Related papers: Representation for filtration-consistent nonlinear…
In this paper we study a family of nonlinear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued…
In stochastic decision problems, one often wants to estimate the underlying probability measure statistically, and then to use this estimate as a basis for decisions. We shall consider how the uncertainty in this estimation can be…
In this paper, a general model of wireless channels is established based on the physics of wave propagation. Then the problems of inverse scattering and channel prediction are formulated as nonlinear filtering problems. The solutions to the…
The properties of nonlinear PDEs that generate filtered solutions are explored with particular attention given to the constraints on the residual term. The analysis is carried out for nonlinear PDEs with an emphasis on evolution problems…
It has been a well-known problem in the $G$-framework that it is hard to compute the sublinear expectation of the $G$-normal distribution $\hat{\mathbb{E}}[\varphi(X)]$ when $\varphi$ is neither convex nor concave, if not involving any PDE…
We provide a general approach to construct a stochastic process with a given consistent family of finite dimensional distributions under a nonlinear expectation space. We use this approach to construct a generalized Gaussian process under a…
In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for…
This paper is concerned with the problem of nonlinear filter stability of ergodic Markov processes. The main contribution is the conditional Poincar\'e inequality (PI), which is shown to yield filter stability. The proof is based upon a…
There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include…
The theory of backward SDEs extends the predictable representation property of Brownian motion to the nonlinear framework, thus providing a path-dependent analog of fully nonlinear parabolic PDEs. In this paper, we consider backward SDEs,…
This work concerns generalized backward stochastic differential equations, which are coupled with a family of reflecting diffusion processes. First of all, we establish the large deviation principle for forward stochastic differential…
We define $g$-expectation of a distribution as the infimum of the $g$-expectations of all the terminal random variables sharing that distribution. We present two special cases for nonlinear $g$ where the $g$-expectation of distributions can…
We develop a notion of nonlinear expectation --G-expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce…
In this paper, the problem of state estimation, in the context of both filtering and smoothing, for nonlinear state-space models is considered. Due to the nonlinear nature of the models, the state estimation problem is generally intractable…
We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution…
Multi-modal densities appear frequently in time series and practical applications. However, they cannot be represented by common state estimators, such as the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF), which…
This paper is intended to give a probabilistic representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use it connection with…
In this letter, a new filtering technique to solve a nonlinear state estimation problem has been developed. It is well known that for a nonlinear system, the prior and posterior probability density functions (pdf) are non-Gaussian in…
A $G$-normal random variable $X\sim \mathcal{N}(0,[\underline{\sigma}^2,\overline{\sigma}^2])$ does not admit a unique probability law due to volatility uncertainty. For a given test function $\phi$, the $G$-expectation admits the…
We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, $L^p$ data and generators satisfying weak integrability conditions. We deal with equations…