We consider a stochastic control problem which is composed of a controlled stochastic differential equation, and whose associated cost functional is defined through a controlled backward stochastic differential equation. Under appropriate convexity assumptions on the coefficients of the forward and the backward equations we prove the existence of an optimal control on a suitable reference stochastic system. The proof is based on an approximation of the stochastic control problem by a sequence of control problems with smooth coefficients, admitting an optimal feedback control. The quadruplet formed by this optimal feedback control and the associated solution of the forward and the backward equations is shown to converge in law, at least along a subsequence. The convexity assumptions on the coefficients then allow to construct from this limit an admissible control process which, on an appropriate reference stochastic system, is optimal for our stochastic control problem.
In this paper, we consider the fully coupled forward-backward stochastic functional differential equations (FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated backward stochastic differential equations as the backward equations. We will prove the existence and uniqueness theorem for FBSFDEs. As an application, we deal with a quadratic optimal control problem for functional stochastic systems, and get the explicit form of the optimal control by virtue of FBSFDEs.
We derive a stochastic Gronwall lemma with suprema over the paths in the upper bound of the assumed affine-linear growth assumption. This allows applications to Itô processes with coefficients which depend on earlier time points such as stochastic delay equations or Euler-type approximations of stochastic differential equations. We apply our stochastic Gronwall lemma with path-suprema to stochastic functional differential equations and prove a strong convergence rate for coefficient functions which depend on path-suprema.
In this paper, under a one-sided Lipschitz condition on the drift coefficient we adopt (via contraction principle) a exponential approximation argument to investigate large deviations for neutral stochastic functional differential equations.
By the approximation method introduced in \cite{FYW}, the existence and uniqueness are proved for a class of distribution-dependent stochastic functional differential equations (DDSFDEs). Moreover, combining the Harnack and shift-Harnack inequalities for classical stochastic functional differential equations with Girsanov's theorem, Harnack and shift-Harnack inequalities are obtained for the non-linear semigroup $P_t^*$ associated to the functional solution.
By using Girsanov transformation and martingale representation, Talagrand-type transportation cost inequalities, with respect to both the uniform and the $L^2$ distances on the global free path space, are established for the segment process associated to a class of neutral functional stochastic differential equations. Neutral functional stochastic partial differential equations are also investigated.
By using Malliavin calculus, Bismut derivative formulae are established for a class of stochastic (functional) differential equations driven by fractional Brownian motions. As applications, Harnack type inequalities and strong Feller property are presented.
In this paper, we study a new type of stochastic functional differential equations which is called hybrid pantograph stochastic functional differential equations. We investigate several moment properties and sample properties of the solutions to the equations by using the method of multiple Lyapunov functions, such as the moment exponential stability, almost sure exponential stability and almost sure polynomial stability, etc.
This work is devoted to deriving the Onsager-Machlup action functional for stochastic partial differential equations with (non-Gaussian) Levy process as well as Gaussian Brownian motion. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. This enables the investigation of the most probable transition path for infinite dimensional stochastic dynamical systems modeled by stochastic partial differential equations, by minimizing the Onsager-Machlup action functional.
In this article we consider multiplicative operator-valued white noise functionals related to a stochastic flow. A generalization of the Krylov-Veretennikov expansion is presented. An analog of such expansion for the Arratia flow is derived.
In the paper, we consider a type of stochastic differential equations driven by G-Lévy processes. We prove that a kind of their additive functionals has path independence and extend some known results.
We establish new general sufficient conditions for the existence of an invariant measure for stochastic functional differential equations and for exponential or subexponential convergence to the equilibrium. The obtained conditions extend Veretennikov--Khasminskii conditions for SDEs and are optimal in a certain sense.
In this paper, we prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations. As an application, we derive a functional iterated logarithm law for the solutions of multivalued stochastic differential equations.
In this paper, we investigate suffcient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDE). Moreover, the results extend the ones in the linear expectation case [1] and nonlinear expectation framework [8].
By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two results on existence and uniqueness of solutions on an open domain are presented.
In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.
In this paper, we study the convergence of the Euler-Maruyama numerical solutions for pantograph stochastic functional differential equations which was proposed in [11]. We also show that the numerical solutions have the properties of almost surely polynomial stability and exponential stability with the help of semi-martingale convergence theorem.
