Gopinath Kallianpur

Stochastic evolution equations driven by nuclear-space-valued martingales

This paper presents a theory of stochastic evolution equations for nuclear-space-valued processes and provides a unified treatment of several examples from the field of applications. (C0, 1) reversed evolution systems on countably Hilbertian nuclear spaces are also investigated.

Infinite dimensional stochastic differential equation models for spatially distributed neurons

The membrane potential of spatially distributed neurons is modeled as a random field driven by a generalized Poisson process. Approximation to an Ornstein-Uhlenbeck type process is established in the sense of weak convergence of the induced measures in Skorokhod space.

Approximations to the solution of the zakai equation using multiple wiener and stratonovich integral expansions

A system of mtegro-differential equations, for the kernels in the multiple Wiener integral (MWI) representation for the solution of the Zakai equation, is derived. Approximations for the conditiona...

A finitely additive white noise approach to nonlinear filtering

An approach to nonlinear filtering theory is developed in which finitely additive white noise replaces the Wiener process in the observation process model. The important case when the signal is a Markov process independent of the noise is investigated in detail. The theory turns out to be simpler than the current theory based on the stochastic calculus. Stochastic partial differential equations are replaced by partial differential equations in which the observation (in the finitely additive set up) occurs as a parameter. Theorems on existence and uniqueness of solutions are obtained. The white noise approach has the advantage that it provides a robust solution to the filtering problem. Furthermore, the robust theory based on the Ito calculus can be recovered from the results of this paper.

Diffusion equations in duals of nuclear spaces

A stochastic Galerkin method is used to establish the existence of a solution to a martingale problem posed by an Ito type stochastic differential equation for processes taking values in the dual of a nuclear space. Uniqueness of the strong solution is also shown using the monotonicity condition. An application to the motion of random strings is discussed.

Robustness of the nonlinear filter

In the nonlinear filtering model with signal and observation noise independent, we show that the filter depends continuously on the law of the signal. We do not assume that the signal process is Markov and prove the result under minimal integrability conditions. The analysis is based on expressing the nonlinear filter as a Wiener functional via the Kallianpur-Striebel Bayes formula.

Gaussian random fields

Nonanticipative representations of Gaussian random fields equivalent to the two-parameter Wiener process are defined, and necessary and sufficient conditions for their existence derived. When such representations exist they provide examples of canonical representations of multiplicity one. In contrast to the one-parameter case, examples are given where nonanticipative representations do not exist. Nonanticipative representations along increasing paths are also studied.

Propagation of chaos and the McKean-Vlasov equation in duals of nuclear spaces

An interacting system ofn stochastic differential equations taking values in the dual of a countable Hilbertian nuclear space is considered. The limit (in probability) of the sequence of empirical measures determined by the above systems asn tends to ∞ is identified with the law of the unique solution of the McKean-Vlasov equation. An application of our result to interacting neurons is briefly discussed. The propagation of chaos result obtained in this paper is shown to contain and improve the well-known finite-dimensional results.

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained.

The existence and uniqueness of solutions of nuclear space-valued stochastic differential equations driven by poisson random measures

In this paper, we study stochastic differential equations (SDEs) on duals of nuclear spaces driven by Poisson random measures. The existence of a weak solution is obtained by the Galerkin method. For uniqueness, a class of :-valued processes which are called Good processes are introduced. An equivalence relation is established between SDEs driven by Poisson random measures and those by Good processes. The uniqueness is established by extending the Yamada-Watanabe argument to the SDEs driven by Good processes. This is an extension to discontinuous infinite dimensional SDEs of work done by G. Kallianpur, I. Mitoma and R. Wolpert for nuclear space valued diffusions.