arXiv subjects

Sonja Cox

Publications and source records attributed to Sonja Cox.

21 records · Page 2Linked to original sources

arXiv2010

Vector-valued stochastic delay equations - a semigroup approach

Let E be a type 2 UMD Banach space, H a Hilbert space and let p be in [1,\infty). Consider the following stochastic delay equation in E: dX(t) = AX(t) + CX_t + b(X(t),X_t)dW_H(t), t>0; X(0) = x_0; X_0 = f_0. Here A : D(A) -> E is the generator of a C_0-semigroup, the operator C is given by a Riemann-Stieltjes integral, B : E x L^p(-1,0;E) -> γ(H,E) is a Lipschitz function and W_H is an H-cylindrical Brownian motion. We prove that a solution to \eqref{SDE1} is equivalent to a solution to the corresponding stochastic Cauchy problem, and use this to prove the existence, uniqueness and continuity of a solution.

math.FA↗

arXiv2010

Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems

We study the splitting scheme associated with the linear stochastic Cauchy problem dU(t) = AU(t) dt + dW(t), where A is the generator of an analytic C_0-semigroup S={S(t)} on a Banach space E and W={W(t)} is a Brownian motion with values in a fractional domain space E_\b associated with A. We prove that if \a,\b,\g,þ\ge 0 are such that \g + þ< 1 and max[0,(\a-\b+þ)] + \g < 1/2, then the approximate solutions U_n (where n is the number of time steps) converge to the solution U in the Holder space C^\g([0,T];E_\a), both in L^p-means and almost surely, with rate 1/n^þ.

math.NA↗

arXiv2008

Some remarks on tangent martingale difference sequences in $L^1$-spaces

Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant $C_{p,X}$ depending only on X and p exists such that for any two X-valued martingales f and g with tangent martingale difference sequences one has \[\E\|f\|^p \leq C_{p,X} \E\|g\|^p (*).\] This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either f or g satisfy the so-called (CI) condition. However, for some applications it suffices to assume that (*) holds whenever g satisfies the (CI) condition. We show that the class of Banach spaces for which (*) holds whenever only g satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space L^1. We state several problems related to (*) and other decoupling inequalities.

math.PR↗