1 Notation In these notes we shall in general use standard notation. For every n ∈ N Bn denotes the Borel algebra on R and if (Ω,F , P ) is a probability space, X : Ω→ R a random variable, then we let X(P ) denote the distribution measure (the image measure) on R of X, e.g. X(P )(A) = P (X−1(A)) for allA ∈ Bn. (1.1) If n ∈ N, we let 〈·, ·〉 denote the canonical inner product on R. Hence for all ...

This is a set of lecture notes based on a graduate course given at the Taught Course Centre in Mathematics in 2011. The course is based on a selection of material from my book with Yuval Peres, entitled Brownian motion, which was published by Cambridge University Press in 2010. 1 Lévy's construction of Brownian motion and modulus of continuity Much of probability theory is devoted to describing...

In this paper, we study a class of impulsive neutral stochastic functional integro-differential equations with infinite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion (fBm) with Hurst parameter H 2 ð1=2; 1Þ in the Hilbert space. We prove the existence and uniqueness of the mild solution for this kind of equations with the coeffici...

Multifractional Brownian motion (mBm), denoted here by X, is one of the paradigmatic examples of a continuous Gaussian process whose pointwise Hölder exponent depends on the location. Recall that X can be obtained (see e.g. [BJR97, AT05]) by replacing the constant Hurst parameter H in the standard wavelet series representation of fractional Brownian motion (fBm) by a smooth function H(·) depend...

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [5] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

In finance one of the primary issues is managing risk. Related to this issue and maybe for hedging, investors are naturally interested in the expected values of supremum, infimum, maximum gain and maximum loss of risky assets and the relations between them. Price of a risky asset, stock, can be modeled using Brownian motion and fractional Brownian motion. In this study, we first present the mar...