Do you want to publish a course? Click here

A probabilistic point of view for the exact or approximated computation of the solution to Kolmogorov hypoelliptic equations

82   0   0.0 ( 0 )
 Added by Pierre Etore
 Publication date 2021
  fields
and research's language is English
 Authors Pierre Etore




Ask ChatGPT about the research

In this work, we propose a method for solving Kolmogorov hypoelliptic equations based on Fourier transform and Feynman-Kac formula. We first explain how the Feynman-Kac formula can be used to compute the fundamental solution to parabolic equations with linear or quadratic potential. Then applying these results after a Fourier transform we deduce the computation of the solution to a a first class of Kolmogorov hypoelliptic equations. Then we solve partial differential equations obtained via Feynman-Kac formula from the Ornstein-Uhlenbeck generator. Also, a new small time approximation of the solution to Kolmogorov hypoelliptic equations is provided. We finally present the results of numerical experiments to check the practical efficiency of this approximation.



rate research

Read More

101 - Alessia E. Kogoj 2016
We show how to apply harmonic spaces potential theory in the study of the Dirichlet problem for a general class of evolution hypoelliptic partial differential equations of second order. We construct Perron-Wiener solution and we provide a sufficient condition for the regularity of the boundary points. Our criterion extends and generalizes the classical parabolic-cone criterion for the Heat equation due to Effros and Kazdan.
74 - Carlo Marinelli 2019
We prove a maximum principle for mild solutions to stochastic evolution equations with (locally) Lipschitz coefficients and Wiener noise on weighted $L^2$ spaces. As an application, we provide sufficient conditions for the positivity of forward rates in the Heath-Jarrow-Morton model, considering the associated Musiela SPDE on a homogeneous weighted Sobolev space.
In $mathbb R^d$, $d geq 3$, consider the divergence and the non-divergence form operators begin{equation} tag{$i$} - abla cdot a cdot abla + b cdot abla, end{equation} begin{equation} tag{$ii$} - a cdot abla^2 + b cdot abla, end{equation} where $a=I+c mathsf{f} otimes mathsf{f}$, the vector fields $ abla_i mathsf{f}$ ($i=1,2,dots,d$) and $b$ are form-bounded (this includes the sub-critical class $[L^d + L^infty]^d$ as well as vector fields having critical-order singularities). We characterize quantitative dependence on $c$ and the values of the form-bounds of the $L^q rightarrow W^{1,qd/(d-2)}$ regularity of the resolvents of the operator realizations of ($i$), ($ii$) in $L^q$, $q geq 2 vee ( d-2)$ as (minus) generators of positivity preserving $L^infty$ contraction $C_0$ semigroups. The latter allows to run an iteration procedure $L^p rightarrow L^infty$ that yields associated with ($i$), ($ii$) $L^q$-strong Feller semigroups.
190 - M. Kunze , L. Lorenzi , A. Rhandi 2013
Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.
We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-Morton model via Musielas stochastic PDE.
comments
Fetching comments Fetching comments
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا