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## Publications

Publications (99)

The mean-field stochastic partial differential equation (SPDE) corresponding to a mean-field super-Brownian motion (sBm) is obtained and studied. In this mean-field sBm, the branching-particle lifetime is allowed to depend upon the probability distribution of the sBm itself, producing an SPDE whose space-time white noise coefficient has, in additio...

We obtain the Holder continuity and joint Hölder continuity in space and time for the random field solution to the parabolic Anderson equation \((\partial_t-\frac{1}{2}\Delta)u=u\diamond\dot{W}\) in d-dimensional space, where Ẇ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density µ(ξ). We assu...

This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solutio...

This paper provides several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. First and higher order power variations are used to estimate the volatility parameter. The almost sure conv...

We consider a $d$-dimensional branching particle system in a random environment. Suppose that the initial measures converge weakly to a measure with bounded density. Under the Mytnik-Sturm branching mechanism, we prove that the corresponding empirical measure $X_t^n$ converges weakly in the Skorohod space $D([0,T];M_F(\mathbb{R}^d))$ and the limit...

We show that the random field solution to the parabolic Anderson equation $(\partial_t-\frac12 \Delta)u=u\diamond \dot{W}$ is jointly H\"older continuous in space and time.

In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively based on variance computations and on path-wise consid...

We derive the strong consistency of the least squares estimator for the drift coefficient of a fractional stochastic differential system. The drift coeffcient is one-sided dissipative Lipschitz and the driving noise is additive and fractional with Hurst parameter $H \in (\frac{1}{4}, 1)$. We assume that continuous observation is possible. The main...

We present an explicit solution triplet (Y,Z,K) to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process Y is expressed by an integral whose kernel is explicitly given. The processes Z and K are expressed by Hida–Malliavin derivatives involving Y.

In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the form \begin{align*} dY(t) =-f(t,Y(t),Z(t),K(t),\mathbb{E}[\varphi(Y(t),Z(t),K(t,\cdot ))])dt+Z(t)dB(t) + {\textstyle\int_{\mathbb{R}_{0}}} K(t,\zeta)\tilde{N}(dt,d\zeta), \end{align*} where $B$ is a Brownian motion, $\tilde{N}$ is the compensat...

We investigate the sharp density $\rho(t,x; y)$ of the solution $u(t,x)$ to stochastic partial differential equation $\frac{\partial }{\partial t} u(t,x)=\frac12 \Delta u(t,x)+u\diamond \dot W(t,x)$, where $\dot W$ is a general Gaussian noise and $\diamond$ denotes the Wick product. We mainly concern with the asymptotic behavior of $\rho(t,x; y)$ w...

This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter \(H \in (\frac {1}{4}, \frac {1}{2})\) in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of...

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 ach...

This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod...

In this paper, we consider a class of backward doubly stochastic differential equations (BDSDE for short) with general terminal value and general random generator. Those BDSDEs do not involve any forward diffusion processes. By using the techniques of Malliavin calculus, we are able to establish the $L^p$-H\"{o}lder continuity of the solution pair....

This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2})$ in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the...

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and les...

In this paper, we study two variations of the time discrete Taylor schemes
for rough differential equations and for stochastic differential equations
driven by fractional Brownian motions. One is the incomplete Taylor scheme
which excludes some terms of an Taylor scheme in its recursive computation so
as to reduce the computation time. The other on...

We study the propagation of high peaks (intermittency front) of the solution
to a stochastic heat equation driven by multiplicative centered Gaussian noise
in $\mathbb{R}^d$. The noise is assumed to have a general homogeneous
covariance in both time and space, and the solution is interpreted in the
senses of the Wick product. We give some estimates...

This paper studies the weak and strong solutions to the stochastic
differential equation $ dX(t)=-\frac12 \dot W(X(t))dt+d\mathcal{B}(t)$, where
$(\mathcal{B}(t), t\ge 0)$ is a standard Brownian motion and $W(x)$ is a two
sided Brownian motion, independent of $\mathcal{B}$. It is shown that the
It\^o-McKean representation associated with any Browni...

This paper studies the nonlinear one-dimensional stochastic heat equation
driven by a Gaussian noise which is white in time and which has the covariance
of a fractional Brownian motion with Hurst parameter
1/4\textless{}H\textless{}1/2 in the space variable. The existence and
uniqueness of the solution u are proved assuming the nonlinear coefficien...

