We obtain upper and lower Gaussian-type bounds on the density of each component $Y^i_t$ of the solution $Y_t$ to a multidimensional non-Markovian backward SDE. Our approach is based on the Nourdin-Viens formula and a stochastic version of Wazewski's theorem on the positivity of the components of a solution to an ODE. Furthermore, we apply our results to stochastic gene expression; namely, we estimate the density of the law of the amount of protein generated by a gene in a gene regulatory network.
In this note, we give a necessary and sufficient condition under which the comparison theorem holds for multidimensional stochastic differential equations (SDEs) with jumps and for matrix-valued SDEs with jumps.
In this paper, we obtained the strong convergence of Wong-Zakai approximations of reflected SDEs in a general multidimensional domain giving an affirmative answer to the question posed in [ES].
We show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.
The large deviations principles are established for a class of multidimensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and anticipated.
Multidimensional Stable driven McKean-Vlasov SDEs with distributional interaction kernel -- a regularization by noise perspective
University of Nantes（南特大学）；HSE University（高等经济大学）；Université d’Evry Val d’Essonne（埃夫里-瓦勒德埃松大学）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS (UMR 8071)（法国国家科学研究中心）
We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.
In this paper, we address the problem of existence and uniqueness of a global classical solution to a multidimensional stochastic Burgers equation without gradient-type assumptions on the force or the initial condition. The equation is first transformed to a random PDE, and then solved via the associated forward-backward SDE. Additionally, we obtain a new a priori gradient estimate valid for a large class of second-order quasilinear parabolic PDEs which becomes an important tool in our approach. Also, we study the stochastic Burgers equation in the vanishing viscosity limit.
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…
The purpose of the article is to address the limiting behavior of the solutions of stochastic differential equations driven by a pointy $d$-dimensional gradient as the intensity of the underlying Brownian motion tends to $0$. By pointy gradient, we here mean that the drift derives from a potential that is ${\mathcal C}^{1,1}$ on any compact subset that does not contain the origin. As a matter of fact, the corresponding deterministic version of the differential equation may have an infinite number of solutions when initialized from $0_{{\mathbb R}^d}$, in which case the limit theorem proved in …
We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $σ$, satisfying Krylov--Röckner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,σ^i)$, both for Itô and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^1-b^2$. We then discuss several applications of these results to McKean--Vlasov SDEs, criteria for strong compactness of solutions and Wong--Zakai type theorems.
Let X be the solution of the multidimensional stochastic differential equationdX(t) = b(t, X(t)) dt + sigma(t, X(t)) dW(t)\, with X(0)=x where W is a standard Brownian motion. We show that when b is measurable and sigma is in an appropriate Sobolev space, the law of X satisfies a uniform quadratic transportation inequality.
Rio gave a concise bound for the central limit theorem in the Vaserstein distances, which is a ratio between some higher moments and some powers of the variance. As a corollary, it gives an estimate for the normal approximation of the small jumps of the Lévy processes, and Fournier applied that to the Euler approximation of Lévy-driven stochastic differential equations. However both results are restricted to the one-dimensional case. It will be shown in this article that, following Davie's idea, one can generalise Rio's result to multidimensional case, and the number of moments required is ind…
In this paper, we prove a comparison theorem for multidimensional G-SDEs. Moreover we obtain respectively the sufficient conditions and necessary conditions of the monotonicity and order-preservation for two multidimensional G-diffusion processes. Finally, we give some applications.
Forward-backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced almost 30 years ago, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of a solution, show existence and uniqueness of a strong solution of the first FBSDE, and weak existence for the second. We establish a link with PDE theory via a nonlinear Fe…
In this paper, by employing Gaussian type estimate of heat kernel, we establish Krylov's estimate and Khasminskill's estimate for EM algorithm. As applications, by taking Zvonkin's transformation into account, we investigate convergence rate of EM algorithm for a class of multidimensional SDEs under integrability conditions, where the drifts need not to be piecewise Lipschitz and are much more singular in some sense.
