Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-bifractional Brownian motion

2020 ◽  
pp. 1-19
Author(s):  
Nenghui Kuang ◽  
Ying Li
Keyword(s):  
Brownian Motion ◽  
Quadratic Variation ◽  
Bifractional Brownian Motion

The sub-bifractional Brownian motion

2013 ◽  
Vol 50 (1) ◽  
pp. 67-121 ◽  
Author(s):  
Charles El-Nouty ◽  
Jean-Lin Journé
Keyword(s):  
Brownian Motion ◽  
Integral Test ◽  
Bifractional Brownian Motion

The sub-bifractional Brownian motion, which is a quasi-helix in the sense of Kahane, is presented. The upper classes of some of its increments are characterized by an integral test.

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 Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

2020 ◽  
Vol 12 (1) ◽  
pp. 128-145
Author(s):  
Abdelmalik Keddi ◽  
Fethi Madani ◽  
Amina Angelika Bouchentouf
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonparametric Estimation ◽  
Unknown Function ◽  
Asymptotic Properties ◽  
Kernel Estimator ◽  
Trend Function ◽  
Bifractional Brownian Motion ◽  
Numerical Example

AbstractThe main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type {\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}}where {{\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}}} is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.

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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