The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to \[
{\rm d}Z_t=σ(Z_{t-}){\rm d} X_t
\]driven by a (symmetric) $α$-stable Lévy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $α\in(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.
The path independence of additive functionals for SDEs driven by the G-Brownian motion is characterized by nonlinear PDEs. The main result generalizes the existing ones for SDEs driven by the standard Brownian motion.
A new proof of pathwise uniqueness for SDEs with Sobolev diffusion and integrable drift term is introduced by extending a method from E. Fedrizzi and F. Flandoli (Pathwise uniqueness and continuous dependence of SDEs with non-regular drift, Stochastics 83 (2011), pp. 241--257) to the case of nonconstant diffusion.
Sufficient and necessary conditions are presented for the order preservation of path-distribution dependent SDEs. Differently from the corresponding study of distribution independent SDEs, to investigate the necessity of order preservation for the present model we need to construct a family of probability spaces in terms of the ordered pair of initial distributions.
In this paper, we study (strong and weak) existence and uniqueness of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributionsof a continuous function.
In this paper, existence and uniqueness are proved for path-dependent McKean-Vlasov type SDEs with integrability conditions. Gradient estimates and Harnack type inequalities are derived in the case that the coefficients are Dini continuous in the space variable. These generalize the corresponding results derived for classical functional SDEs with singular coefficients.
In this paper, the well-posedness for one-dimensional path dependent McKean-Vlasov SDEs with $α$($α\geq \frac{1}{2}$)-Hölder continuous diffusion is investigated. Moreover, the associated quantitative propagation of chaos in the sense of Wasserstein distance, total variation distance as well as relative entropy is studied.
To characterize the regularity of distribution-path dependent SDEs in the initial distribution which varies in the class of probability measures on the path space, we introduce the intrinsic and Lions derivatives for probability measures on Banach spaces, and prove the chain rule of the Lions derivative for the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is i…
By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.
We derive the optimal rate of convergence for the mean squared error at the terminal point for anticipating linear stochastic differential equations, where the integral is interpreted in Skorohod sense. Although alternative proof techniques are needed, our results can be seen as generalizations of the corresponding results for Itō SDEs. As a key tool we extend optimal approximation results for vectors of correlated Wiener integrals to general random vectors, which contain the solutions of our Skorohod SDEs.
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.
In this paper we develop via Girsanov's transformation a perturbation argument to investigate weak convergence of Euler-Maruyama (EM) scheme for path-dependent SDEs with Hölder continuous drifts. This approach is available to other scenarios, e.g., truncated EM schemes for non-degenerate SDEs with finite memory or infinite memory. Also, such trick can be applied to study weak convergence of truncated EM scheme for a range of stochastic Hamiltonian systems with irregular coefficients and with memory, which are typical degenerate dynamical systems. Moreover, the weak convergence of path-dependen…
Let P2(Rd) be the space of probability measures on Rd with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space Rd*P2(Rd) equipped with the usual derivative in space variable and Lions derivative in distribution. These PDEs are solved by using probabilis- tic arguments developed from [2]. In particular, the path independence of the Girsanov transformation killing the drift term is identified with a nonlinear PDE on Rd*P2(Rd), which includes corresponding results derived earlier for the classical SDEs as special …
A new proof of a pathwise uniqueness result of Krylov and Röckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.
Here we study stochastic differential equations with a reflecting boundary condition. We provide sufficient conditions for pathwise uniqueness and non-explosion property of solutions in a framework admitting non-Lipschitz continuous coefficients and non-smooth domains.
For $α\in (0,1)$, we consider stochastic differential equations driven by one-sided stable processes of order $α$: \[dX_t= ϕ(X_{t-})\ dZ_t.\] We prove that pathwise uniqueness holds for this equation under the assumptions that $ϕ$ is continuous, non-decreasing and positive on $\R$. A counterexample is given to show that the positivity of $ϕ$ is crucial for pathwise uniqueness to hold.
Sufficient and necessary conditions are presented for the comparison theorem of path dependent $G$-SDEs. Different from the corresponding study in path independent $G$-SDEs, a probability method is applied to prove these results. Moreover, the results extend the ones in the linear expectation case.
