Related papers: Martingales with Independent Increments
Some classes of increment martingales, and the corresponding localized classes, are studied. An increment martingale is indexed by the real line and its increment processes are martingales. We focus primarily on the behavior as time goes to…
This paper provides the time-dependent $L^2$-martingale representation of the forward stochastic integral where the driving noise is the Riemann-Liouville fractional Brownian motion with parameter $\frac{1}{2} < H < 1$ and the integrand is…
An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of…
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using the Taylor expansion, is…
We construct a class of one-dimensional diffusion processes on the particles of branching Brownian motion that are symmetric with respect to the limits of random martingale measures. These measures are associated with the extended extremal…
We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively…
We investigate the global well-posedness and asymptotic behavior of $L^2$-solutions to stochastic nonlinear Schr\"odinger equations with multiplicative noise driven by continuous square integrable martingales with density. Our approach…
In this paper we derive novel change of variable formulas for stochastic integrals w.r.t. a time-changed Brownian motion where we assume that the time-change is a general increasing stochastic process with finitely many jumps in a bounded…
In this paper we study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moment. For the sake of applications we derive the self-normalized version of this theorem.…
In this paper, we assume that the filtration $\bb F$ is generated by a $d$-dimensional Brownian motion $W=(W_1,\cdots,W_d)'$ as well as an integer-valued random measure $\mu(du,dy)$. The random variable $\ttau$ is the default time and $L$…
We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends…
It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous…
Determinantal process is a dynamical extension of a determinantal point process such that any spatio-temporal correlation function is given by a determinant specified by a single continuous function called the correlation kernel.…
We show that solutions of free stochastic differential equations with regular drifts and diffusion coefficients, when considered backwards in time, still satisfy free SDEs for an explicit free Brownian motion and drift. We also study the…
We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of…
We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such…
Consider $\mathbb{G}$ the progressive enlargement of a filtration $\mathbb{F}$ with a random time $\tau$. Assuming that, in $\mathbb{F}$, the martingale representation property holds, we examine conditions under which the martingale…
This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic…
We derive the explicit form of the martingale representation for square-integrable processes that are martingales with respect to the natural filtration of the super-Brownian motion. This is done by using a weak extension of the Dupire…
This short note explores the maximum-entropy walk on the unit interval that is a median-martingale. That is, the median of its next state is equal to its current state. The stationary distribution of this walk is the arcsine distribution,…