In this paper we consider a general class of second order stochastic partial
differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is
white in time and it has a homogeneous spatial covariance. Using the techniques
of Malliavin calculus we derive the smoothness of the density of the solution
at a fixed number of points $(t,x_1),...

This paper is concerned with a class of stochastic differential equations
with state-dependent switching. The Malliavin calculus is used to study the
smoothness of the density of the solutions under the Hormander type conditions.
Moreover, the strong Feller property of the process is obtained by using the
Bismut formula. Theirreducibility of the se...

In this paper, we study the stochastic wave equations in the three spatial dimensions driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path Hölder continuity of the solution both in time variable and in space variables. The conditions are given either in terms of the mean Hölder continuity of...

For a stochastic differential equation driven by a fractional Brownian motion
with Hurst parameter $H> \frac12$ it is known that the existing (naive) Euler
scheme has the rate of convergence $n^{1-2H}$, which means no convergence to
zero of the error when $H$ is formally set to $\frac 12$ (the standard Brownian
motion case). In this paper we introd...

This paper studies singular mean field control problems and singular mean
field stochastic differential games. Both sufficient and necessary conditions
for the optimal controls and for the Nash equilibrium are obtained. Under some
assumptions the optimality conditions for singular mean-field control are
reduced to a reflected Skorohod problem, whos...

This paper considers the problem of partially observed optimal control for
forward stochastic systems which are driven by Brownian motions and an
independent Poisson random measure with a feature that the cost functional is
of mean-field type. When all the system coefficients and the objective
performance functionals are allowed to be random, possi...

Consider a Gaussian stationary sequence with unit variance X. Assume that the
central limit theorem holds for a weighted sum of elements of the form f(X_k),
where f designates a finite sum of Hermite polynomials. Then we prove that the
uniform convergence of the density of this weighted sum towards the standard
Gaussian density also holds true, und...

This paper studies the stochastic heat equation with multiplicative noises of
the form uW, where W is a mean zero Gaussian noise and the differential element
uW is interpreted both in the sense of Skorohod and Stratonovich. The existence
and uniqueness of the solution are studied for noises with general time and
spatial covariance structure. Feynma...

Applying an upper bound estimate for small $L^{2}$ ball probability for
fractional Brownian motion (fBm), we prove the non-degeneracy of some Sobolev
pseudo-norms of fBm.

In this paper, we study the stochastic wave equations in the spatial
dimension 3 driven by a Gaussian noise which is white in time and correlated in
space. Our main concern is the sample path H\"older continuity of the solution
both in time variable and in space variables. The conditions are given either
in terms of the mean H\"older continuity of...

For a stochastic differential equation driven by a fractional Brownian motion
with Hurst parameter $H> \frac12$ it is known that the classical Euler scheme
has the rate of convergence $2H-1$. In this paper we introduce a new numerical
scheme which is closer to the classical Euler scheme for diffusion processes,
in the sense that it has the rate of...

The aim of this paper is to establish the uniform convergence of the
densities of a sequence of random variables, which are functionals of an
underlying Gaussian process, to a normal density. Precise estimates for the
uniform distance are derived by using the techniques of Malliavin calculus,
combined with Stein's method for normal approximation. W...

We consider the Kyle-Back model for insider trading, with the difference that the classical Brownian motion noise of the noise traders is replaced by the noise of a fractional Brownian motion B
H
with Hurst parameter \({H>\frac{1}{2}}\) (when \({H=\frac{1}{2}, B^H}\) coincides with the classical Brownian motion). Heuristically, for \({H>\frac{1}{2}...

In this paper, a Feynman-Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H < 1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman-Ka...

We obtain a maximum principle for stochastic control problem of general
controlled stochastic differential systems driven by fractional Brownian
motions (of Hurst parameter $H>1/2$). This maximum principle specifies a system
of equations that the optimal control must satisfy (necessary condition for the
optimal control). This system of equations co...

In this paper we compute the $\frac 43$-variation of the derivative of the
self-intersection Brownian local time $\gamma_t=\int_0^t \int_0^u \delta
'(B_u-B_s)dsdu\,, t\ge 0$, applying techniques from the theory of fractional
martingales.

We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish t...

