Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBoundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes II
139
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 modelling 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.
rate research
Read More
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 modelling 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.
In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problem with non standard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with non standard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation.
Let $mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $mathcal{X}$, let $F:mathcal{X}rightarrowmathcal{X}$ be a (smooth enough) function and let ${W(t)}_{tgeq 0}$ be a $mathcal{X}$-valued cylindrical Wiener process. For $alphain [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2alpha-1}:Q^{1-2alpha}(mathcal{X})subseteqmathcal{X}rightarrowmathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation begin{gather} left{begin{array}{ll} dX(t,x)=big(AX(t,x)+F(X(t,x))big)dt+ Q^{alpha}dW(t), & t>0; X(0,x)=xin mathcal{X}, end{array} right. end{gather} and in its associated transition semigroup begin{align} P(t)varphi(x):=E[varphi(X(t,x))], qquad varphiin B_b(mathcal{X}), tgeq 0, xin mathcal{X}; end{align} where $B_b(mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(mathcal{X}, u)$, where $ u$ is the unique invariant probability measure of eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincare inequalities and we study the maximal Sobolev regularity for the stationary equation [lambda u-N_2 u=f,qquad lambda>0, fin L^2(mathcal{X}, u);] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(mathcal{X}, u)$.
In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=int_{mathbb{R}^n}int_{mathbb{R}^n}e^{i(x-y)cdotxi}sigma(x+tau(y-x),xi)u(y)dydxi, $$ where $tau:mathbb{R}^ntomathbb{R}^n$ is a general function. In particular, for the linear choices $tau(x)=0$, $tau(x)=x$, and $tau(x)=frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $tau$ and here we investigate the corresponding calculus in the model case of $mathbb{R}^n$. We also give examples of nonlinear $tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.
In this paper we study time-inhomogeneo
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
