Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

Multi-Dimensional G–Brownian Motion and Related Stochastic Calculus under G–Expectation

We develop a notion of nonlinear expectation —-G–expectation—generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G–normal distributions. With this nonlinear distribution we can introduce our G–expectation under which the canonical process is a multi–dimensional G–Brownian motion. We then establish the related stochastic calculus, especially st...

G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Itô Type

We introduce a notion of nonlinear expectation —-G–expectation—generated by a nonlinear heat equation with a given infinitesimal generator G. We first discuss the notion of G–standard normal distribution. With this nonlinear distribution we can introduce our G–expectation under which the canonical process is a G–Brownian motion. We then establish the related stochastic calculus, especially stoc...

Brownian Motion and Stochastic Calculus

This note is about Doob decomposition and the basics of Square integrable martingales Contents 1 Doob-Meyer Decomposition 1 2 Square Integrable Martingales 4 Brownian Motion and Stochastic Calculus Continuout Time Submartingales Usually it’s su¢ ce to only discuss submartingales by symmetry in de…nition and techniques are the same. 1 Doob-Meyer Decomposition Doob-meyer decomposition clears the ...

G–Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty

Fractional Brownian motion: stochastic calculus and applications

Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H ∈ (0, 1) called the Hurst index. In this note we will survey some facts about the stochastic calculus with respect to fBm using a pathwise approach and the techniques of the Malliavin calculus. Some applications in turbulence and finance will be discussed. Math...