scholarly journals A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion

2016 ◽  
Vol 23 (2) ◽  
pp. 141-150
Author(s):  
Soufiane Aazizi
Keyword(s):  
Brownian Motion ◽  
Simple Proof ◽  
Quadratic Variation ◽  
Subfractional Brownian Motion

scholarly journals Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Yuquan Cang ◽  
Junfeng Liu ◽  
Yan Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Nonparametric Regression ◽  
Local Time ◽  
Kernel Function ◽  
Malliavin Calculus ◽  
Bandwidth Parameter ◽  
Subfractional Brownian Motion ◽  
Normal Law

We study the asymptotic behavior of the sequenceSn=∑i=0n-1K(nαSiH1)(Si+1H2-SiH2),asntends to infinity, whereSH1andSH2are two independent subfractional Brownian motions with indicesH1andH2, respectively.Kis a kernel function and the bandwidth parameterαsatisfies some hypotheses in terms ofH1andH2. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motionSH1. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.

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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