We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Levy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelli
ng domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided Levy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corresponds to fast-forwarding, i.e. removing the time the process spends outside the domain. We treat all combinations of killing, reflecting and fast-forwarding boundary conditions. In Part I we show wellposedness of the backward and forward Cauchy problems with a one-sided pseudo-differential operator with boundary conditions as generator. We do so by showing convergence of Feller semigroups based on grid point approximations of the modified Levy process. In Part II we show that the limiting Feller semigroup is indeed the semigroup associated with the modified Levy process by showing continuity of the modifications with respect to the Skorokhod topology.
We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial different
ial equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Holder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.
We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Levy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelli
ng domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided Levy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corresponds to fast-forwarding, i.e. removing the time the process spends outside the domain. We treat all combinations of killing, reflecting and fast-forwarding boundary conditions. In Part I we show wellposedness of the backward and forward Cauchy problems with a one-sided pseudo-differential operator with boundary conditions as generator. We do so by showing convergence of Feller semigroups based on grid point approximations of the modified Levy process. In Part II we show that the limiting Feller semigroup is indeed the semigroup associated with the modified Levy process by showing continuity of the modifications with respect to the Skorokhod topology.
We consider a system of stochastic Allen-Cahn equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative Gaussian noise driven stochastic Allen-Cahn equation is given with possibly different potential barr
ier heights supplemented by a continuity condition and a Kirchhoff-type law in the vertices. Using the semigroup approach for stochastic evolution equations in Banach spaces we obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. We also prove more precise space-time regularity of the solution.
The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic
differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford-Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Frechet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.
This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solu
tions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.
We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posednes
s of the associated Cauchy problems in $C_0(Omega)$ and $L_1(Omega)$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.
The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in $mathbb{R}^d$ is considered. The differential operator is given by the fractional power $L^beta$, $betain(0,
1)$, of an integer order elliptic differential operator $L$ and is therefore non-local. Its inverse $L^{-beta}$ is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power operator $L^{-beta}$ is approximated by a weighted sum of non-fractional resolvents $(I + t_j^2 L)^{-1}$ at certain quadrature nodes $t_j>0$. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way, an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation, the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for $L=kappa^2-Delta$, $kappa > 0$, with homogeneous Dirichlet boundary conditions on the unit cube $(0,1)^d$ in $d=1,2,3$ spatial dimensions for varying $betain(0,1)$ attest the theoretical results.
We consider autonomous stochastic ordinary differential equations (SDEs) and weak approximations of their solutions for a general class of sufficiently smooth path-dependent functionals f. Based on tools from functional It^o calculus, such as the fun
ctional It^o formula and functional Kolmogorov equation, we derive a general representation formula for the weak error $E(f(X_T)-f(tilde X_T))$, where $X_T$ and $tilde X_T$ are the paths of the solution process and its approximation up to time T. The functional $f:C([0,T],R^d)to R$ is assumed to be twice continuously Frechet differentiable with derivatives of polynomial growth. The usefulness of the formula is demonstrated in the one dimensional setting by showing that if the solution to the SDE is approximated via the linearly time-interpolated explicit Euler method, then the rate of weak convergence for sufficiently regular f is 1.
We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under a
ppropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean-$p$ Holder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space-time (Sobolev-Holder) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family.