Consider a multidimensional SDE of the form $X_t = x+\int_{0}^{t} b(X_{s-})ds+\int{0}^{t} f(X_{s-})dZ_s$ where $(Z_s)_{s\ge 0}$ is a symmetric stable process. Under suitable assumptions on the coefficients the unique strong solution of the above equation admits a density w.r.t. the Lebesgue measure and so does its Euler scheme. Using a parametrix approach, we derive an error expansion at order 1 w.r.t. the time step for the difference of these densities.
We obtain sufficient condition for SDEs to evolve in the positive orthant. We use comparison theorem arguments to achieve this. As a result we prove the existence of a unique strong solution for a class of multidimensional degenerate SDEs with non-Lipschitz diffusion coefficients.
In this paper we obtain Gaussian-type lower bounds for the density of solutions to stochastic differential equations (SDEs) driven by a fractional Brownian motion with Hurst parameter $H$. In the one-dimensional case with additive noise, our study encompasses all parameters $H\in(0,1)$, while the multidimensional case is restricted to the case $H>1/2$. We rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools.
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.
We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and Øksendal in [7] for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô's SDEs, we consider for any finite partition of the time interval $[0,T]$ a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional ca…
We establish well-posedness results for multidimensional non degenerate $α$-stable driven SDEs with time inhomogeneous singular drifts in $\mathbb{L}^r-{\mathbb B}_{p,q}^{-1+γ}$ with $γ<1$ and $α$ in $(1,2]$, where $\mathbb{L}^r$ and ${\mathbb B}_{p,q}^{-1+γ} $ stand for Lebesgue and Besov spaces respectively. Precisely, we first prove the well-posedness of the corresponding martingale problem and then give a precise meaning to the dynamics of the SDE. This allows us in turn to define an ad hoc notion of weak solution, for which well-posedness holds as well. Our results rely on the smoothing p…
We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d $\ge$ 1. Namely, we study the case where the stable index of the driving process Z is $α$ = 1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Zt = (Z 1 t ,. .. , Z d t) and Z 1 ,. .. , Z d are independent one dimensional Cauchy processes. Our approach relies on L p-estimates for stable operators and uses perturbative arguments. 1. Statement of the problem and main results W…
This paper develops a new efficient scheme for approximations of expectations of the solutions to stochastic differential equations (SDEs). In particular, we present a method for connecting approximate operators based on an asymptotic expansion with multidimensional Malliavin weights to compute a target expectation value precisely. The mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin calculus. Moreover, numerical experiments for option pricing under local and stochastic volatility models confirm the effectiveness of our scheme. Especially, our weak approximati…
We combine the rough path theory and stochastic backward error analysis to develop a new framework for error analysis on numerical schemes. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein scheme for stochastic differential equations driven by multiplicative multidimensional fractional Brownian motion with Hurst parameter $H\in(\frac14,\frac12)$ is $(2H-\frac12)^-$ for sufficiently smooth coefficients, which is optimal in the sense that it is consistent with the result of the corresponding implementable approximation of the Lévy area of fractional …
Implementation of Strong Numerical Methods of Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with Non-Commutative Noise Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions and Multiple Fourier-Legendre Series
This paper proves joint convergence of the approximation error for several stochastic integrals with respect to local Brownian semimartingales, for nonequidistant and random grids. The conditions needed for convergence are that the Lebesgue integrals of the integrands tend uniformly to zero and that the squared variation and covariation processes converge. The paper also provides tools which simplify checking these conditions and which extend the range for the results. These results are used to prove an explicit limit theorem for random grid approximations of integrals based on solutions of mu…
A result of A.M. Davie [Int. Math. Res. Not. 2007] states that a multidimensional stochastic equation $dX_t = b(t, X_t)\,dt + dW_t$, $X_0=x$, driven by a Wiener process $W= (W_t)$ with a coefficient $b$ which is only bounded and measurable has a unique solution for almost all choices of the driving Brownian path. We consider a similar problem when $W$ is replaced by a Lévy process $L= (L_t)$ and $b$ is $β$-Hölder continuous in the space variable, $ β\in (0,1)$. We assume that $L_1$ has a finite moment of order $θ$, for some $θ>0$. Using also a new càdlàg regularity result for strong solutions,…
In this work, by using Levi's parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the existence and uniqueness of strong solutions for stochastic differential equation driven by Markov process with irregular coefficients, whose generator is a non-local and non-symmetric Lévy type operator.
