Related papers: Stochastic Calculus for the Theta Process
This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path $\boldsymbol{\omega}$ with Hurst index…
We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic…
We consider minimization of stochastic functionals that are compositions of a (potentially) non-smooth convex function $h$ and smooth function $c$ and, more generally, stochastic weakly-convex functionals. We develop a family of stochastic…
Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in…
We study the limit of the joint distribution of a multidimensional Generalized Tempered Stable (GTS) process and its quadratic covariation process when the stable index tends to two. Under a proper scaling, the GTS processes converges to a…
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions:…
We present a simple stochastic quadrant model for calculating the transport and de- position of heavy particles in a fully developed turbulent boundary layer based on the statistics of wall-normal fluid velocity fluctuations obtained from a…
This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that…
A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and…
We study the statistical inference problem for a complex $\alpha$-fractional Brownian bridge process $Z$ defined by the stochastic differential equation \[ \mathrm{d}Z_t = -\alpha \frac{Z_t}{T - t} \mathrm{d}t + \mathrm{d}\zeta_t, \quad t…
We derive the classical Hamilton-Jacobi equation from first principles as the natural description for smooth stochastic processes when one neglects stochastic velocity fluctuations. The Schr\"{o}dinger equation is shown to be the natural…
This primer explains how continuous-time stochastic processes (precisely, Brownian motion and other Ito diffusions) can be defined and studied on manifolds. No knowledge is assumed of either differential geometry or continuous-time…
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…
We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, $F(t)$, has a nontrivial quartic variation. This process,…
Stochastic processes play a fundamental role in physics, mathematics, engineering and finance. One potential application of quantum computation is to better approximate properties of stochastic processes. For example, quantum algorithms for…
This paper provides an existence-and-uniqueness theorem characterizing the stochastic integral with respect to a Wiener process. The integral is represented as a mapping from the space of measurable and adapted pathwise locally integrable…
In the paper we study stochastic convolution appearing in Volterra equation driven by so called L\'evy process. By L\'evy process we mean a process with homogeneous independent increments, continuous in probability and cadlag.
We present a method for incorporating a stochastic point of view into physics exercises of mathematics education. The core of our method is the randomization of some inputs, the system model used does not differ from what we would use in…
Several versions of It\^{o}'s formula have been obtained in the context of the functional stochastic calculus. Here, we revisit this topic in two ways. First, by defining a notion of derivative along a functional, we extend the setting of…
We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that…