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 [7] 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 ...

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2. Equivalently it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough pat...

Fundamental to many applications in financial engineering is the normal (Gaussian) distribution. It is the building block for simulating such basic stochastic processes as Brownian motion and geometric Brownian motion. In this section, we will go over algorithms for generating univariate normal rvs and learn how to use such algorithms for constructing sample paths of Brownian motion and geometr...

2) and 3) together can be summarized by: If t0 = 0 < t1 < t2 < · · · < tk, then the increment rvs B(ti) − B(ti−1), i ∈ {1, . . . k}, are independent with B(ti) − B(ti−1) ∼ N(0, ti − ti−1) (normal with mean 0 and variance ti − ti−1). In particular, B(ti) − B(ti−1) is independent of B(ti−1) = B(ti−1)−B(0). If we only wish to simulate B(t) at one fixed value t, then we need only generate a unit no...

— In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show some applications of SLND in studying sample path properties of (N, d)-Gaussian random fields. The...

We show the existence, for any k 2 N, of processes which have the same k-marginals as Brownian motion, although they are not Brownian motions. For k = 4, this proves a conjecture of Stoyanov. The law ~ P of such a \weak Brownian motion of order k" can be constructed to be equivalent to Wiener measure P on C0; 1]. On the other hand, there are weak Brownian motions of arbitrary order whose law is...

Fundamental to many applications in financial engineering is the normal (Gaussian) distribution. It is the building block for simulating such basic stochastic processes as Brownian motion and geometric Brownian motion. In this section, we will go over algorithms for generating univariate normal rvs and learn how to use such algorithms for constructing sample paths of Brownian motion and geometr...