62 research outputs found
Stability of the stochastic heat equation in
We consider the white-noise driven stochastic heat equation on
with Lipschitz-continuous drift and diffusion
coefficients and . We derive an inequality for the
-norm of the difference between two solutions. Using some
martingale arguments, we show that this inequality provides some {\it a priori}
estimates on solutions. This allows us to prove the strong existence and
(partial) uniqueness of weak solutions when the initial condition belongs only
to , and the stability of the solution with respect to this initial
condition. We also obtain, under some conditions, some results concerning the
large time behavior of solutions: uniqueness of the possible invariant
distribution and asymptotic confluence of solutions
Weak order for the discretization of the stochastic heat equation
In this paper we study the approximation of the distribution of
Hilbert--valued stochastic process solution of a linear parabolic stochastic
partial differential equation written in an abstract form as driven by a Gaussian
space time noise whose covariance operator is given. We assume that
is a finite trace operator for some and that is
bounded from into for some . It is not required
to be nuclear or to commute with . The discretization is achieved thanks to
finite element methods in space (parameter ) and implicit Euler schemes in
time (parameter ). We define a discrete solution and for
suitable functions defined on , we show that |\E \phi(X^N_h) - \E
\phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) \noindent where . Let us note that as in the finite dimensional case the rate of
convergence is twice the one for pathwise approximations
Absolute continuity for some one-dimensional processes
We introduce an elementary method for proving the absolute continuity of the
time marginals of one-dimensional processes. It is based on a comparison
between the Fourier transform of such time marginals with those of the one-step
Euler approximation of the underlying process. We obtain some absolute
continuity results for stochastic differential equations with H\"{o}lder
continuous coefficients. Furthermore, we allow such coefficients to be random
and to depend on the whole path of the solution. We also show how it can be
extended to some stochastic partial differential equations and to some
L\'{e}vy-driven stochastic differential equations. In the cases under study,
the Malliavin calculus cannot be used, because the solution in generally not
Malliavin differentiable.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ215 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Volatility Uncertainty Quantification in a Stochastic Control Problem Applied to Energy
This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solutions to a system of second order parabolic non-linear PDEs. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We provide an example of the methodology in the context of a swing contract (energy contract with flexibility in purchasing energy power), this allows us to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model.This research is part of the Chair Financial Risks of the Risk Foundation, the Finance for Energy Market Research Centre (FiME) and the ANR project CAESARS (ANR-15-CE05-0024)
Numerical methods for stochastic partial differential equations with multiples scales
A new method for solving numerically stochastic partial differential
equations (SPDEs) with multiple scales is presented. The method combines a
spectral method with the heterogeneous multiscale method (HMM) presented in [W.
E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585,
2005]. The class of problems that we consider are SPDEs with quadratic
nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis,
Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation
which describes the effective dynamics at long time scales can be rigorously
derived for both advective and diffusive time scales. Our method, based on
micro and macro solvers, allows to capture numerically the amplitude equation
accurately at a cost independent of the small scales in the problem. Numerical
experiments illustrate the behavior of the proposed method.Comment: 30 pages, 5 figures, submitted to J. Comp. Phy
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