Least squares volatility change point estimation for partially observed diffusion processes

A one dimensional diffusion process $X={X_t, 0\leq t \leq T}$, with drift $b(x)$ and diffusion coefficient $\sigma(\theta, x)=\sqrt{\theta} \sigma(x)$ known up to $\theta>0$, is supposed to switch volatility regime at some point $t^\in (0,T)$. On the basis of discrete time observations from $X$, the problem is the one of estimating the instant of change in the volatility structure $t^$ as well as the two values of $\theta$, say $\theta_1$ and $\theta_2$, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length $\Delta_n$ with $n\Delta_n=T$. To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant.

💡 Research Summary

The paper investigates the problem of detecting and estimating a single change point in the volatility structure of a one‑dimensional diffusion process observed at discrete, high‑frequency time points. The underlying stochastic differential equation is
 dXₜ = b(Xₜ) dt + √θ σ(Xₜ) dWₜ,
where the diffusion coefficient is separable: σ(θ, x)=√θ σ(x). The volatility parameter θ is assumed to take the value θ₁ up to an unknown time t*∈(0,T) and θ₂ thereafter. Observations are taken at equally spaced times t_i = iΔ_n, i=0,…,n, with Δ_n = T/n and the asymptotic regime n→∞, Δ_n→0 (high‑frequency sampling).

Methodology. By applying the Euler scheme, the authors construct the normalized increments
 Z_i = (X_{i+1}−X_i−b(X_i)Δ_n) / (√Δ_n σ(X_i)),
which under the model are i.i.d. N(0,θ). The squared increments Z_i² therefore have expectation θ. The change‑point estimator is defined via a least‑squares criterion: for each candidate k (the index of the change point) compute the sample means of Z_i² before and after k, denoted \barθ₁(k) and \barθ₂(k). The objective function is
 U_k = Σ_{i=1}^k (Z_i²−\barθ₁)² + Σ_{i=k+1}^n (Z_i²−\barθ₂)²,
and the estimated change point \hat k minimizes U_k. Algebraic manipulation shows that minimizing U_k is equivalent to maximizing
 V_k = √