This paper focuses on controllability results of stochastic delay partial functional integro-differential equations perturbed by fractional Brownian motion. Sufficient conditions are established using the theory of resolvent operators combined with a fixed point approach for achieving the required result. An example is provided to illustrate the theory.
The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic density dependent predator-prey model with Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, weak persistence in the mean and extinction of a population densities in the considered stochastic predator-prey model.
In this article, the path independent property of additive functionals of McKean-Vlasov stochastic differential equations with jumps is characterised by nonlinear partial integro-differential equations involving $L$-derivatives with respect to probability measures introduced by P.-L. Lions. Our result extends the recent work [16] by Ren and Wang where their concerned McKean-Vlasov stochastic differential equations are driven by Brownian motions.
We consider linear n-th order stochastic differential equations on [0,1], with linear boundary conditions supported by a finite subset of [0,1]. We study some features of the solution to these problems, and especially its conditional independence properties of Markovian type.
Several versions of Itô's formula have been obtained in the context of the functional stochastic calculus. Here, we revisit this topic in two ways. First, by defining a notion of derivative along a functional, we extend the setting of the (semimartingale) functional Itô's formula and corresponding calculus. Second, for Lévy processes, an optimal local-time based Itô's formula is obtained. Some quick applications are then given.
This paper aims at developing a systematic study for the weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.
In this paper, we present an overview of the recent developments of functional quantization of stochastic processes, with an emphasis on the quadratic case. Functional quantization is a way to approximate a process, viewed as a Hilbert-valued random variable, using a nearest neighbour projection on a finite codebook. A special emphasis is made on the computational aspects and the numerical applications, in particular the pricing of some path-dependent European options.
In this paper we study the controllability of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.
We present an alternative proof for the existence of solutions of stochastic functional differential equations satisfying a global Lipschitz condition. The proof is based on an approximation scheme in which the continuous path dependence does not go up to the present: there is a memory gap. Strong convergence is obtained by closing the gap. Such approximation is particularly useful when extending stochastic models with discrete delay to models with continuous full finite memory.
In this paper, we study a functional fully coupled forward-backward stochastic differential equations (FBSDEs). Under a new type of integral Lipschitz and monotonicity conditions, the existence and uniqueness of solutions for functional fully coupled FBSDEs is proved. We also investigate the relationship between the solution of functional fully coupled FBSDE and the classical solution of the path-dependent partial differential equation (P-PDE). When the solution of the P-PDE has some smooth and regular properties, we solve the related functional fully coupled FBSDE and prove the P-PDE has a un…
We introduce a stochastic system of interacting particles which is expected to furnish as the number of particles goes to infinity a stochastic approach of the 2-D Keller-Segel model. In this note, we prove existence and some uniqueness for the stochastic model for the parabolic-elliptic Keller-Segel equation, for all regimes under the critical mass. Prior results for existence and weak uniqueness have been very recently obtained by N. Fournier and B. Jourdain [6].
Sufficient and necessary conditions are presented for the order-preservation of stochastic functional differential equations on $\R^d$ with non-Lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency of the conditions extends and improves some known comparison theorems derived recently for one-dimesional equations and multidimensional equations without delay, and the necessity is new even in these special situations.
We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [DebRom2014], by proving local estimate for the density, existence for the density with summable drift, and by improving the regularity of the density.
One proves existence and uniqueness of strong solutions to stochastic porous media equations under minimal monotonicity conditions on the nonlinearity. In particular, we do not assume continuity of the drift or any growth condition at infinity.
In this work, neutral stochastic functional differential equations with infinite delay (NSFDEwID) has been studied. The existence and uniqueness of solutions to NSFDEwID at the state space $ C_{r} $ under the local weak monotone condition, the weak coercivity condition and the global condition on the neutral term have been investigated. In addition, the $ \mathcal{L}^{2} $ and exponential estimates have been studied of NSFDEwID.
In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $C_0$-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates t…
In the framework of stochastic functional differential equations (SFDE's) and the corresponding calculus developed in the recent years by F. Yan and S. Mohammed, we provide a series of representation formulae for a variety of highly degenerate functional differential equations of the type of the Feynman-Kac formulae. More precisely, we study the stochastic process satisfying regular SFDE's with killing and absorbing boundary, we give the differential equation to be solved in order to compute the distribution of the first exit time from a regular domain and apply our results to a model describi…
The paper is devoted to the existence of integral functionals $\int_0^\infty f(X(t))\,{\mathrm{d}t}$ for several classes of processes in $\mathbb{R}$ with $d\ge 3$. Some examples such as Brownian motion, fractional Brownian motion, compound Poisson process, Markov processes admitting densities of transitional probabilities are considered.
