The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type

We consider the integral equation driven by a standard Brownian motion and fractional Brownian motion (fBm). Since fBm is not a semimartingale, we cannot use the semimartingale theory to define an integral with respect to the fBm. Furthermore, a well-developed theory of stochastic differential equations is not applicable to solve it. Existence and uniqueness conditions are obtained for a solution in the space of continuous functions with q-bounded variation, q>2.     

Convergence of delay differential equations driven by fractional Brownian motion

In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L ^p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.     

Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

In this paper, we consider stochastic differential equations with non-negativity constraints, driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first study an ordinary integral equation, where the integral is defined in the Young sense, and we prove an existence result and the boundedness of the solutions. Then, we apply this result pathwise to solve the stochastic problem.     

Non-densely defined impulsive neutral stochastic functional differential equations driven by fBm in Hilbert space with infinite delay

We study a class of non-densely defined impulsive neutral stochastic functional differential equations driven by an independent cylindrical fractional Brownian motion (fBm) with Hurst parameter H∈ (1/2, 1) in the Hilbert space. We prove the existence and uniqueness of the integral solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions. The results are obtained by using the method of successive approximation.     

Global attractiveness and exponential decay of neutral stochastic functional differential equations driven by fBm with Hurst parameter less than 1/2

We are concerned with a class of neutral stochastic functional differential equations driven by fractional Brownian motion (fBm) in the Hilbert space. We obtain the global attracting sets of this kind of equations driven by fBm with Hurst parameter \hslash \in (0, 1/2): Especially, some suffcient conditions which ensure the exponential decay in the p-th moment of the mild solution of the considered equations are obtained. In the end, one example is given to illustrate the feasibility and effectiveness of results obtained.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

Smooth Density for Some Nilpotent Rough Differential Equations

In this note, we provide a nontrivial example of a 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 measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris-type lemma in the rough paths context.     

Controlled differential equations as Young integrals: A simple approach

The theory of rough paths allows one to define controlled differential equations driven by a path which is irregular. The most simple case is the one where the driving path has finite p-variations with 1?p<2, in which case the integrals are interpreted as Young integrals. The prototypal example is given by stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2. Using simple computations, we give the main results regarding this theory - existence, uniqueness, convergence of the Euler scheme, flow property … - which are spread out among several articles.     

Delay Equations with Non-negativity Constraints Driven by a Hölder Continuous Function of Order \beta \in \left(\frac13,\frac12\right)

In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a Hölder continuous function of order \(\beta \in (\frac13,\frac12)\) . We also obtain a bound for the supremum norm of this solution. As an application, we get these results for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H \(\in (\frac13,\frac12)\) .     

Pathwise definition of second-order SDEs

In this article, a class of second-order differential equations on [0,1], driven by a γ-Hölder continuous function for any value of γ∈(0,1) 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 that we have produced coincides with the one which would be obtained with Malliavin calculus tools, and prove that the law of the solution is absolutely continuous with respect to the Lebesgue measure.     

Solving of second order nonlinear PDE problems by using artificial controls with controlled error

In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region inR ². We transfer this problem to a new problem of second order nonlinear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error inL ₁ is controlled.     

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.     

Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay

In this paper, we consider a class of fractional neutral stochastic functional differential equations with infinite delay driven by a cylindrical fractional Brownian motion (fBm) in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.     

Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion

In this paper, we consider a class of stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than 1/2. The existence of local random unstable manifolds is shown if the linear parts of these SPDEs are hyperbolic. For this purpose we introduce a modified Lyapunov-Perron transform, which contains stochastic integrals. By the singularities inside these integrals we obtain a special Lyapunov-Perron's approach by treating a segment of the solution over time interval [0,1] as a starting point and setting up an infinite series equation involving these segments as time evolves. Using this approach, we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium.     

Numerical methods for a class of jump–diffusion systems with random magnitudes

Stochastic differential delay equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. In this paper, we shall deal with convergence of the semi-implicit Euler method for nonlinear stochastic differential delay equations with random jump magnitudes and show that the approximate solutions strongly converge to the exact solutions with the order 1 − 1/q (q > 1). This result is more general than what they deal with the jump of deterministic magnitude.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

Existence and Uniqueness of Invariant Measures for SPDEs with Two Reflecting Walls

In this article, we study stochastic partial differential equations with two reflecting walls h ¹ and h ², driven by space-time white noise with non-constant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. The strong Feller property is also obtained.     

Identification for Linear Stochastic Systems Driven by Fractional Brownian Motion

Abstract

We apply Grenander's method of sieves to the problem of identification or estimation of the “drift” function for linear stochastic systems driven by a fractional Brownian motion (fBm). We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximise the likelihood function.     

Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process

We discuss some inference problems associated with the fractional Ornstein–Uhlenbeck (fO–U) process driven by the fractional Brownian motion (fBm). In particular, we are concerned with the estimation of the drift parameter, assuming that the Hurst parameter $H$ is known and is in $[1/2, 1)$ . Under this setting we compute the distributions of the maximum likelihood estimator (MLE) and the minimum contrast estimator (MCE) for the drift parameter, and explore their distributional properties by paying attention to the influence of $H$ and the sampling span $M$ . We also deal with the ordinary least squares estimator (OLSE) and examine the asymptotic relative efficiency. It is shown that the MCE is asymptotically efficient, while the OLSE is inefficient. We also consider the unit root testing problem in the fO–U process and compute the power of the tests based on the MLE and MCE.     

Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion

This paper deals with the existence and uniqueness of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion B ^H , with Hurst parameter H ∈ (1/2, 1). We use the theory of resolvent operators developed by R. Grimmer to show the existence of mild solutions. An example is provided to illustrate the results of this work.