The Bessel process in low dimension (0 $\le$ $δ$ $\le$ 1) is not an It{ô} process and it is a semimartingale only in the cases $δ$ = 1 and $δ$ = 0. In this paper we first characterize it as the unique solution of an SDE with distributional drift or more precisely its related martingale problem. In a second part, we introduce a suitable notion of path-dependent Bessel processes and we characterize them as solutions of path-dependent SDEs with distributional drift.
We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given $n$-dimensional path-dependent SDE into a suitable $L_p$-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate Hörmander's bracket condition in $\mathbb R^n$ for non-anticipative SDE coefficients defi…
In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index $H\in (1/2,1)$.
We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.
In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path--dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama--Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super--linear growth condition), Gaussian two--sided bound and Hölder conti…
In this paper, we are interested in least squares estimator for a class of path-dependent McKean-Vlasov stochastic differential equations (SDEs). More precisely, we investigate the consistency and asymptotic distribution of the least squares estimator for the unknown pa- rameters involved by establishing an appropriate contrast function. Comparing to the existing results in the literature, the innovations of our paper lie in three aspects: (i) We adopt a tamed Euler-Maruyama algorithm to establish the contrast function under the monotone condition, under which the Euler-Maruyama scheme no long…
We consider the Stochastic Differential Equation $X_t = X_0 + \int_0^t b(s,X_s) ds + B_t$, in $\mathbb{R}^d$. We give an example of a drift $b$ such that there does not exist a weak solution, but there exists a solution for almost every realization of the Brownian motion $B$. We also give an explicit example of a drift such that the SDE has a pathwise unique weak solution, but path-by-path uniqueness (i.e. uniqueness of solutions to the ODE for almost every realization of the Brownian motion) is lost. These counterexamples extend the results obtained in arXiv:2001.02869 to dimension $d=1$.
We consider a process given as the solution of a stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Explicit and optimal bounds for the Lebesgue density of that process at any given time are derived. The bounds and their optimality is shown by identifying the worst case stochastic differential equation. Then we generalise our findings to a larger class of diffusion coefficients.
This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time t can depend on the whole path of the forward process up to time t. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling fiel…
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 show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we prove a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality.
Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod's selection theorem.
Generalization of the Itô-Wentzel formula for the generalized Itô's SDEs (Itô's GSDEs) system with not centered measure is constructed. This construction is based on the basis of the stochastic kernel of integral transformation. The Itô's GSDEs system for the kernel of the integral invariant is constructed. The concept of a stochastic first integral of the Itô's GSDEs system with not centered measure is introduced. The conditions for the random function, that it's the first integral of the set Itô's GSDEs system, are defined.
We study a class of stochastic integral equations with jumps under non-Lipschitz conditions. We use the method of Euler approximations to obtain the existence of the solution and give some sufficient conditions for the strong uniqueness.
In this paper, we derive a characterization theorem for the path-independent property of the density of the Girsanov transformation for {\it degenerated} stochastic differential equations (SDEs), extending the characterization theorem of \cite{twwy} for the non-degenerated SDEs. We further extends our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalizes the corresponding characterization theorem established in \cite{hqwu}.
