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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.

60E10 Characteristic functions; other transforms
60J60 Diffusion processes
62F10 Point estimation
Full Text: DOI
[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 Berlin, Heidelberg · Zbl 0936.60024
[6] 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, (), 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 Berlin · Zbl 0830.60025
[18] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), 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 Berlin, Heidelberg
[22] Liptser, R.S.; Shiryaev, A.N., Statistics of random processes II. applications, (2001), 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 Berlin · Zbl 0999.60004
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