Volume 19, Issue 1

This paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 ≤ m < n be fixed positive integers, some open domain, and h : a function yielding a full partitioning of D into a family, denoted M ( h ), of lower-dimensional surfaces/manifolds via inverse mapping h –1 as D = ∪ M ( h ), where M ( h ) = d { h –1 ( t ) : t in range( h) }, noting each h –1 ( t ) can also be considered the solution set of all X in D of the simultaneous equations h ( X ) = t . Let X be a random vector (rv) over D having a probability density function (pdf) ƒ. Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh ( X )/ dX over D , etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M ( h ) that satisfies a list of natural desirable properties? More generally, for fixed positive integer r , we can pose a similar question concerning rv ψ ( X ), when ψ : is some bounded a.e. continuous function, not necessarily admitting a pdf.

In this paper we consider a backward stochastic differential equation driven by an infinite dimensional martingale. Our aim is to derive the existence and uniqueness of the solution to such an equation. The filtration we consider is an arbitrary right continuous one not necessarily the natural filtration of a Brownian motion, which is furnished usually for the theory of BSDEs. This in particular allows us to study more applications, for example the maximum principle for an optimal control of a stochastic system.

We study one-dimensional stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion. We prove some properties of the solutions of such equations and of the corresponding Euler scheme. We obtain the convergence rate of the Euler scheme for diffusions with weak singularity at zero.

We consider a stock price Z t whose dynamics follows a geometric Brownian motion living on the standard Gaussian white noise space. We regard the risk-free interest rate r and volatility σ as independent variables of the stock price. We show that the partial derivatives of the stock price with respect to r and σ satisfy equations which involve the Gross Laplacian and the number operator of the stock price. Introducing an operator transferring white noise functionals to generalized functionals of square of white noise, we give equations for the stock price including the Lévy Laplacian and the Volterra Laplacian. Moreover we prove that those equations characterize the stock price up to a constant only depending on time t .