Free Brownian motion, free stochastic calculus, and random matrices

1996 ◽  
pp. 1-19 ◽  
Author(s):  
Philippe Biane
Keyword(s):  
Brownian Motion ◽  
Random Matrices ◽  
Stochastic Calculus ◽  
Free 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

scholarly journals REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950104
Author(s):  
KAMIL KALETA ◽  
MARIUSZ OLSZEWSKI ◽  
KATARZYNA PIETRUSKA-PAŁUBA
Keyword(s):  
Brownian Motion ◽  
Transition Probability ◽  
Strong Feller Property ◽  
Feller Property ◽  
Reflected Diffusion ◽  
Geometric Conditions ◽  
Regularity Properties ◽  
Probability Densities ◽  
Free Brownian Motion ◽  
Strong Feller

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

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.

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