Volatility change point detection for linear parabolic SPDEs

TL;DR

This work develops a rigorous volatility change point test for linear parabolic SPDEs observed at high frequency in space and time. By projecting the SPDE onto a spectral basis and constructing a CUSUM-type statistic T_n from approximately recovered coordinate processes, the authors prove that under no-change (H0) the statistic converges to the Kolmogorov distribution (supremum of a Brownian bridge), while under change (H1) the test is consistent. Two practical methodologies are presented for the one-dimensional case, and the approach is extended to two spatial dimensions, with simulations confirming finite-sample performance. The results offer a theoretically solid and implementable tool for detecting volatility shifts in spatio-temporal SPDE models across various disciplines.

Abstract

We consider change point detection for the volatility in second order linear parabolic stochastic partial differential equations based on high frequency spatio-temporal data. We give a test statistic to detect changes in the volatility based on change point analysis for diffusion processes and derive the asymptotic null distribution of the test statistic. We also show that the test is consistent. Moreover, we provide some examples and then perform numerical simulations of the proposed test statistic.

Figures (7)

• Figure 1: $\tau=1$ (No change points).
• Figure 2: $\tau=0.5$, $\sigma_2^\dag = 1.5$.
• Figure 3: $\tau=0.5$, $\sigma_2^\dag = 1.7$.
• Figure 4: $\tau=0.1$, $\sigma_2^\dag = 1.8$.
• Figure 5: $\tau=0.3$, $\sigma_2^\dag = 1.8$.

Theorems & Definitions (38)

• Remark 1
• Theorem 2.1
• Remark 2
• Remark 3
• Theorem 2.2
• Proposition 5.1
• Proposition 5.2
• proof : Proof of Proposition \ref{['prop1']}
• proof : Proof of Proposition \ref{['prop2']}
• Proposition 5.3