10 research outputs found
On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations: On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations
We are interested in a nonlinear filtering problem motivated by an
information-based approach for modelling the dynamic evolution of a
portfolio of credit risky securities.
We solve this
problem by `change of measure method\\\'' and show the existence of the
density of the unnormalized conditional distribution which is a
solution to the Zakai equation. Zakai equation is a linear SPDE
which, in general, cannot be solved analytically. We apply Galerkin
method to solve it numerically and show the convergence of Galerkin
approximation in mean square. Lastly, we design an adaptive Galerkin
filter with a basis of Hermite polynomials and we present numerical
examples to illustrate the effectiveness of the proposed method. The
work is closely related to the paper Frey and Schmidt (2010)
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows
This paper presents a nonparametric statistical modeling method for
quantifying uncertainty in stochastic gradient systems with isotropic
diffusion. The central idea is to apply the diffusion maps algorithm to a
training data set to produce a stochastic matrix whose generator is a discrete
approximation to the backward Kolmogorov operator of the underlying dynamics.
The eigenvectors of this stochastic matrix, which we will refer to as the
diffusion coordinates, are discrete approximations to the eigenfunctions of the
Kolmogorov operator and form an orthonormal basis for functions defined on the
data set. Using this basis, we consider the projection of three uncertainty
quantification (UQ) problems (prediction, filtering, and response) into the
diffusion coordinates. In these coordinates, the nonlinear prediction and
response problems reduce to solving systems of infinite-dimensional linear
ordinary differential equations. Similarly, the continuous-time nonlinear
filtering problem reduces to solving a system of infinite-dimensional linear
stochastic differential equations. Solving the UQ problems then reduces to
solving the corresponding truncated linear systems in finitely many diffusion
coordinates. By solving these systems we give a model-free algorithm for UQ on
gradient flow systems with isotropic diffusion. We numerically verify these
algorithms on a 1-dimensional linear gradient flow system where the analytic
solutions of the UQ problems are known. We also apply the algorithm to a
chaotically forced nonlinear gradient flow system which is known to be well
approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11
The Zakai equation of nonlinear filtering for jump-diffusion observation: existence and uniqueness
This paper is concerned with the nonlinear filtering problem for a general
Markovian partially observed system (X,Y), whose dynamics is modeled by
correlated jump-diffusions having common jump times. At any time t, the
sigma-algebra generated by the observation process Y provides all the available
information about the signal X. The central goal of stochastic filtering is to
characterize the filter which is the conditional distribution of X, given the
observed data. It has been proved in Ceci-Colaneri (2012) that the filter is
the unique probability measure-valued process satisfying a nonlinear stochastic
equation, the so-called Kushner-Stratonovich equation (KS-equation). In this
paper the aim is to describe the filter in terms of the unnormalized filter,
which is solution to a linear stochastic differential equation, called the
Zakai equation. We prove equivalence between strong uniqueness for the solution
to the Kushner Stratonovich equation and strong uniqueness for the solution to
the Zakai one and, as a consequence, we deduce pathwise uniqueness for the
solutions to the Zakai equation by applying the Filtered Martingale Problem
approach (Kurtz-Ocone (1988), Kurtz-Nappo (2011), Ceci-Colaneri (2012)). To
conclude, we discuss some particular cases.Comment: 29 page
On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations: On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations
We are interested in a nonlinear filtering problem motivated by an
information-based approach for modelling the dynamic evolution of a
portfolio of credit risky securities.
We solve this
problem by `change of measure method\\\'' and show the existence of the
density of the unnormalized conditional distribution which is a
solution to the Zakai equation. Zakai equation is a linear SPDE
which, in general, cannot be solved analytically. We apply Galerkin
method to solve it numerically and show the convergence of Galerkin
approximation in mean square. Lastly, we design an adaptive Galerkin
filter with a basis of Hermite polynomials and we present numerical
examples to illustrate the effectiveness of the proposed method. The
work is closely related to the paper Frey and Schmidt (2010)
On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations: On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations
We are interested in a nonlinear filtering problem motivated by an
information-based approach for modelling the dynamic evolution of a
portfolio of credit risky securities.
We solve this
problem by `change of measure method\\\'' and show the existence of the
density of the unnormalized conditional distribution which is a
solution to the Zakai equation. Zakai equation is a linear SPDE
which, in general, cannot be solved analytically. We apply Galerkin
method to solve it numerically and show the convergence of Galerkin
approximation in mean square. Lastly, we design an adaptive Galerkin
filter with a basis of Hermite polynomials and we present numerical
examples to illustrate the effectiveness of the proposed method. The
work is closely related to the paper Frey and Schmidt (2010)