On the Nonlinear Filtering Equations for Superprocesses in Random Environment
In the paper we define the Dawson-Watanabe type superprocesses in random environment as solutions to the related martingale problems. An environment is modelled by a finite state time homogeneous Markov process with the given transition probability intens
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Abstract In the paper we define the Dawson-Watanabe type superprocesses in random environment as solutions to the related martingale problems. An environment is modelled by a finite state time homogeneous Markov process with the given transition probability intensity matrix. A system of nonlinear stochastic equations is derived for a posteriori probabilities. Reduced system of linear equations is also obtained. Keywords Covariance operator • Cylindrical martingale • Dawson-Watanabe superprocess • Martingale problem • Nonlinear filtering • Random environment • Reduced equation • Stochastic integral Mathematics Subject Classification (2010): 60J70, 60K37
1 Introduction Dawson-Watanebe type of superprocesses in Rd arise as the scaling limits of branching particle systems, which undergo near critical branching and Markov spatial motions, characterized via Prokorov’s relative compactness criterion as unique solutions to the related martingale problems (see, e.g., [9]). Restricting ourselves to the finite variance state depending branching mechanisms and diffusions with jumps spatial motions, we shall arrive to the following definition.
B. Grigelionis () Institute of Mathematics and Informatics, Vilnius University, Akademijos str. 4, LT-08663 Vilnius, Lithuania e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 16, © Springer-Verlag Berlin Heidelberg 2013
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Let Cb .Rd / D ff W Rd ! R1 ; f is continuous and boundedg; Cb2 .Rd / D ff W Rd ! R1 ; f is C 2 with bounded partials of order 2 or lessg; equipped with the topology of uniform convergence on compact sets, .Af /.x/ D
d X
bj .x/
j D1
Z C Rd nfxg
d ıf 1 X ı2f .x/ C aj k .x/ .x/ ıxj 2 ıxj ıxk j;kD1
3 X ıf 4f .y/ f .x/ 1jxyj1 .y/ .yj xj / .x/5˘.x; dy/; x 2 Rd ; ıx j j D1 2
with coefficients, satisfying the standard Lipschitz continuity and linear growth conditions, A W Cb2 .Rd / ! Cb .Rd /; measures on Rd , endowed with the topology of weak MF .Rd / be a space of finite R convergence, .f / D f .x/.dx/, W Rd ! Œ0; 1/ and hj W Rd ! R1 ; j D 0; Rd
1; : : : ; N are continuous bounded functions, 1A is an indicator function. Let .˝; F ; F; P/ be a stochastic basis, F D fFt ; t 0g, Mloc .P; F/ be the class c of .P; F/ – local martingales, Mloc .P; F/ be the class of continuous .P; F/ – local martingales. (For used terminology and notations from stochastic analysis see, e.g., [5, 7]). Consider a stochastic process f.t ; Xt /; t 0g, taking values in f0; 1; : : : ; N g MF .Rd /, where ft ; t 0g is a finite state time homogeneous .P; F/ – Markov chain with the transition probability intensity matrix D ..j; k//0j;kN and fXt ; t 0g is an F – adapted MF .Rd / – valued process such that, for each f 2 Cb2 .Rd /, Z
t
Mt .f / WD Xt .f / X0 .f /
Xs .Af C hs f /ds; 0
is a continuous .P; F/ – local martingale, satisfying Zt hM.f /it D
Xs .f 2 /ds; 0
t 0:
t 0;
On the Nonline
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