In this paper we study backward stochastic differential equations with
general terminal value and general random generator. In particular, we do not
require the terminal value be given by a forward diffusion equation. The
randomness of the generator does not need to be from a forward equation,
either. Motivated from applications to numerical simula...

We present a mathematical model for a Black–Scholes market driven by fractional Brownian motion B H (t) with Hurst parameter [Formula: see text]. The interpretation of the integrals with respect to B H (t) is in the sense of Itô (Skorohod–Wick), not pathwise (which is known to lead to arbitrage).
We find explicitly the optimal consumption rate and...

The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion B H (t); 1/2 < H < 1.
We show that if we use an Itô type of stochastic integration with respect to B H (t) (as developed in Ref. 8), then the cor...

We prove a central limit theorem for an additive functional of the
$d$-dimensional fractional Brownian motion with Hurst index
$H\in(\frac{1}{1+d},\frac{1}{d})$, using the method of moments, extending the
result by Papanicolaou, Stroock and Varadhan in the case of the standard
Brownian motion.

In this paper, we establish a version of the Feynman-Kac formula for
multidimensional stochastic heat equation driven by a general semimartingale.
This Feynman-Kac formula is then applied to study some nonlinear stochastic
heat equations driven by nonhomogenous Gaussian noise: First, it is obtained an
explicit expression for the Malliavin derivativ...

This paper deals with the problems of consistency and strong consistency of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. Both the central limit theorem and the Berry-Esséen bounds for these estimators are obtained by using the Stein's method via Malliavin cal...

The H\"older continuity of the solution to a nonlinear stochastic partial
differential equation arising from one dimensional super process is obtained.
It is proved that the H\"older exponent in time variable is as close as to 1/4,
improving the result of 1/10 in a recent paper by Li et al [3]. The method is
to use the Malliavin calculus. The H\"ol...

In this note, we provide a non trivial example of differential equation
driven by a fractional Brownian motion with Hurst parameter 1/3 < H < 1/2,
whose solution admits a smooth density with respect to Lebesgue's measure. The
result is obtained through the use of an explicit representation of the
solution when the vector fields of the equation are...

In this article, we derive a Stratonovich and Skorohod type change of
variables formula for a multidimensional Gaussian process with low H\"older
regularity (typically lower than 1/4). To this aim, we combine tools from rough
paths theory and stochastic analysis.

We study a least squares estimator for the Ornstein-Uhlenbeck process, , driven by fractional Brownian motion BH with Hurst parameter . We prove the strong consistence of (the almost surely convergence of to the true parameter [theta]). We also obtain the rate of this convergence when 1/2<=H<3/4, applying a central limit theorem for multiple Wiener...

The purpose of this note is to prove a central limit theorem for the $L^3$-modulus of continuity of the Brownian local time using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^3$-modulus of the Brownian local time.

The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR}, using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time.

We study the problem of parameter estimation for generalized Ornstein-Uhlenbeck processes driven by [alpha]-stable noises, observed at discrete time instants. Least squares method is used to obtain an asymptotically consistent estimator. The strong consistency and the rate of convergence of the estimator have been studied. The estimator has a highe...

In this paper we establish a version of the Feynman-Kac formula for the stochastic heat equation with a multiplicative fractional Brownian sheet. We prove the smoothness of the density of the solution, and the H\"older regularity in the space and time variables.

This paper deals with the problems of consistence and strong consistence of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. A central limit theorem for these estimators is also obtained by using the Malliavin calculus.

This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormali...

We study general linear and nonlinear backward stochastic differential equations driven by fractional Brownian motions. The existence and uniqueness of the solutions are obtained under some mild assumptions. In the nonlinear case we obtain an inequality of the type similar to in the classical backward stochastic differential equations. This leads t...

In this paper we apply Clark–Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). As a consequence, we derive the existence of some exponential moments for this random variable.

In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter . Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely cont...

We study a stochastic control problem in which the state process is described by a stochastic differential equation (SDE) that is driven by a Brownian motion and a Poisson random measure, and is a. ne in both the state and the control. The performance functional is quadratic in the state and the control. All the coefficients are allowed to be rando...

We study optimal stopping problems for some functionals of Brownian motion in the case when the decision whether or not to stop before (or at) time t is allowed to be based on the -advanced information Ft+ , where Fs is the -algebra generated by Brownian motion up to time s, s , > 0 being a fixed constant. Our approach involves the forward integral...

