Stochastic calculus on distorted Brownian motion

1988 ◽  
Vol 29 (1) ◽  
pp. 207-209 ◽  
Author(s):  
Dražen Pantić
Keyword(s):  
Brownian Motion ◽  
Stochastic Calculus ◽  
Distorted Brownian Motion

Stochastic Calculus for Fractional Brownian Motion and Applications

2008 ◽  
Author(s):  
Francesca Biagini ◽  
Yaozhong Hu ◽  
Bernt Øksendal ◽  
Tusheng Zhang
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Calculus

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.

CENTRAL LIMIT THEOREMS FOR WEIGHTED SUMS OF LINEAR PROCESSES: LP -APPROXIMABILITY VERSUS BROWNIAN MOTION

2009 ◽  
Vol 25 (3) ◽  
pp. 748-763 ◽  
Author(s):  
Kairat T. Mynbaev
Keyword(s):  
Brownian Motion ◽  
Central Limit ◽  
Asymptotic Expansions ◽  
Limit Theorems ◽  
Stochastic Calculus ◽  
Central Limit Theorems ◽  
Weighted Sums ◽  
Linear Processes

Standardized slowly varying regressors are shown to be Lp-approximable. This fact allows us to provide alternative proofs of asymptotic expansions of nonstochastic quantities and central limit results due to P.C.B. Phillips, under a less stringent assumption on linear processes. The recourse to stochastic calculus related to Brownian motion can be completely dispensed with.

Sign in / Sign up

Export Citation Format

Share Document