We derive conditional Gaussian type identities of the form E [ exp ( i ∫ T 0 utdBt ) ∣∣∣∣ ∫ T 0 |ut|dt ] = exp ( − 2 ∫ T 0 |ut|dt ) , for Brownian stochastic integrals, under conditions on the process (ut)t∈[0,T ] specified using the Malliavin calculus. This applies in particular to the quadratic Brownian integral ∫ t 0 ABsdBs under the matrix condition A †A2 = 0, using a characterization of Yo...

We consider the problem of computing L1-distances between every pair of probability densities from a given family. We point out that the technique of Cauchy random projections [Ind06] in this context turns into stochastic integrals with respect to Cauchy motion. For piecewise-linear densities these integrals can be sampled from if one can sample from the stochastic integral of the function x 7→...

We discuss stochastic representations of advection diffusion equations with variable diffusivity, stochastic integrals of motion and generalized relative entropies.

In this article we consider the stochastic heat equation in [0, T ]× Rd, driven by a sequence (β)k of i.i.d. fractional Brownian motions of index H > 1/2 and random multiplication functions (g)k. The stochastic integrals are of Hitsuda-Skorohod type and the solution is interpreted in the weak sense. Using Malliavin calculus techniques, we prove the existence and uniqueness of the solution in a ...

We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Lévy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of it...

Using fractional calculus we define integrals of the form ∫ b a f(xt)dyt, where x and y are vector-valued Hölder continuous functions of order β ∈ ( 1 3 , 1 2 ) and f is a continuously differentiable function such that f ′ is λ-Höldr continuous for some λ > 1 β − 2. Under some further smooth conditions on f the integral is a continuous functional of x, y, and the tensor product x ⊗ y with respe...

Probability theory is an important tool of modeling randomness in a practical problem. But besides randomness, in the real world, there exists other kind of uncertainties such as impreciseness or vagueness. Set-valued functions are employed to model the impreciseness in applied field such as in Economics, control theory (see for example [1]). Integrals of set-valued functions have been received...

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of chaos space of the random field leads to two suc...