Brownian Motion Reflection Principle

[Emmanuelle Thaller]

Westart with Brownian motion. The reflection principle, as described briefly in this post from the depths of history, is a classical technique for studying the maximum of Brownian motion. Roughly speaking, we exploit the fact that , but we then apply this at the hitting time of a particular positive value, using the Strong Markov Property.

Our main concern is conditioning to stay above zero. Let be some complete if cumbersome notation for a Brownian bridge B from (0,x) to (t,y). Then another simple transformation of the previous result gives

A useful technique will be the viewpoint of random walk as the values taken by Brownian motion at a sequence of stopping times. This Skorohod embedding is slightly less obvious when considering a general random walk bridge inside a general Brownian bridge, but is achievable. We want to study quantities like

As earlier, the second is obtained from the first by integrating over suitable y. This function has to account for the extra complications when either end-point is near zero, for which the event where the Brownian motion goes negative without the random walk going negative requires additional care.

(At a more general level, it is meaningful to describe the random walk conditioned on staying positive forever. Although this would a priori require conditioning on an event of probability zero, it can be handled formally as an example of an h-transform.)

The order of taking limits is truly crucial. We can also obtain a distributional scaling limit at time n under conditioning to stay non-negative up to time n. But then this is the size-biased normal distribution (the Rayleigh distribution), rather than the square-size-biased normal distribution we say in this setting. And we can exactly see why. Relative to the normal distribution which applies in the absence of conditioning, we require size-biasing to account for the walk staying positive up to time m, and then also size-biasing to account for the walk staying positive for the rest of time (or up to n in the limit if you prefer).

The asymptotics for were the crucial step, for which only heuristics are present in this post. It remains the case that estimates of this kind form the crucial step in other more exotic conditioning scenarios. This is immediately visible (even if the random walk notation is rather exotic) in, for example, Proposition 2.2 of [CHL17], of which we currently require a further level of generalisation.

We present a reflection principle for a wide class of symmetric random motions with finite velocities. We propose a deterministic argument which is then applied to trajectories of stochastic processes. In the case of symmetric correlated random walks and the symmetric telegraph process, we provide a probabilistic result recalling the classical reflection principle for Brownian motion, but where the initial velocity has a crucial role. In the case of the telegraph process we also present some consequences which lead to further reflection-type characteristics of the motion.

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account.Find out more about saving content to Dropbox.

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account.Find out more about saving content to Google Drive.

Do you have any conflicting interests? *Conflicting interests helpClose Conflicting interests help Please list any fees and grants from, employment by, consultancy for, shared ownership in or any close relationship with, at any time over the preceding 36 months, any organisation whose interests may be affected by the publication of the response. Please also list any non-financial associations or interests (personal, professional, political, institutional, religious or other) that a reasonable reader would want to know about in relation to the submitted work. This pertains to all the authors of the piece, their spouses or partners.

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical reflection principle to work. We call this method a weak reflection principle and show that it provides solutions to many problems for which the classical reflection principle is typically used. In addition, unlike the classical reflection principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak reflection principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the Lvy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak reflection principle in financial mathematics, computational methods and inverse problems.

I am trying to simulate a one-dimensional Brownian motion with a barrier, i.e. it is not allowed to go below (for above) a certain value, in which case it is reflected as if it hit a wall. Unfortunately, the functions in Mathematica (Ito process, Weiner process, transformed process) do not allow this simple extension natively. I have searched the whole WWW to no avail.

Its transition distribution follows from the mirror principle: It is the sum of the distribution with start point x0 and .-x0. This distribution is an even function, solves the diffusion equation on the half line, has the delta distribuition at x0 and has derivative zero at the boundary indicating that the diffusion current over the boundary is zero.

FRE-GY 6231 Stochastic Calculus in Finance1.5 Credits This course provides the mathematical foundations of option pricing and credit risk models. The techniques covered include filtrations, arithmetic and geometric Brownian motion, first passage time, the reflection principle, the stochastic Ito integral, Ito differential calculus, change of probability measure, martingales, Stochastic Differential Equations and the connection with Partial Differential Equations. Some financial applications to European options and bonds will be presented.

Prerequisite(s): Matriculation into a graduate program sponsored by the Department of Finance & Risk Engineering, or permission of the Department and FRE-GY 6083

Weekly Lecture Hours: 1.5 Weekly Lab Hours: 0 Weekly Recitation Hours: 0

A second course in stochastic processes and applications to insurance. Markov chains (discrete and continuous time), processes with jumps; Brownian motion and diffusions; Martingales; stochastic calculus; applications in insurance and finance. Content: Stochastic processes in discrete and continuous time; Markov chains: Markov property, Chapman-Kolmogorov equation, classification of states, stationary distribution, examples of infinite state space; filtrations and conditional expectation; discrete time martingales: martingale property, basic examples, exponential martingales, stopping theorem, applications to random walks; Poisson processes: counting processes, definition as counting process with independent and stationary increments, compensated Poisson process as martingale, distribution of number of events in a given time interval as well as inter-event times, compound Poisson process, application to ruin problem for the classical risk process via Gerber's martingale approach; Markov processes: Kolmogorov equations, solution of those in simple cases, stochastic semigroups, birth and death chains, health/sickness models, stationary distribution; Brownian motion: definition and basic properties, martingales related to Brownian motion, reflection principle, Ito-integral, Ito's formula with simple applications, linear stochastic differential equations for geometric Brownian motion and the Ornstein-Uhlenbeck process, first approach to change of measure techniques, application to Black-Scholes model. The items in the course content that also appear in the content of ST227 are covered here at greater depth. However, ST227 is not a pre-requisite for this course.

This course will be delivered through a combination of classes and lectures totalling a minimum of 29 hours across Michaelmas Term. This year, some or all of this teaching may be delivered through a combination of virtual classes and flipped-lectures delivered as short online videos. This course includes a reading week in Week 6 of Michaelmas Term.

Please note that during 2020/21 academic year some variation to teaching and learning activities may be required to respond to changes in public health advice and/or to account for the situation of students in attendance on campus and those studying online during the early part of the academic year. For assessment, this may involve changes to mode of delivery and/or the format or weighting of assessments. Changes will only be made if required and students will be notified about any changes to teaching or assessment plans at the earliest opportunity.

Prerequisite: An introductory probability course such as MATH 4710, BTRY 4080, ORIE 3600, ECON 3190. General proficiency in calculus and linear algebra.If unsure about your preparation, please discuss it with me.

Three extra topics not covered this semester are renewal processes, queueing theory, and the connection between random walks and electrical networks. If you are interested to learn more about renewal processes and queueing theory, check Chapter 3 and sections 4.5-4.6 of the textbook. For random walks and electrical networks, there is a very nice introduction in Chapter 9 of Markov chains and mixing times by Levin, Peres, and Wilmer, and much more information in Probability on trees and networks by Lyons and Peres.

3a8082e126