随机热方程的拟似然估计

Quasi-Likelihood Estimation ofStochastic Heat Equation

摘要: 我们使用拟似然方法研究了随机热方程的参数估计问题，该方程为：$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}u(t,x)=\frac{1}{2}\mathrm{\Delta }u(t,x)+\sqrt{\theta }\dot{W}(t,x)(t\ge 0,x\in \mathbb{R})$$ 其初始条件u(0, x) = 0， ˙W 是时空白噪声，Δ 为拉普拉斯算子。假设解关于时间可以离散观测， 我们给出了参数θ 的估计量，并基于Malliavin 微积分得到估计量的渐近行为。

Abstract: In this paper, we investigate the parameter estimation of stochastic heat equation by using the quasi-likelihood method. The equation is given by: $$\frac{\mathrm{\partial }}{\mathrm{\partial }t}u(t,x)=\frac{1}{2}\mathrm{\Delta }u(t,x)+\sqrt{\theta }\dot{W}(t,x)(t\ge 0,x\in \mathbb{R})$$ with u(0, x) = 0, where ˙W is a space-time white noise and Δ is Lapalacian. Assuming that the temporal process can be discretely observed, we provide an estimator for the parameter θ and derive the asymptotic behavior of the estimator based on Malliavin calculus.