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

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