Abstract
A Bayes‐type formula is derived for the nonlinear filter where the observation contains both general Gaussian noise as well as Cox noise whose jump intensity depends on the signal. This formula extends the well‐known Kallianpur‐Striebel formula in the classical non‐linear filter setting. We also discuss Zakai‐type equations for both the unnormalized conditional distribution as well as unnormalized conditional density in case the signal is a Markovian jump diffusion.
Highlights
The general filtering setting can be described as follows
Assume a partially observable process X, Y Xt, Yt 0≤t≤T ∈ R2 defined on a probability space Ω, F, P
The real valued process Xt stands for the unobservable component, referred to as the signal process or system process, whereas Yt is the observable part, called observation process
Summary
The general filtering setting can be described as follows. Assume a partially observable process X, Y Xt, Yt 0≤t≤T ∈ R2 defined on a probability space Ω, F, P. Information about Xt can only be obtained by extracting the information about X that is contained in the observation Yt in a best possible way In filter theory this is done by determining the conditional distribution of Xt given the information σ-field FtY generated by Ys, 0 ≤ s ≤ t. The objective of the paper is in a first step to extend the Kallianpur-Striebel Bayes type formula to the generalized filter setting from above. In a second step we derive a Zakai-type measure valued stochastic differential equations for the unnormalized conditional distribution of the filter. For this purpose we assume the signal process X to be a Markov process with generator Ot : Lt Bt given as.
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