Abstract
In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.
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Dawson, P., Gailis, R., Meehan, A.: Detecting disease outbreaks using a combined Bayesian network and particle filter approach. J. Theor. Biol. 370, 171–183 (2015)
Mitter, S.K.: On the analogy between mathematical problems of nonlinear filtering and quantum physics. Ricerche Automat. 10(2), 163–216 (1979)
Yang, T., Huang, G., Mehta, P.G.: Joint probabilistic data association-feedback particle filter for multiple target tracking applications. In: 2012 American Control Conference (ACC), pp. 820–826. IEEE (2012)
Anadranistakis, M., Lagouvardos, K., Kotroni, V., Elefteriadis, H.: Correcting temperature and humidity forecasts using Kalman filtering: potential for agricultural protection in northern Greece. Atmos. Res. 71(3), 115–125 (2004)
Das, S.: Computational business analytics. CRC Press, Boca Raton (2013). https://doi.org/10.1201/b16358
Young, L., Young, J.: Statistical ecology (1998). https://doi.org/10.1007/978-1-4757-2829-3
Rüdiger, F., Thorsten, S.: Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering. Finance Stoch. 16(1), 105–133 (2012). https://doi.org/10.1007/s00780-011-0153-0
Frey, R., Schmidt, T., Xu, L.: On Galerkin approximations for the Zakai equation with diffusive and point process observations. SIAM J. Numer. Anal. 51(4), 2036–2062 (2013). https://doi.org/10.1137/110837395
Brémaud, P.: A Martingale approach to point processes, vol. 345. University of California, Berkeley (1972)
Frey, R., Schmidt, T.: Pricing corporate securities under noisy asset information. Math. Finance 19(3), 403–421 (2009). https://doi.org/10.1111/j.1467-9965.2009.00374.x
Aggoun, L.: Robust filtering and detection of an insurance model. Stoch. Dyn. 7(1), 91–102 (2007). https://doi.org/10.1142/S0219493707001949
Ceci, C., Colaneri, K., Cretarola, A.: Local risk-minimization under restricted information on asset prices. Electron. J. Probab. 20, 96–30 (2015). https://doi.org/10.1214/EJP.v20-3204
Qiao, H., Duan, J.: Nonlinear filtering of stochastic dynamical systems with Lévy noises. Adv. in Appl. Probab. 47(3), 902–918 (2015). https://doi.org/10.1239/aap/1444308887
Fernando, B.P.W., Hausenblas, E.: Nonlinear filtering with correlated Lévy noise characterized by copulas. Braz. J. Probab. Stat. 32(2), 374–421 (2018). https://doi.org/10.1214/16-BJPS347
Zakai, M.: On the optimal filtering of diffusion processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 11(3), 230–243 (1969)
Bensoussan, A., Glowinski, R., Răşcanu, A.: Approximation of the Zakai equation by the splitting up method. SIAM J. Control. Optim. 28(6), 1420–1431 (1990). https://doi.org/10.1137/0328074
Florchinger, P., le Gland, F.: Time-discretization of the Zakai equation for diffusion processes observed in correlated noise. Stochastics and Stochastic Rep. 35(4), 233–256 (1991). https://doi.org/10.1080/17442509108833704
Gyöngy, I., Krylov, N.: On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31(2), 564–591 (2003). https://doi.org/10.1214/aop/1048516528
Ito, K.: Approximation of the Zakai equation for nonlinear filtering. SIAM J. Control. Optim. 34(2), 620–634 (1996). https://doi.org/10.1137/S0363012993254783
Bao, F., Cao, Y., Webster, C., Zhang, G.: A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations. SIAM/ASA J. Uncertain. Quantif. 2(1), 784–804 (2014). https://doi.org/10.1137/140952910
Florchinger, P., Le Gland, F.: Time-discretization of the Zakai equation for diffusion processes observed in correlated noise. Stochastics Stochastics Rep. 35(4), 233–256 (1991). https://doi.org/10.1080/17442509108833704
