The randomly stopped geometric Brownian motion (original) (raw)

STUDY ON GEOMETRIC BROWNIAN MOTION WITH APPLICATIONS

The purpose of this paper is to give a detail study on geometric Brownian motion with some applications. Keywords- Geometric Brownian motion, Ito lemma, delays geometric Brownian motion, geometric Brownian with jump.

A note on the distribution of integrals of geometric Brownian motion

Statistics & Probability Letters, 2001

The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At := t 0 exp{Zs}ds, t ≥ 0, where {Zs : s ≥ 0} is a one-dimensional Brownian motion with drift coefficient µ and diffusion coefficient σ 2. In particular, both expected values of the form v(t, x) := Ef (x + At), f homogeneous, as well as the probability density a(t, y)dy := P (At ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.

On the probability of hitting a constant or a time-dependent boundary for a geometric Brownian motion with time-dependent coefficients

Applied Mathematical Sciences

This paper presents exact analytical formulae for the crossing of a constant onesided or two-sided boundary by a geometric Brownian motion with timedependent, non-random, drift and diffusion coefficients, under the assumption that the drift coefficient is a constant multiple of the diffusion coefficient, as well as approximate analytical formulae for general time-dependent, non-random, drift and diffusion coefficients and general time-dependent, non-random, boundaries. The numerical implementation of these formulae is very simple.

On The Validity of The Geometric Brownian Motion Assumption

The Engineering Economist, 2005

The geometric Brownian motion (GBM) process is frequently invoked as a model for such diverse quantities as stock prices, natural resource prices, and the growth in demand for products or services. We discuss a process for checking whether a given time series follows the GBM process. Methods to remove seasonal variation from such a time series are also analyzed. Of four industries studied, the historical time series for usage of established services meet the criteria for a GBM, however the data for growth of emergent services do not.

Estimation tail parameter for Geometric Brownian motion

Al-Qadisiyah Journal Of Pure Science, 2021

Right-tailed distributions are very important in many applications. There are many studies estimating the tail index. In this paper, we will estimate the tail parameter using the three (the Direct, Bootstrap and Double Bootstrap) methods. Our aim is to illustrate the best way to estimate the -stable with using simulation and real data for the daily Iraqi financial market dataset.

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

Entropy

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate ...

The Brownian motion

2019

This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field. It also includes mathematical definitions and the hidden stories behind the terms discussing why the theories are presented in specific ways.

On the relation between the distributions of stopping time and stopped sum with applications

Let TT\T be a stopping time associated with a sequence of independent random variables Z1,Z2,...Z_{1},Z_{2},...Z1,Z2,... . By applying a suitable change in the probability measure we present relations between the moment or probability generating functions of the stopping time TTT and the stopped sum %S_{T}=Z_{1}+Z_{2}+...+Z_{T}. These relations imply that, when the distribution of STS_{T}ST\ is known, then the distribution of TTT\ is also known and vice versa. Applications are offered in order to illustrate the applicability of the main results, which also have independent interest. In the first one we consider a random walk with exponentially distributed up and down steps and derive the distribution of its first exit time from an interval (−a,b).(-a,b).(a,b). In the second application we consider a series of samples from a manufacturing process and we let Zi,igeq1Z_{i},i\geq 1Zi,igeq1, denoting the number of non-conforming products in the iii-th sample. We derive the joint distribution of the random vector (T,ST)(T,S_{T})(T,ST), wh...