This paper provides a an introduction to some basic properties of Brownian motion. In particular, it shows that Brownian motion exists, that Brownian motion is nowhere differentiability, and that Brownian motion has finite quadratic variation.

(To Appear) Stochastic Calculus for Brownian Motion on a Brownian Fracture By Davar Khoshnevisan* & Thomas M. Lewis University of Utah & Furman University Abstract. The impetus behind this work is a pathwise development of stochastic integrals with respect to iterated Brownian motion. We also provide a detailed analysis of the variations of iterated Brownian motion. These variations are linked ...

In the present work, a mathematical model is developed and analyzed to study the influence of nanoparticle concentration through Brownian motion and thermophoresis diffusion. The governing system of PDEs is transformed into a coupled non-linear ODEs by using suitable variables. The converted equations are then solved by using robust shooting method with the help of MATLAB (bvp4c). The impacts o...

Assuming that the underlying stock follows Fractional Brownian motion and that stochastic interest rate meets the Vasicek model of interest rates, this paper establishes pricing model of Warrant Bonds and deduces the pricing formula of Warrant Bonds by utilizing risk-neutral valuation theory. Finally, this paper analyzes influence of concerned parameters of pricing model on the value of Warrant...

A mathematical model to price convertible bonds involving mixed fractional Brownian motion with jumps is presented. We obtain a general pricing formula using the risk neutral pricing principle and quasi-conditional expectation. The sensitivity of the price to changing various parameters is discussed. Theoretical prices from our jump mixed fractional Brownian motion model are compared with the p...

We investigated three models of Brownian motors which convert rotational diffusion into directed translational motion by switching on and off a potential. In the first model a spatially asymmetric potential generates directed translational motion by rectifying rotational diffusion. It behaves much like a conventional flashing ratchet. The second model utilizes both rotational diffusion and drif...

As a hot research topic nanofluids have attracted great interest from researchers worldwide. Although nanofluids are found to exhibit higher thermal conductivity compared to their base fluids, the underlying mechanisms for the enhancement are still debated and not fully understood. In addition, there has been little agreement among different studies and no widely accepted model is also availabl...