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Temporal quartic variation for non-linear stochastic heat equations with piecewise constant coefficients

Yongkang Li, Huisheng Shu, Litan Yan

Abstract

We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study the limit behavior of the temporal quartic variation of the mild solution. As an application, we deduce a consistent estimator based on corresponding results.

Temporal quartic variation for non-linear stochastic heat equations with piecewise constant coefficients

Abstract

We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study the limit behavior of the temporal quartic variation of the mild solution. As an application, we deduce a consistent estimator based on corresponding results.

Paper Structure

This paper contains 3 sections, 11 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Temporal Variation

Key Result

Lemma 2.1

For every $0\leq s<t\leq T$ and $x, y\in\mathbb{R}$, we have with $C_{a_1, a_2}:=\frac{1+|\beta|}{\sqrt{2\pi}}(\frac{1}{\sqrt{a_1}}+\frac{1}{\sqrt{a_2}})$.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6: Gronwall's lemma
  • ...and 14 more