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Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions. (English) Zbl 1215.60015
Summary: We consider a process \((X_t^{(\alpha)})_{t\in[0,T)}\) given by the SDE \(dX_t^{(\alpha)}= \alpha b(t) X_t^{(\alpha)}\,dt+\sigma(t)\,dB_t\), \(t\in [0,T)\), 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 \(\sigma :[0,T)\rightarrow (0,\infty )\) are continuously differentiable functions. Assuming \(\frac{d}{dt}(\frac{b(t)}{\sigma(t)^2})= -2K\frac{b(t)^2}{\sigma(t)^2}\), \(t\in[0,T)\), with some \(K\in\mathbb R\), we derive an explicit formula for the joint Laplace transform of \(\int_0^t \frac{b(s)^2}{\sigma(s)^2} (X_2^{(\alpha)})^2\,ds\) 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) \(\widehat{\alpha}_t\) of \(\alpha \) can be expressed in terms of these random variables. As an application, we show that in case of \(\alpha =K\), \(K\neq 0\),
\[ \sqrt{I_K(t)} \big(\widehat{\alpha}_t-K\big) \overset{\mathcal L}= -\frac{\text{sign}(K)}{\sqrt{2}} \frac{\int_0^1 W_s\,dW_s}{\int_0^1(W_s)^2\,ds}, \quad \forall t\in(0,T), \]
where \(I_K(t)\) denotes the Fisher information for \(\alpha \) contained in the observation \(X_s^{(K)})_{s\in[0,t]}\), \((W_s)_{s\in[0,1]}\) is a standard Wiener process and \(\overset{\mathcal L}=\) denotes equality in distribution. We also prove asymptotic normality of the MLE \(\widehat{\alpha_t}\) of \(\alpha \) as \(t\uparrow T\) for \(\text{sign}(\alpha-K)= \text{sign}(K)\), \(K\neq 0\). 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 \(dX_t^{(\alpha)}= \frac{\alpha}{T-t} X_t^{(\alpha)}dt+dB_t\), \(t\in[0,T)\), with initial condition \(X_0^{(\alpha)}=0\). In case of \(\alpha >0\), this process is known as an \(\alpha \)-Wiener bridge, and in case of \(\alpha =1\), this is the usual Wiener bridge.

MSC:
60E10 Characteristic functions; other transforms
60J60 Diffusion processes
62F10 Point estimation
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