Exact variations for stochastic heat equations with piecewise constant coefficients and application to parameter estimation

2020 ◽  
Vol 100 ◽  
pp. 77-106
Author(s):  
M. Zili ◽  
E. Zougar
Keyword(s):  
Parameter Estimation ◽  
Heat Equations ◽  
Piecewise Constant ◽  
Constant Coefficients ◽  
Stochastic Heat Equations

scholarly journals A note on parameter estimation for discretely sampled SPDEs

2019 ◽  
Vol 20 (03) ◽  
pp. 2050016 ◽  
Author(s):  
Igor Cialenco ◽  
Yicong Huang
Keyword(s):  
Parameter Estimation ◽  
Bounded Domain ◽  
Asymptotic Properties ◽  
Mesh Size ◽  
Fixed Time ◽  
Estimation Problem ◽  
Joint Estimation ◽  
Heat Equations ◽  
Whole Space ◽  
Stochastic Heat Equations

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

scholarly journals Fractional stochastic heat equation with piecewise constant coefficients

2020 ◽  
Vol 21 (01) ◽  
pp. 2150002
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Mounir Zili ◽  
Eya Zougar
Keyword(s):  
Diffusion Coefficient ◽  
Heat Equation ◽  
Divergence Form ◽  
Stochastic Heat Equation ◽  
Piecewise Constant ◽  
Constant Diffusion Coefficient ◽  
Infinite Dimensional ◽  
Order Elliptic Operator ◽  
Constant Diffusion ◽  
Constant Coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

scholarly journals A weak convergence approach to hybrid LQG problems with infinite control weights

2002 ◽  
Vol 15 (1) ◽  
pp. 1-21
Author(s):  
G. George Yin ◽  
Jiongmin Yong
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Small Parameter ◽  
Limit Problem ◽  
Discrete Events ◽  
Linear Quadratic ◽  
Piecewise Constant ◽  
Constant Coefficients ◽  
The Cost ◽  
Random Times

This work is concerned with a class of hybrid LQG (linear quadratic Gaussian) regulator problems modulated by continuous-time Markov chains. In contrast to the traditional LQG models, the systems have both continuous dynamics and discrete events. In lieu of a model with constant coefficients, these coefficients vary with time and exhibit piecewise constant behavior. At any time t, the system follows a stochastic differential equation in which the coefficients take one of the m possible configurations where m is usually large. The system may jump to any of the possible configurations at random times. Further, the control weight in the cost functional is allowed to be indefinite. To reduce the complexity, the Markov chain is formulated as singularly perturbed with a small parameter. Our effort is devoted to solving the limit problem when the small parameter tends to zero via the framework of weak convergence. Although the limit system is still modulated by a Markov chain, it has a much smaller state space and thus, much reduced complexity.

scholarly journals Some properties of non-linear fractional stochastic heat equations on bounded domains

2017 ◽  
Vol 102 ◽  
pp. 86-93 ◽  
Author(s):  
Mohammud Foondun ◽  
Ngartelbaye Guerngar ◽  
Erkan Nane
Keyword(s):  
Heat Equations ◽  
Bounded Domains ◽  
Non Linear ◽  
Stochastic Heat Equations
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