| Tsirelson's stochastic di erential equation is called \celebrated and mysterious" by Rogers and Williams bd16ce. This note aims at making it a little more celebrated and a little less mysterious. Using a deterministic time-change, we translate the study of Tsirelson's equation into the study of \eternal" Brownian motion on the circle. This allows us to show that the ltration generated by any ...

where (Yt,Zt) are unknown predictable processes. We will assume that f is a Lipschitz function with respect to its arguments throughout this paper. Since this equation has its important applications into control theory and mathematical finance, many mathematicians are not satisfied merely by descriptive existence theorems. They are also interested in constructing the numerical solutions. In ord...

We consider a stochastic control problem which is composed of a controlled stochastic differential equation, and whose associated cost functional is defined through a controlled backward stochastic differential equation. Under appropriate convexity assumptions on the coefficients of the forward and the backward equations we prove the existence of an optimal control on a suitable reference stoch...

In many applications (engineering, management, economy) one is led to control problems for stochastic systems : more precisely the state of the system is assumed to be described by the solution of stochastic differential equations and the control enters the coefficients of the equation. Using the dynamic programming principle E. Bellman [6] explained why, at least heuristically, the optimal cos...

We propose a novel stochastic method to generate paths conditioned to start in an initial state and end in a given final state during a certain time t(f). These paths are weighted with a probability given by the overdamped Langevin dynamics. We show that these paths can be exactly generated by a non-local stochastic differential equation. In the limit of short times, we show that this complicat...

Supplementary Material A. Background on Fokker-Planck Equation The Fokker-Planck equation (FPE) associated with a given stochastic differential equation (SDE) describes the time evolution of the distribution on the random variables under the specified stochastic dynamics. For example, consider the SDE: dz = g(z)dt+N (0, 2D(z)dt), (16) where z ∈ R, g(z) ∈ R, D(z) ∈ Rn×n. The distribution of z go...

Some stochastic control systems that are described by stochastic differential equations with a fractional Brownian motion are considered. The solutions of these systems are defined by weak solutions. These weak solutions are obtained by the transformation of the measure for a fractional Brownian motion by a Radon-Nikodym derivative. This weak solution approach is used to solve a control problem...

Height-diameter models are classically analyzed by fixed or mixed linear and non-linear regression models. In order to possess the among-plot variability, we propose stochastic differential equations that are deduced from the standard deterministic dynamic ordinary differential equations by adding the process variability to the growth dynamic. The advantage of the stochastic differential equati...