Le Gland, F.: Splitting-up approximation for SPDEs and SDEs with application to nonlinear filtering. In: Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991). Lect. Notes Control Inf. Sci., vol. 176, pp. 177–187. Springer, Berlin (1992). https://doi.org/10.1007/BFb0007332
Xu, L.: On Galerkin approximations for the Zakai equation with diffusive and point process observations. PhD thesis, Leipzig, Univ., Diss., 2011 (2010)
Protter, P.E.: Stochastic integration and differential equations. Stochastic Modelling and Applied Probability, vol. 21, p. 419. Springer, Berlin (2005). https://doi.org/10.1007/978-3-662-10061-5. Second edition. Version 2.1, Corrected third printing
Jacod, J., Shiryaev, A.: Limit theorems for stochastic processes, vol. 288. Springer, Heidelberg (2013)
Pardoux, E.: Équations du filtrage non linéaire, de la prédiction et du lissage. Stochastics 6(3–4), 193–231 (1981/82). https://doi.org/10.1080/17442508208833204
Bain, A., Crisan, D.: Fundamentals of stochastic filtering. Stochastic Modelling and Applied Probability, vol. 60, p. 390. Springer, New York(2009). https://doi.org/10.1007/978-0-387-76896-0
Shreve, S.E.: Stochastic calculus for finance. II. Springer Finance, p. 550. Springer, New York (2004). Continuous-time models
Liptser, R.S., Shiryaev, A.N.: Statistics of random processes. I, expanded edn. Applications of Mathematics (New York), vol. 5, p. 427. Springer, Berlin (2001). General theory, Translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability
Bensoussan, A.: On a general class of stochastic partial differential equations. Stochastic Hydrology & Hydraulics 1(4), 297–302 (1987)
Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3(2), 127–167 (1979). https://doi.org/10.1080/17442507908833142
Da Prato, G., Kunstmann, P.C., Lasiecka, I., Lunardi, A., Schnaubelt, R., Weis, L.: Functional analytic methods for evolution equations. Lecture Notes in Mathematics, vol. 1855, p. 472. Springer, Berlin (2004). https://doi.org/10.1007/b100449. Edited by M. Iannelli, R. Nagel and S. Piazzera
Gawarecki, L., Mandrekar, V.: Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations. Probability and its Applications (New York), p. 291. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-16194-0
Milstein, G.N.: Numerical integration of stochastic differential equations. Mathematics and its Applications, vol. 313, p. 169. Kluwer Academic Publishers Group, Dordrecht (1995). https://doi.org/10.1007/978-94-015-8455-5. Translated and revised from the 1988 Russian original
Kanagawa, S.: Error estimations for the Euler-Maruyama approximate solutions of stochastic differential equations. Monte Carlo Methods Appl. 1(3), 165–171 (1995). https://doi.org/10.1515/mcma.1995.1.3.165
Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations, pp. 8–12. Springer, New York (2005)
Shen, J., Tang, T., Wang, L.-L.: Spectral methods: algorithms, analysis and applications, vol. 41. Springer, London (2011)
Funding
Yongkui Zou’s research was partly supported by the National Key R &D Program (2020YFA0714101, 2020YFA0713601), NSFC (12171199, 11971198), Jilin Provincial Department of Science and Technology (20210201015GX). Fengshan Zhang’s research was partially supported by the Postdoctoral Fellowship Program of CPSF under Grant Number (GZC20232911), and Yanzhao Cao’s research was partially supported by the US Department of Energy, Office of Science, Advanced Scientific Computing Research with grant number DE-SC0022253.
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Zhang, F., Zou, Y., Chai, S. et al. Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations. Adv Comput Math 50, 73 (2024). https://doi.org/10.1007/s10444-024-10169-w
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DOI: https://doi.org/10.1007/s10444-024-10169-w