The quadratic variation of Brownian motion on a time scale

2012 ◽  
Vol 82 (9) ◽  
pp. 1677-1680 ◽  
Author(s):  
David Grow ◽  
Suman Sanyal
Keyword(s):  
Brownian Motion ◽  
Time Scale ◽  
Quadratic Variation

Observation of Brownian motion on the time scale of hydrodynamic interactions

1993 ◽  
Vol 70 (2) ◽  
pp. 242-245 ◽  
Author(s):  
M. H. Kao ◽  
A. G. Yodh ◽  
D. J. Pine
Keyword(s):  
Brownian Motion ◽  
Time Scale ◽  
Hydrodynamic Interactions

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (02) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density. In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

First-order calculus and option pricing

2014 ◽  
Vol 01 (01) ◽  
pp. 1450009 ◽  
Author(s):  
Peter Carr
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Option Pricing ◽  
Stochastic Calculus ◽  
Quadratic Variation ◽  
Modern Theory ◽  
Barrier Options ◽  
Barrier Option ◽  
First Order ◽  
Itô Calculus

The modern theory of option pricing rests on Itô calculus, which is a second-order calculus based on the quadratic variation of a stochastic process. One can instead develop a first-order stochastic calculus, which is based on the running minimum of a stochastic process, rather than its quadratic variation. We focus here on the analog of geometric Brownian motion (GBM) in this alternative stochastic calculus. The resulting stochastic process is a positive continuous martingale whose laws are easy to calculate. We show that this analog behaves locally like a GBM whenever its running minimum decreases, but behaves locally like an arithmetic Brownian motion otherwise. We provide closed form valuation formulas for vanilla and barrier options written on this process. We also develop a reflection principle for the process and use it to show how a barrier option on this process can be hedged by a static postion in vanilla options.

scholarly journals DIFFERENTIAL FORMULAS OF STOCHASTIC FUNCTIONS

2009 ◽  
Vol 12 (7) ◽  
pp. 29-34
Author(s):  
Dam Ton Duong
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Quadratic Variation ◽  
Stochastic Functions

Based on the quadratic variation theorem of the Brownian motion, we have established the basic rules of stochastic differetial calculus operations. Theorem 1. If X,Y, are positive-valued stochastic processes satisfying respectively the following stochastic differenntial equations Then a, b R: Where Theorem 2 Suppose is the Hermite type stochastic process of then

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