We prove an existence and uniqueness result for quasilinear Stochastic PDEs with Obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier.
In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion (GSDEs) with integral-Lipschitz conditions on their coefficients.
In the present paper, a stochastic Taylor expansion of some functional applied to the solution process of an Itô or Stratonovich stochastic differential equation with a multi-dimensional driving Wiener process is given. Therefore, the multi-colored rooted tree analysis is applied in order to obtain a transparent representation of the expansion which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting. Further, some estimates for the mean--square and the mean truncation errors are given.
In this paper we shall establish an existence and uniqueness result for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter $H > \frac{1}{2} and a multidimensional standard Brownian motion under a weaker condition than the Lipschitz one.
In this note, we shall consider the existence of invariant measures for a class of infinite dimensional stochastic functional differential equations with delay whose driving semigroup is eventually norm continuous. The results obtained are applied to stochastic heat equations with distributed delays which appear in such terms having the highest order partial derivatives. In the systems, the associated driving semigroups are generally non eventually compact.
The existence-uniqueness theory for solutions to stochastic dynamic systems is always a significant theme and has received a huge attention. The objective of this article is to study the mentioned theory for stochastic functional differential equations (SFDEs) driven by G-Lévy process. The existence-uniqueness theorem for solutions to SFDEs driven by G-Lévy process has been determined. The error estimation between the exact solution and Picard approximate solutions has been shown. In addition, the exponential estimate has been derived.
One proves that the stochastic porous media equation in 3-D has a unique nonnegative solution for nonnegative initial data in $H^{-1}(\mathcal O)$ if the nonlinearity is monotone and has polynomial growth.
In this article we investigate the controllability for neutral stochastic functional integro-differential equations with finite delay, driven by a fractional Brownian motion with Hurst parameter lesser than $1/2$ in a Hilbert space. We employ the theory of resolvent operators combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result
In this note we prove the existence of a density for the law of the solution for 1-dimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
Consider stochastic functional differential equations, whose coefficients depend on past histories. The solution determines a non-Markov process. In the present paper, we shall obtain the existence of smooth densities for joint distributions of solutions, under the uniformly elliptic condition on the diffusion coefficients, via the Malliavin calculus. As an application, we shall study the computations of the Greeks on options associated with the asset price dynamics models with delayed effects.
In this paper we consider the controllability of certain class of non-autonomous neutral evolution stochastic functional differential equations, with time varying delays, driven by a fractional Brownian motion in a separable real Hilbert space. Sufficient conditions for controllability are obtained by employing a fixed point approach. A practical example is provided to illustrate the viability of the abstract result of this work.
In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by a fractional Brownian motion in a Hilbert space. We prove an existence and uniqueness result for the mild solution by means of the Banach fixed point principle. A practical example is provided to illustrate the viability of the abstract result of this work.
Under proper scaling and distributional assumptions, we prove the convergence in the Skorokhod space endowed with the M_1-topology of a sequence of stochastic integrals of a deterministic function driven by a time-changed symmetric α-stable Lévy process. The time change is given by the inverse β-stable subordinator.
In this paper, we study the existence and uniqueness of mild solution for a stochastic neutral partial functional integro-differential equation with delay in a Hilbert space driven by a fractional Brownian motion and with non-deterministic diffusion coefficient. We suppose that the linear part has a resolvent operator. We also establish a sufficient condition for the existence of the density of a function of the solution. An example is provided to illustrate the results of this work
Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Final…
The aim of this work is to prove existence and uniqueness of $L^{2}-$solutions of stochastic fractional partial differential equations in one spatial dimension. We prove also the equivalence between several notions of $L^{2}-$solutions. The Fourier transform is used to give meaning to SFPDEs. This method is valid also when the diffusion coefficient is random.
We show the continuity of a specific cost functional $J(ϕ) = \mathbb{E} \sup_{ t \in [0,T]}(φ(\mathcal{L}[t,u_ϕ(t), ϕ(t)]))$ of the SNSE in 2D on an open bounded nonperiodic domain $\mathcal{O}$ with respect to a special set of feedback controls $\{ϕ_n\}_{n \geq 0}$, where $φ(x) =\log(1 + x)^{1-ε}$ with $0 < ε<1$.
