## 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
Full Text:

### References:

 [1] Albanese, C.; Lawi, S., Laplace transforms for integrals of Markov processes, Markov Process. Related Fields, 11, 4, 677-724 (2005) · Zbl 1090.60068 [2] Arató, M., Linear Stochastic Systems with Constant Coefficients. A Statistical Approach, Lecture Notes in Control and Inform. Sci., vol. 45 (1982), Springer · Zbl 0544.93060 [3] Barczy, M.; Iglói, E., Karhunen-Loève expansions of alpha-Wiener bridges, Cent. Eur. J. Math., 9, 1, 65-84 (2011) · Zbl 1228.60047 [4] Barczy, M.; Pap, G., Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes, J. Statist. Plann. Inference, 140, 6, 1576-1593 (2010) · Zbl 1185.62147 [5] Bishwal, J. P.N., Parameter Estimation in Stochastic Differential Equations (2007), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0936.60024 [6] M.J. Bobkoski, Hypothesis testing in nonstationary time series, PhD dissertation, University of Wisconsin, 1983.; M.J. Bobkoski, Hypothesis testing in nonstationary time series, PhD dissertation, University of Wisconsin, 1983. [7] Borodin, A. N.; Salminen, P., Handbook of Brownian Motion - Facts and Formulae (2002), Birkhäuser · Zbl 1012.60003 [8] Brennan, M. J.; Schwartz, E. S., Arbitrage in stock index futures, J. Business, 63, 1, S7-S31 (1990) [9] Deheuvels, P.; Martynov, G., Karhunen-Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, (Progr. Probab., vol. 55 (2003), Birkhäuser Verlag: Birkhäuser Verlag Basel), 57-93 · Zbl 1048.60021 [10] Deheuvels, P.; Peccati, G.; Yor, M., On quadratic functionals of the Brownian sheet and related processes, Stochastic Process. Appl., 116, 3, 493-538 (2006) · Zbl 1090.60020 [11] Delyon, B.; Hu, Y., Simulation of conditioned diffusion and application to parameter estimation, Stochastic Process. Appl., 116, 11, 1660-1675 (2006) · Zbl 1107.60046 [12] Es-Sebaiy, K.; Nourdin, I., Parameter estimation for $$α$$-fractional bridges (2011) · Zbl 1268.62099 [13] Feigin, P. D., Some comments concerning a curious singularity, J. Appl. Probab., 16, 2, 440-444 (1979) · Zbl 0409.62082 [14] Florens-Landais, D.; Pham, H., Large deviations in estimation of an Ornstein-Uhlenbeck model, J. Appl. Probab., 36, 1, 60-70 (1999) · Zbl 0978.62070 [15] Gao, F.; Hannig, J.; Lee, T.-Y.; Torcaso, F., Laplace transforms via Hadamard factorization, Electron. J. Probab., 8, 13 (2003), 20 pp · Zbl 1064.60061 [16] Hurd, T. R.; Kuznetsov, A., Explicit formulas for Laplace transforms of stochastic integrals, Markov Process. Related Fields, 14, 2, 277-290 (2008) · Zbl 1149.60021 [17] Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes (2003), Springer-Verlag: Springer-Verlag Berlin · Zbl 0830.60025 [18] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0734.60060 [19] Kleptsyna, M. L.; Le Breton, A., Statistical analysis of the fractional Ornstein-Uhlenbeck type process, Stat. Inference Stoch. Process., 5, 3, 229-248 (2002) · Zbl 1021.62061 [20] Kleptsyna, M. L.; Le Breton, A., A Cameron-Martin type formula for general Gaussian processes - a filtering approach, Stochastics and Stochastics Reports, 72, 3-4, 229-250 (2002) · Zbl 1002.60031 [21] Liptser, R. S.; Shiryaev, A. N., Statistics of Random Processes I. General Theory (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1008.62072 [22] Liptser, R. S.; Shiryaev, A. N., Statistics of Random Processes II. Applications (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0591.60039 [23] Luschgy, H., Local asymptotic mixed normality for semimartingale experiments, Probab. Theory Related Fields, 92, 2, 151-176 (1992) · Zbl 0768.62067 [24] Mansuy, R., On a one-parameter generalization of the Brownian bridge and associated quadratic functionals, J. Theoret. Probab., 17, 4, 1021-1029 (2004) · Zbl 1063.60049 [25] Prakasa Rao, B. L.S., Semimartingales and Their Statistical Inference (1999), Chapman & Hall/CRC · Zbl 0960.62090 [26] Tanaka, K., Time Series Analysis, Nonstationary and Noninvertible Distribution Theory, Wiley Ser. Probab. Stat. (1996) · Zbl 0861.62062 [27] Yor, M., Exponential Functionals of Brownian Motion and Related Processes (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 0999.60004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.