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

Abstract

We consider a process ${(X_t^{(\alpha )})}_{t\in [0,T)}$ given by the SDE $\mathrm{d}{X}_t^{(\alpha )}=\alpha b(t)X_t^{(\alpha )}\mathrm{d}t+\sigma (t)\mathrm{d}{B}_t$, $t\in [0,T)$, with initial condition $X_0^{(\alpha )}=0$, where $T\in (0,\infty ]$, $\alpha \in \mathbb{R}$, ${(B_t)}_{t\in [0,T)}$ is a standard Wiener process, $b:[0,T)\to \mathbb{R}\setminus \{0\}$ and $\sigma :[0,T)\to (0,\infty )$ are continuously differentiable functions. Assuming $\frac{\mathrm{d}}{\mathrm{d}t}(\frac{b(t)}{\sigma {(t)}^2})=-2K\frac{b{(t)}^2}{\sigma {(t)}^2}$, $t\in [0,T)$, with some $K\in \mathbb{R}$, we derive an explicit formula for the joint Laplace transform of ${\int }_0^t\frac{b{(s)}^2}{\sigma {(s)}^2}{(X_s^{(\alpha )})}^2\mathrm{d}s$ and ${(X_t^{(\alpha )})}^2$ for all $t\in [0,T)$ and for all $\alpha \in \mathbb{R}$. Our motivation is that the maximum likelihood estimator (MLE) ${\hat{\alpha }}_t$ of α can be expressed in terms of these random variables. As an application, we show that in case of $\alpha =K$, $K\ne 0$,

$$\sqrt{I_K(t)}({\hat{\alpha }}_t-K)\stackrel{\mathcal{L}}{=}-\frac{\mathrm{sign}(K)}{\sqrt{2}}\frac{{\int }_0^1{W}_s\mathrm{d}{W}_s}{{\int }_0^1{(W_s)}^2\mathrm{d}s},\forall t\in (0,T),$$

where $I_K(t)$ denotes the Fisher information for α contained in the observation ${(X_s^{(K)})}_{s\in [0,t]}$, ${(W_s)}_{s\in [0,1]}$ is a standard Wiener process and $\stackrel{\mathcal{L}}{=}$ denotes equality in distribution. We also prove asymptotic normality of the MLE ${\hat{\alpha }}_t$ of α as $t\uparrow T$ for $\mathrm{sign}(\alpha -K)=\mathrm{sign}(K)$, $K\ne 0$. As an example, for all $\alpha \in \mathbb{R}$ and $T\in (0,\infty )$, we study the process ${(X_t^{(\alpha )})}_{t\in [0,T)}$ given by the SDE $\mathrm{d}{X}_t^{(\alpha )}=-\frac{\alpha }{T-t}{X}_t^{(\alpha )}\mathrm{d}t+\mathrm{d}{B}_t$, $t\in [0,T)$, with initial condition $X_0^{(\alpha )}=0$. In case of $\alpha >0$, this process is known as an α-Wiener bridge, and in case of $\alpha =1$, this is the usual Wiener bridge.