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.     

Besov regularity for the generalized local time of the indefinite Skorohod integral

Let (t∈[0,1]) be the indefinite Skorohod integral on the canonical probability space (Ω,F,P), and let Lt(x) (t∈[0,1], x∈R) be its the generalized local time introduced by Tudor in [C.A. Tudor, Martingale-type stochastic calculus for anticipating integral processes, Bernoulli 10 (2004) 313-325]. We prove that the generalized local time, as function of x, has the same Besov regularity as the Brownian motion, as function of t, under some conditions imposed on the anticipating integrand u.     

The stochastic wave equation with fractional noise: A random field approach

We consider the linear stochastic wave equation with spatially homogeneous Gaussian noise, which is fractional in time with index H>1/2$H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in Dalang (1999) [10], where the noise is white in time. Under this condition, we show that the solution is L²(Ω)$L^2\left(\Omega \right)$-continuous. Similar results are obtained for the heat equation. Unlike in the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Aging for interacting diffusion processes

We study the aging phenomenon for a class of interacting diffusion processes {Xt(i),i∈Zd}. In this framework we see the effect of the lattice dimension d on aging, as well as that of the class of test functions f(Xt) considered. We further note the sensitivity of aging to specific details, when degenerate diffusions (such as super random walk, or parabolic Anderson model), are considered. We complement our study of systems on the infinite lattice, with that of their restriction to finite boxes. In the latter setting we consider different regimes in terms of box size scaling with time, as well as the effect that the choice of boundary conditions has on aging. The key tool for our analysis is the random walk representation for such diffusions.     

Homeomorphic property of solutions of SDE driven by countably many Brownian motions with non-Lipschitzian coefficients

We study m-dimensional SDE , where {Wi}_i?1 is an infinite sequence of independent standard d-dimensional Brownian motions. The existence and pathwise uniqueness of strong solutions to the SDE was established recently in [Z. Liang, Stochastic differential equations driven by countably many Brownian motions with non-Lipschitzian coefficients, Preprint, 2004]. We will show that the unique strong solution produces a stochastic flow of homeomorphisms if the modulus of continuity of coefficients is less than , ?∈[0,1) with ?(−1)=1, and the coefficients are compactly supported.     

The equivalence between uniqueness and continuous dependence of solutions for FBSDEs with continuous monotone coefficients

In this paper we study one kind of coupled forward-backward stochastic differential equation. With some particular choice for the coefficients, if one of them satisfies a uniform growth condition and they are accordingly monotone, then we obtain the equivalence between the uniqueness of solution and its continuous dependence on x and ξ, where x is the initial value of the forward component and ξ is the terminal value of the backward component.     

Ito's- and Tanaka's-type formulae for the stochastic heat equation: The linear case

In this paper, we consider the linear stochastic heat equation with additive noise in dimension one. Then, using the representation of its solution X as a stochastic convolution of the cylindrical Brownian motion with respect to an operator-valued kernel, we derive Itô's- and Tanaka's-type formulae associated to X.     

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if b is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined ε within the time horizon T. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

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.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F_n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F_n to zero; (ii) the covariance matrix of F_n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a R^d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.     

Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation

We present new results regarding the existence of density of the real-valued solution to a 3-dimensional stochastic wave equation. The noise is white in time and with a spatially homogeneous correlation whose spectral measure μ satisfies that , for some . Our approach is based on the mild formulation of the equation given by means of Dalang's extended version of Walsh's stochastic integration; we use the tools of Malliavin calculus. Let S₃ be the fundamental solution to the 3-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral and upper bounds for in terms of powers of t. These estimates are crucial in the analysis of the Malliavin variance, which can be done by a comparison procedure with respect to smooth approximations of the distribution-valued function S₃(t) obtained by convolution with an approximation of the identity.     

The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier-Stokes equations in unbounded domains

In this paper we consider the existence and uniqueness of weak energy solutions to a stochastic 2-dimensional non-Lipschitz Navier-Stokes equation perturbed by the cylindrical Wiener process W(t) in a bounded or unbounded domain D with the smooth boundary ∂D or D=R²: