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

Abstract

We consider a process $(X_t{)}_{t\in [0,T)}$ given by the SDE $d{X}_t=\alpha b(t)X_{t}dt+\sigma (t)d{B}_t$, $t\in [0,T)$, 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 $\sign(α-K)=\sign(K)$, $K\ne 0$, and for $\alpha =K$, $K\ne 0$. As an example, we examine the so-called $\alpha$-Wiener bridges given by SDE $d{X}_t=-\frac{\alpha }{T-t}{X}_{t}dt+d{B}_t$, $t\in [0,T)$, with initial condition $X_0=0$.

27 pages, references are updated

References in corpus (1)

• Laplace Transforms for Integrals of Markov Processes

Cited by in corpus (2)

• Karhunen-Loeve expansions of alpha-Wiener bridges
• alpha-Wiener bridges: singularity of induced measures and sample path properties