Many phylogenetic comparative methods that are currently widely used in the scientific literature assume a Brownian motion model for trait evolution, but the suitability of that model is rarely tested, and a number of important factors might affect whether this model is appropriate or not. For instance, we might expect evolutionary change in adaptive radiations to be driven by the availability ...

The falling and settling of solid particles in gases and liquids is a natural phenomenon happens in many industrial processes. This phenomenon has altered pure forced convection to a combination of heat conduction and heat convection in a flow over a plate. In this paper, the coupling of conduction (inside the plate) and forced convection of a non-homogeneous nanofluid flow (over a flat plate) ...

Applying the theory of voter model and the theory of stopping time, we investigate the statistical properties of the fluctuations of interfaces model that defined from the voter model. We show that the probability distributions of the fluctuations, under some conditions, converge to the corresponding distribution of a geometric Brownian motion. Key-Words: Fluctuation; interface; voter model; st...

We study some path transformations related to Pitman’s theorem on Brownian motion and the three dimensional Bessel process. We relate these to Littelmann path model, and give applications to representation theory and to Brownian motion in a Weyl chamber.

The purpose of this paper is to present a survey of recent developments concerning the distributions of occupation times of Brownian motion and their applications in mathematical finance. The main result is a closed form version for Akahori’s generalized arc-sine law which can be exploited for pricing some innovative types of options in the Black & Scholes model. Moreover a straightforward proo...

We study drawdowns and rallies of Brownian motion. A rally is defined as the difference of the present value of the Brownian motion and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. This paper determines the probability that a drawdown of a units precedes a rally of b units. We apply this result to examine stock market c...

We define weighted fractional Brownian sheets, which are a class of Gaussian random fields with four parameters that include fractional Brownian sheets as special cases, and we give some of their properties. We show that for certain values of the parameters the weighted fractional Brownian sheets are obtained as limits in law of occupation time fluctuations of a stochastic particle model. In co...

We prove a Donsker type approximation theorem for the fractional Brownian motion in the case H > 1/2. Using this approximation we construct an elementary market model that converges weakly to the fractional analogue of the Black–Scholes model. We show that there exist arbitrage opportunities in this model. One such opportunity is constructed explicitly.

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...

The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copenhagen, who effectively created a model of Brownian motion while studying time...