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

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

2008-10-16

arxiv.org/abs/0810.2930v2

Journal of Mathematical Analysis and Applications, 380(2), 2011, 405-424

Mathematics

Probability

27 pages, references are updated

Scientific paper

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 $X_0=0$.

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