Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

TL;DR

This paper derives explicit formulas for the Laplace transforms of certain functionals of time-inhomogeneous diffusions, enabling analysis of maximum likelihood estimators and applications to alpha-Wiener bridges.

Contribution

It provides a novel explicit formula for the joint Laplace transform of specific functionals of inhomogeneous diffusions under certain differential equations.

Findings

Explicit Laplace transform formulas derived for functionals of diffusions.

Asymptotic behavior of MLE for the drift parameter analyzed.

Application to alpha-Wiener bridges demonstrated.

Abstract

We consider a process $(X_t)_{t\in[0,T)}$ given by the SDE $dX_t = \alpha b(t)X_t dt + \sigma(t) dB_t$, $t\in[0,T)$, with initial condition $X_0=0$, where $T\in(0,\infty]$, $\alpha\in R$, $(B_t)_{t\in[0,T)}$ is a standard Wiener process, $b:[0,T)\to R\setminus\{0\}$ and $\sigma:[0,T)\to(0,\infty)$ are continuously differentiable functions. Assuming that $b$ and $\sigma$ satisfy a certain differential equation we derive an explicit formula for the joint Laplace transform of $\int_0^t\frac{b(s)^2}{\sigma(s)^2}(X_s)^2 ds$ and $(X_t)^2$ for all $t\in[0,T)$. As an application, we study asymptotic behavior of the maximum likelihood estimator of $\alpha$ for $\sign(\alpha-K)=\sign(K)$, $K\ne0$, and for $\alpha=K$, $K\ne0$. As an example, we examine the so-called $\alpha$-Wiener bridges given by SDE $dX_t = -\frac{\alpha}{T-t}X_t dt + dB_t$, $t\in[0,T)$, with initial condition $X_0=0$.