In a work of van Gaans (2005a) stochastic integrals are regarded as L2-curves. In Filipović and Tappe (2008) we have shown the connection to the usual Itô-integral for càdlàg-integrands. The goal of this note is to complete this result and to provide the full connection to the Itô-integral. We also sketch an application to stochastic partial differential equations.

In this paper, we introduce the idea of stochastic integrals with respect to an increasing process in the G-framework and extend G-Itô’s formula. Moreover, we study the solvability of the scalar valued stochastic differential equations driven by G-Brownian motion with reflecting boundary conditions (RGSDEs).

These are lecture notes for the class Stochastic Calculus offered at the Courant Institute in the Fall Semester of 2012. It is a graduate level class. Students should have a solid background in probability and linear algebra. The topic selection is guided in part by the needs of our MS program in Mathematics in Finance. But it is not focused entirely on the Black Scholes theory of derivative pr...

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This paper employs an elementary method to derive the Milstein...

The limit theorems for certain stochastic processes generated by permanents of random matrices of independent columns with exchangeable components are established. The results are based on the martingale decomposition of a random permanent function similar to the one known for U -statistics and on relating the components of this decomposition to some multiple stochastic integrals.

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained. AMS 2000 Mathematics Subject Classification: Primary 31C25; Secondary 60J57, 60J55, 60H05.

For system identification, the ordinary differential equations (ODEs) model is popular for its accuracy and effectiveness. Consequently, the ODEs model is extended to the stochastic differential equations (SDEs) model to tackle the stochastic case intuitively. But the existence of stochastic integral is a rigid barrier. We simply transform the SDEs to their corresponding stochastic difference e...

In this paper, a slimple linear high-gain integral-derivative observer is presented based on singular perturbation technique. The proposed integral-derivative observer can estimate synchronously the multiple integrals and derivatives of a signal. The merits of the presented integralderivative observer include its synchronous estimation of integrals and derivatives, simple implementation, suffic...