Related papers: Stochastic Calculus via Stopping Derivatives

Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any…

We present here a simple method for computing the large deviation of long time average for stochastic jump processes. We show that the computation of the rate function can be reduced to that of a partial differential equation governing the…

We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path…

Stochastic gradient descent is an optimisation method that combines classical gradient descent with random subsampling within the target functional. In this work, we introduce the stochastic gradient process as a continuous-time…

We derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is…

Classical Stokes' drift is the small time-averaged drift velocity of suspended non-diffusing particles in a fluid due to the presence of a wave. We consider the effect of adding diffusion to the motion of the particles, and show in…

The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the…

In studying randomized search heuristics, a frequent quantity of interest is the first time a (real-valued) stochastic process obtains (or passes) a certain value. The processes under investigation commonly show a bias towards this goal,…

We approximate stochastic processes in finite dimension by dynamical systems. We provide trajectorial estimates which are uniform with respect to the initial condition for a well chosen distance. This relies on some non-expansivity property…

A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a…

Timeseries generated from a dynamical source can often be modeled as sample paths of a stochastic differential equation (SDE). The timeseries thus reflects the motion of a particle which flows along the direction provided by a drift /…

Stochastic processes are proposed whose master equations coincide with classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors based on the Goldstein-Kac telegraph process, the model describes the motion of particles…

This paper contributes to the study of relative martingales. Specifically, for a closed random set $H$, they are processes null on $H$ which decompose as $M=m+v$, where $m$ is a c\`adl\`ag uniformly integrable martingale and, $v$ is a…

Stochastic integrals are defined with respect to a collection $P = (P_i; \, i \in I)$ of continuous semimartingales, imposing no assumptions on the index set $I$ and the subspace of $\mathbb{R}^I$ where $P$ takes values. The integrals are…

We develop a stochastic model for Lagrangian velocity as it is observed in experimental and numerical fully developed turbulent flows. We define it as the unique statistically stationary solution of a causal dynamics, given by a stochastic…

In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about…

We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…

We give a stochastic extension of the Brane Calculus, along the lines of recent work by Cardelli and Mardare. In this presentation, the semantics of a Brane process is a measure of the stochastic distribution of possible derivations. To…

Many time series are effectively generated by a combination of deterministic continuous flows along with discrete jumps sparked by stochastic events. However, we usually do not have the equation of motion describing the flows, or how they…

The concept of impedance, which characterises the current response to a periodical driving, is introduced in the context of stochastic transport. In particular, we calculate the impedance for an exactly solvable model, namely the stochastic…