Let X be a Markov process with generator A and let Y (t) = γ(X(t)). The conditional distribution πt of X(t) given σ(Y (s) : s ≤ t) is characterized as a solution of a filtered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of Rogers and Pitman on Markov functions. Applications include uniqueness of filtering equations, exchangeability of the s...

Suppose we are given a computably enumerable object. We interested in the strength of oracles that can compute an object approximates this c.e. It turns out many cases arising from algorithmic randomness or computable analysis, resulting highness property is either close to, equivalent to being PA complete. examine, for example, majorizing martingale by oracle-computable martingale, computing l...

The stochastic exponential Zt = exp{Mt − M0 − (1/2)〈M,M〉t} of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where Mt = R t 0 b(Yu) dWu and Y is a one-dimensional diffusion driven by a Brownian motion W . Furthermore, we provide a necessary and sufficient condition for Z ...

We explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory. We prove that a fuzzy market is viable if and only if an equivalent martingale measure exists, from which we construct the forward probability measure and under which the discounted price of a default-free bond is a martingale. We model the evolution of credit migration of a d...

Theorem 1 (Jewett-Krieger). Let (X,A , μ, S) be an ergodic dynamical system. Then there is a compact space Y and a homeomorphism T on Y , which is uniquely ergodic and minimal, such that if ν denotes the unique T -invariant probability measure on Y and B the Borel σalgebra of Y , the ergodic dynamical system (Y,B, ν, T ) is equivalent to (X,A , μ, S). In addition, (Y,B, ν, T ) can be constructe...