In this article, a class of second order differential equations on [0,1], driven by a general Hölder continuous function and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks to Young integration techniques. We then study the differentiability of the solution with respect to the driving process and consider the case where the equation is driven by a fractional Brownian motion, with two aims in mind: show that the solution we have produced coincides with the one which would be obtained with Malliavin calculus tools, and prove that th…
We consider a process given as the solution of a one-dimensional stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Hölder continuity of the Lebesgue density of that process at any given time is achieved using a different approach than the classical ones in the literature. Namely, the Hölder regularity of the densities is obtained via a control problem by identifying the stochastic differential equation with the worst global Hölder constant. Then we generalise our findings to a larger class of diffusion coefficients. The…
In this paper, we develop a general methodology to prove weak uniqueness for stochastic differential equations with coefficients depending on some path-functionals of the process. As an extension of the technique developed by Bass \& Perkins [BP09] in the standard diffusion case, the proposed methodology allows one to deal with processes whose probability laws are singular with respect to the Lebesgue measure. To illustrate our methodology, we prove weak existence and uniqueness in two examples : a diffusion process with coefficients depending on its running symmetric local time and a diffusio…
In this paper, we will focus - in dimension one - on the SDEs of the type dX_t=s(X_t)dB_t+b(X_t)dt where B is a fractional Brownian motion. Our principal motivation is to describe one of the simplest theory - from our point of view - allowing to study this SDE, and this for any Hurst index H between 0 and 1. We will consider several definitions of solution and we will study, for each one of them, in which condition one has existence and uniqueness. Finally, we will examine the convergence or not of the canonical scheme associated to our SDE, when the integral with respect to fBm is defined usi…
We present a new pathwise approximation scheme for stochastic differential equations driven by multidimensional Brownian motion which does not require the simulation of Lévy area and has a Wasserstein convergence rate better than the Euler scheme's strong error rate of $O(\sqrt{h})$, where $h$ is the step-size. By using rough path theory we avoid imposing any non-degenerate Hörmander or ellipticity assumptions on the vector fields of the SDE, in contrast to the similar papers of Alfonsi, Davie, and Malliavin et al. The scheme is based on the log-ODE method with the Lévy area increments replace…
We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove existence and uniqueness of mild solutions, continuity of mild solutions with respect to perturbations of all the data of the system, Gâteaux differentiability of generic order n of mild solutions with respect to the starting point, continuity of the Gâteaux derivatives with respect to all the data. The analysis is performed for generic spaces of paths that do not necessarily coincide with the space of continuous functions.
By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker--Planck equations for probability measures $(μ_t)_{t \geq 0}$ on the path space $\mathcal C:=C([-r_0,0];\mathbb R^d),$ is analyzed: $$\partial_t μ(t)=L_{t,μ_t}^*μ_t,\ \ t\ge 0,$$ where $μ(t)$ is the image of $μ_t$ under the projection $\mathcal C\niξ\mapsto ξ(0)\in\mathbb R^d$, and $$L_{t,μ}(ξ):= \frac 1 2\sum_{i,j=1}^d a_{ij}(t,ξ,μ)\frac{\partial^2} {\partial_{ξ(0)_i} \partial_{ξ(0)_j}} +\sum_{i=1}^d b_i(t,ξ,μ)\frac{\partial}{\partial_{ξ(0)_i}},\ \ t\ge 0, ξ\in \mathcal C, μ\…
We study optimal stochastic control problem for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients, and gain functionals are path-dependent, and importantly we do not make any ellipticity assumption on the SDE. We develop a controls randomization approach, and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent versio…
In this paper, motivated by modelling currency exchange markets with matrix-valued stochastic processes, matrix-valued stochastic differential equations (SDEs) are formulated. This is done based on the matrix trace, as for the purpose of modelling currency exchange markets. To be more precise, we set up a Hilbert space structure for $n\times n$ square matrices via the trace of the Hadamard product of two matrices. With the help of this framework, one can then define stochastic integral of Itô type and Itô SDEs. Two types of sufficient conditions are discussed for the existence and uniqueness o…
Based on a recent result on characterising the path-independence of the Girsanov transformation for non-Lipschnitz stochastic differential equations (SDEs) with jumps on $R^d$, in this paper, we extend our consideration of characterising the path-indpendent property from finite-dimensional SDEs with jumps to stochastic evolution equations with jumps in Hilbert spaces. This is done via Galerkin type finite-dimensional approximations of the infinite-dimensional stochastic evolution equations with jumps in the manner that one could then link the characterisation of the path-independence for finit…
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 present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the funda…
In this paper, we get some convergence rates in total variation distance in approximating discretized paths of L{é}vy driven stochastic differential equations, assuming that the driving process is locally stable. The particular case of the Euler approximation is studied. Our results are based on sharp local estimates in Hellinger distance obtained using Malliavin calculus for jump processes.
By using a classical truncated argument and introducing the local Wasserstein distance, the global existence and uniqueness are proved for the distribution dependent SDEs with local Lipschitz coefficients. Due to the measure dependence, the conditions in the sense of pointwise for classical cases can be simplified to the conditions in the sense of integral. On the basis of the well-posedness, we prove the $r$-th moment exponential stability using the measure dependent Lyapunov functions and the existence and uniqueness of invariant probability measure is studied under the integrated strong mon…