Fractional Brownian motion.- Intrinsic properties of the fractional Brownian motion.- Stochastic calculus.- Wiener and divergence-type integrals for fractional Brownian motion.- Fractional Wick Ito Skorohod (fWIS) integrals for fBm of Hurst index H >1/2.- WickIto Skorohod (WIS) integrals for fractional Brownian motion.- Pathwise integrals for fract...

In this paper we introduce the notion of fractional martingale as the fractional derivative of order $\alpha$ of a continuous local martingale, where $\alpha\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\frac{2}{1+2\alpha}$, under some integrability assumptions on the quadratic variation of the local martingale. A...

We study the optimal portfolio problem for an insider, in the case where the performance is measured in terms of the logarithm of the terminal wealth minus a term measuring the roughness and the growth of the portfolio. We give explicit solutions in some cases. Our method uses stochastic calculus of forward integrals.

The parameter estimation theory for stochastic differential equa-tions driven by Brownian motions or general Lévy processes with finite second moments has been well developed. In this paper, we consider the parameter estimation problem for Ornstein-Uhlenbeck processes driven by α-stable Lévy motions. The classical maximum likelihood method does not...

The aim of this paper is to study the $d$-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter $% H\in (0,1)$ in time. Two types of equations are considered. First we consider the equation in the It\^{o}-Skorohod sense, and la...

This paper presents a simple unified approach to several inequalities for Gaussian and diffusion measures. They include hypercontractive
inequalities, logarithmic Sobolev inequalities, FKG inequalities, and correlation inequalities.

We derive estimates for the solutions to differential equations driven by a
H\"older continuous function of order $\beta>1/2$. As an application we deduce
the existence of moments for the solutions to stochastic partial differential
equations driven by a fractional Brownian motion with Hurst parameter
$H>{1/2}$.

In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. F...

This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic delay equation with fixed delays in the drift and diffusion terms. In this article, we look at models of the stock price described by stochastic functional differential equa...

Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H\"{o}lder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously differentiable function such that $f'$ is $\lambda$-H\"oldr continuous for some $\lambda>\frac1\beta-2$. Under s...

We derive estimates for the solutions to differential equations driven by a H\"older continuous function of order $\beta>1/2$. As an application we deduce the existence of moments for the solutions to stochastic partial differential equations driven by a fractional Brownian motion with Hurst parameter $H>{1/2}$.

Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H\in(0,1). Assume d\geq2. We prove that the renormalized self-intersection local time\ell=\int_0^T\int_0^t\delta(B_t^H-B_s^H) ds dt -E\biggl(\int_0^T\int_0^t\delta (B_t^H-B_s^H) ds dt\biggr) exists in L^2 if and only if H<3/(2d), which generalizes the Varadhan renormalizat...

This paper studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error. The rate of convergence fo...

Let
\(B = { (B_t^{1}, ..., B_t^{d} ),t \geq 0}\) be a d-dimensional fractional Brownian motion with Hurst parameter H and let
\(R_{t} = \sqrt {(B_t^1 )^2 + ... + (B_t^{d} )^{2} }\) be the fractional Bessel process. Itô’s formula for the fractional Brownian motion leads to the equation
\(R_t = \sum_{i = 1}^d ,\int_0^{t} \frac{B_s^{i} }{R_{s} }\ {d}...

A Meyer-Tanaka formula involving weighted local time is derived for fractional Brownian motion and geometric fractional Brownian motion. The formula is applied to the study of the stop-loss-start-gain (SLSG) portfolio in a fractional Black-Scholes market. As a consequence, we obtain a fractional version of the Carr-Jarrow decomposition of the Europ...

This paper is concerned with optimal control of stochastic linear systems involving fractional Brownian motion (FBM). First, as a prerequisite for studying the underlying control problems, some new results on stochastic integrals and stochastic differential equations associated with FBM are established. Then, three control models are formulated and...

We present a white noise calculus for d-parameter fractional Brownian motion B-H (x, omega); x is an element of R-d, omega is an element of Omega with general d-dimensional Hurst parameter H = (H-l,..., H-d) is an element of (0, 1)(d). As an illustration we solve the stochastic Poisson problem DeltaU(x) = -W-H(x); x is an element of D, U = 0 on par...

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note tha...

