for the distribution of a solution to be regular in various senses. We do not impose any specific conditions on the Lévy measure of the noise, and this is the main difference between our method and the known methods by J.Bismut or J.Picard. The main tool in our approach is the stochastic calculus of variations for a Lévy process, based on the time-stretching transformations of the trajectories. Three problems are solved in this framework. First, we prove that if the drift coefficient a is no…
Motivated by the multilevel Monte Carlo method introduced by Giles [5], we study the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are respectively Euler approximations with time steps $1/n$ and $1/nm$ of a given stochastic differential equation $X$ driven by a pure jump Lévy process. In this paper, we prove that this normalized multilevel error converges to different non-trivial limiting processes with various sharp rates $u_{n,m}$ depending on the behavior of the Lévy measure around zero. Our results are consistent with those of Jacod [9] …
We present a new algorithms to discretize a decoupled forward backward stochastic differential equations driven by pure jump Lévy process (FBSDEL in short). The method is built in two steps. Firstly, we approximate the FBSDEL by a forward backward stochastic differential equations driven by a Brownian motion and Poisson process (FBSDEBP in short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps $\eps$ goes to 0. In the second step, we obtain the $L^p$ Hölder continuity of the solution of FBSDEBP and we c…
In this article, we are interested in the strong well-posedness together with the numerical approximation of some one-dimensional stochastic differential equations with a non-linear drift, in the sense of McKean-Vlasov, driven by a spectrally-positive L{é}vy process and a Brownian motion. We provide criteria for the existence of strong solutions under non-Lipschitz conditions of Yamada-Watanabe type without non-degeneracy assumption. The strong convergence rate of the propagation of chaos for the associated particle system and of the corresponding Euler-Maruyama scheme are also investigated. I…
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
In this paper, we first show the well-posedness of the SDEs driven by Lévy noises under mild conditions. Then, we consider the existence and uniqueness of periodic solutions of the SDEs. To establish the ergodicity and uniqueness of periodic solutions, we investigate the strong Feller property and the irreducibility of the corresponding time-inhomogeneous semigroups when both small and large jumps are allowed in the equations. Some examples are presented to illustrate our results.
In this article, we study the effects of the propagation of a non-degenerate Lévy noise through a chain of deterministic differential equations whose coefficients are Hölder continuous and satisfy a weak Hörmander-like condition. In particular, we assume some non-degeneracy with respect to the components which transmit the noise. Moreover, we characterize, for some specific dynamics, through suitable counterexamples , the almost sharp regularity exponents that ensure the weak well-posedness for the associated SDE. As a by-product of our approach, we also derive some Krylov-type estimates for t…
SDE driven by an $α$-stable process, $α\in \lbrack 1,2),$ with Lipshitz continuous coefficient and $β$-Hölder drift is considered. The existence and uniqueness of a strong solution is proved when $β>1-α/2$ by showing that it is $L_{p}$-limit of Euler approximations. The $L_{p}$-error (rate of convergence) is obtained for a nondegenerate truncated and nontruncated driving process. The rate in the case of Lipshitz continuous coefficients is derived as well.
Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Levy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker-Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker-Plank equations for Marcus SDEs i…
The existence and uniqueness are established for McKean-Vlasov SDEs driven by Lévy processes. By using an approximation technique and coupling by change of measures, Harnack inequalities are investigated for McKean-Vlasov SDEs driven by subordinate Brownian motions.
We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a multidimensional Lévy-driven stochastic differential equation and $f$ is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Lévy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions $f$ that are Lipschitz continuous with respect to the supremum nor…
In this article, we consider multilevel Monte Carlo for the numerical computation of expectations for stochastic differential equations driven by Lévy processes. The underlying numerical schemes are based on jump-adapted Euler schemes. We prove stable convergence of an idealised scheme. Further, we deduce limit theorems for certain classes of functionals depending on the whole trajectory of the process. In particular, we allow dependence on marginals, integral averages and the supremum of the process. The idealised scheme is related to two practically implementable schemes and corresponding ce…