We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, \dots, B^{m})$ with Hurst parameter $H \in (\frac 12,1)$. It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes $ B^{1}, \dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slowe…
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.
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in \cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non dens…
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…
The theory of backward SDEs extends the predictable representation property of Brownian motion to the nonlinear framework, thus providing a path-dependent analog of fully nonlinear parabolic PDEs. In this paper, we consider backward SDEs, their reflected version, and their second-order extension, in the context where the final data and the generator satisfy $L^1$-type of integrability condition. Our main objective is to provide the corresponding existence and uniqueness results for general Lipschitz generators. The uniqueness holds in the so-called Doob class of processes, simultaneously under…
In this paper, we prove the existence and uniqueness result of the reflected BSDE with two continuous barriers under monotonicity and general increasing condition on $y$, with Lipschitz condition on $z$.
Under integrability conditions on distribution dependent coefficients, existence and uniqueness are proved for McKean-Vlasov type SDEs with non-degenerate noise. When the coefficients are Dini continuous in the space variable, gradient estimates and Harnack type inequalities are derived. These generalize the corresponding results derived for classical SDEs, and are new in the distribution dependent setting.
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.
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.
This paper is intended to give a probabilistic representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use it connection with reflected generalized backward doubly stochastic differential equation.
This paper is intended to give a representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use its connection with reflected generalized backward doubly stochastic differential equations.
In this paper, we discuss hypercontractivity for the Markov semigroup $P_t$ which is generated by segment processes associated with a range of functional SDEs of neutral type. As applications, we also reveal that the semigroup $P_t$ converges exponentially to its unique invariant probability measure $μ$ in entropy, $L^2 (μ)$ and $\|\cdot\|_{\mbox{var}}$, respectively.
We consider stochastic differential equations (SDEs) driven by Feller processes which are themselves solutions of multivariate Levy driven SDEs. The solutions of these 'iterated SDEs' are shown to be non-Markovian. However, the process consisting of the driving process and the solution is Markov and even Feller in the case of bounded coefficients. The generator as well as the semimartingale characteristics of this process are calculated explicitly and fine properties of the solution are derived via the stochastic symbol. A short simulation study and an outlook in the direction of stochastic mo…
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.
The Harnack and log Harnack inequalities for stochastic differential equation driven by $G$-Brownian motion with multiplicative noise are derived by means of coupling by change of mesure. All of the above results extend the existing ones in the linear expectation setting. Moreover, the gradient estimate generalize the nonlinear results appeared in [11].
In this paper, we discuss exponential mixing property for Markovian semigroups generated by segment processes associated with several class of retarded Stochastic Differential Equations (SDEs) which cover SDEs with constant/variable/distributed time-lags. In particular, we investigate the exponential mixing property for (a) non-autonomous retarded SDEs by the Arzelà--Ascoli tightness characterization of the space $\C$ equipped with the uniform topology (b) neutral SDEs with continuous sample paths by a generalized Razumikhin-type argument and a stability-in-distribution approach and (c) jump-d…
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.
We deal with reflected solutions of anticipated backward doubly stochastic differential equations (RABDSDEs) driven by Teugels martingales associated with Lévy process under a Lipschitz generator where the coefficients of these BDSDEs depend on the future and present value of the solution $\left( Y,Z\right) $. Also we study the existence of a solution for anticipated BDSDEs.
A general maximum principle (necessary and sufficient conditions) for an optimal control problem governed by a stochastic differential equation driven by an infinite dimensional martingale is established. The solution of this equation takes its values in a separable Hilbert space and the control domain need not be convex. The result is obtained by using the adjoint backward stochastic differential equation.
We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.