We find the chaos expansion of local time l(T)((H))(x, (.)) of fractional Brownian motion with Hurst coefficient H is an element of (0, 1) at a point x is an element of R-d. As an application we show that when H(0)d < 1 then l(T)((H))(x, (.)) is an element of L-2(mu). Here mu denotes the probability law of B-(H) and H-0 = max {H-1, ..., H-d}. In pa...

We prove a stochastic maximum principle for controlled processes X(t)=X(u)(t) of the formdX(t)=b(t,X(t),u(t)) dt+[sigma](t,X(t),u(t)) dB(H)(t),where B(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian moti...

The asymptotic behavior as t --> infinity of the solution to the following stochastic heat equations [GRAPHICS] is investigated, where w is a space-time white noise or a space white noise. The use of lozenge means that the stochastic integral of 10 (Skorohod) type is considered. When d = 1, the exact L-2 Lyapunov exponents of the solution are studi...

The concept of probability structure preserving mapping is introduced. The idea is applied to define stochastic integral for fractional Brownian motion (fBm) and to obtain an anticipative Girsanov theorem for fBm. 1 Introduction Fractional Brownian motion (fBm) has received a great deal of attention in recent years. Various authors have developed s...

In this paper a stochastic calculus is given for the fractional Brownian motions that have the Hurst parameter in (1/2, 1). A stochastic integral of Ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment. This integral uses the Wick product and a derivative in the path spac...

Let $X_1(t)$, $\cdots$, $X_n(t)$ be $n$ geometric Brownian motions, possibly correlated. We study the optimal stopping problem: Find a stopping time $\tau^*<\infty$ such that \[ \sup_{\tau}{\Bbb E}^x\Big\{ X_1(\tau)-X_2(\tau)-\cdots -X_n(\tau)\Big\}={\Bbb E}^x \Big\{ X_1(\tau^*)-X_2(\tau^*)-\cdots -X_n(\tau^*)\Big\} , \] the $\sup$ being taken all...

In this paper we provide some sufficient conditions for the Skorohod integral process to have a continuous version. A first set of conditions require the existence of two square integrable derivatives and that the process has moments of order [beta] > 2. Secondly, we prove that the existence of the second derivative can be replaced by an integrabil...

We give a formula of expanding the solution of a stochastic differential equation (abbreviated as SDE) into a finite Ito-Wiener chaos with explicit residual. And then we apply this formula to obtain several inequalities for diffusions such as FKG type inequality, variance inequality and a correlation inequality for Gaussian measure. A simple proof...

Using the Itô-Wiener chaos expansion we prove that the normalized self-intersection local time of planar Brownian motion is not differentiable in the sense of Meyer-Watanabe.

We consider regular partitions of the interval [0,T] into n subintervals based on a sampling density h. If xt is the solution of the original stochastic differential equation and is the approximate solution using the Euler-Maruyama scheme corresponding to a time discretization partition π, we find the limit as a functional of the sampling density h...

We discuss the weak compactness problem related to the self-intersection local time of Brownian motion. We also propose a regular renormalization for self-intersection local time of higher dimensional Brownian motion.

In modelling the pressure p(x,w) at x {element_of} D {contained_in} R{sup d} of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeability k(x,w) {ge} 0, Holden, Lindstrom, Oksendal, Uboe and Zhang [HLOUZ95] studied the following stochastic differential equation div(k(x,w){diamond}{del}p(x,w))=-{line_integral}(x), x...

In this note we prove an interpolation inequality in L p -norm between the Sobolev spaces of order 0 and 2 on the Wiener space. R'esum'e Dans cette note nous d'emontrons une in'egalit'e d'interpolation pour les normes L p entre les espaces de Sobolev d'ordre 0 et 2 sur l'espace de Wiener. Pr'eliminaires- Soit (W; H;¯) l'espace de Wiener classique,...

We consider the approximation problem of stochastic integral with respect to two-parameter Wiener process. We first introduce a kind of symmetric integral and prove it obeys the chain rule. Then we apply an integral formula of bounded variation functions with two variables to show the approximation theorem of stochastic integral in the plane. In pa...

In this paper, we develop two discrete-time strong approximation schemes for solving stochastic differential systems with memory: strong Euler-Maruyama schemes for stochastic delay differential equations (SDDE's) and stochastic functional differential equations (SFDE's) with continuous memory, and a strong Milstein scheme for SDDE's. The convergenc...