Absolute Continuity Of Probability Measures Research Articles

The stochastic maximum principle in optimal control of degenerate diffusions with non-smooth coefficients

For a controlled stochastic differential equation with a finite horizon cost functional, a necessary condition for optimal control of degenerate diffusions with non-smooth coefficients is derived. The main idea is to show that the SDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. We use a technique of Bouleau–Hirsch on absolute continuity of probability measures in order to define the adjoint process on an extension of the initial probability space.

Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations

We study the regularity of the viscosity solution of a quasilinear parabolic partial differential equation with Lipschitz coefficients by using its connection with a forward backward stochastic differential equation (in short FBSDE) and we give a probabilistic representation of the generalized gradient (derivative in the distribution sense) of the viscosity solution. This representation is a kind of nonlinear Feynman–Kac formula. The main idea is to show that the FBSDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. If the diffusion coefficient of the forward equation is uniformly elliptic, we approximate the FBSDE by smooth ones and use Krylov’s estimate to prove the convergence of the derivatives. In the degenerate case, we use techniques of Bouleau–Hirsch on absolute continuity of probability measures.

Density estimation by orthogonal series in an infinite dimensional space: Application to processes of diffusion type I

In this paper, we will be concerned with the estimation of the probability density of a diffusion process with respect to the Wiener measure. Since a diffusion process can be viewed as a random variable taking its values in an infinite dimensional space, this problem can be viewed as a special case of estimating the probability density of a random variable with values in an infinite dimensional space. However, obtaining the absolute continuity of probability measures in infinite dimension is difficult. A density estimator based on a Fourier–Hermite orthogonal series is investigated, which is a generalization of the classical density estimator of a real valued random variable. We establish that our estimator converges in integrated and simple mean square, under suitable conditions. Rates of convergence are also given.

On a criterion of Riemannian distance for singularity and absolute continuity of probability measures

Parametric statistical models, with suitable regularity conditions, have a natural Riemannian manifold structure given by the Fisher information metric. This paper is concerned with the Riemannian distance for geometric conditions on singularity or absolute continuity of probability measures which depend on the parameters.

Distinguishing a Sequence of Random Variables from a Random Translate of Itself

Let $\mathbf{X} = \{X_k\}$ be an i.i.d. real random sequence, let $\epsilon = \{\epsilon_k\}$ be a Rademacher sequence independent of $\mathbf{X}$ and let $\mathbf{a} = \{a_k\}$ be a deterministic real sequence. The aim of this paper is to prove that the mutual absolute continuity of probability measures induced by $\{X_k\}$ and $\{X_k + a_k\epsilon_k\}$ implies $\mathbf{a} \in \ell_4$. This is a generalization of a result of Shepp.

Positive Martingales and Their Induced Measures

Conditions for the absolute continuity of probability measures are given in terms of limits of sequences of Radon-Nikodym derivatives. In the other direction, conditions for the optional stopping theorem for positive martingales are given in terms of properties of their induced measures.

Positive martingales and their induced measures

Conditions for the absolute continuity of probability measures are given in terms of limits of sequences of Radon-Nikodym derivatives. In the other direction, conditions for the optional stopping theorem for positive martingales are given in terms of properties of their induced measures.

ON CONDITIONS FOR ABSOLUTE CONTINUITY OF PROBABILITY MEASURES

A number of sufficient conditions are obtained for the absolute continuity of probability measures under the assumption that their restrictions to some increasing system of σ-algebras are absolutely continuous. An example is given which shows that these conditions are close to being best possible. Bibliography: 7 titles.