Brownian motion in Mathematics

Shizan Fang

University of Bourgogne

Dijon France

华中科技大学

2015.4.23

Brownian motion is the chaotic motion of particles,first observed by Scottish botanist Robert Brown in1827, when he was looking through a microscope atparticles found in pollen grains in water. He notedthat the particles moved through the water but wasnot able to determine the mechanisms that causedthis motion.Of course he did not realized that the motion heobserved will become a source of development of abranch of Mathematic : stochastic calculus.

He is paradoxically a pioneer in Mathematics.

Robert Brown (1773-1858)

Brownian paths in 2D

Brownian paths in 3D

Louis Bachelier (1870-1946)

Theory of speculation

In 1900, in his PhD thesis, Bachelier observed

similar phenomenen in financial market, and

he presented a first stochastic analysis of the

stock and option markets.

His thesis is now the basis of many price model in

Finance, especially for Black-Scholes formula

(1973), Nobel economic prize winner.

Recognition

At beginning, his work was unknown, or not read, or

not understood, sometimes despised. Even P.

Lévy thought that his work was simply wrong.

However thirty years later, his work was recognized

by A. Kolmogorov, a pioner in probability

theory in 1930.

The pionner role and the importance of the work by

L. Bachelier in Probability and Mathematical

finance was emphasized especially by Benoît

Mandebrot (1924-2010).

What he did ?

He considered the chaotic motion observed by R.Brown as a stochastic process (Xt), havingcontinuous paths, which satisfy properties such that:

- Independence of increments- homogeneous: (Xt-Xs) is only a function of (t-s).

He proved that such object was the limit case ofRandom works, his law or distribution enjoyed apartial differential equation, called usually heatequation.

Mutual developement

J. Bachelier was influenced by two schools:

- First by tradition of mathematical physic of J. Fourier (analysis of signals), H. Poincaré (calculus of probability).

- Secondly by speculation in Bourse, a need of a theoretic formulation such as «the difference of gain is in fact of the square root of the time ».

- Here is an example of such a mutual developement.

Nobert Wiener (1894 -1964)

Wiener process

In mathematics, the Wiener process is a continuous time stochastic process named in honor of Nobert Wiener. It is often called standard Brownian motion.

Why?This means that

- It was first N. Wiener who gave a rigouros mathematical construction.

- The notion of Brownian motion was considerably generalized such as Brownian motion of variance σ>0, fractional Brownian motion, Brownian motion in manifolds.

What is said in Wikipedia

In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution.

In applied mathematics, the Wiener process is used to represent the integral of a Gaussian white noise process, and so is useful as a model of noise in electronics engineering, instrument errors in filtering theory and unknown forces in control theory.

Characterisation of Wiener process

i) W0 = 0

ii) The function t → Wt is almost surely everywhere

continuous

iii) Wt has independent increments with Wt−Ws which

follows N(0, t−s) (for 0 ≤ s < t), where N(μ, σ2)

denotes the normal distribution with expected

value μ and variance σ2.

The last condition means that if 0 ≤ s1 < t1 ≤ s2 < t2 then

Wt1−Ws1 and Wt2−Ws2 are independent random

variables, and the similar condition holds for n

increments.

Gaussian random variable

A random variable X is said to be Gaussian, or to have a standard normal distribution if

Then for a probability 0.95, the random variable X is in [-1,98;1,98]; for 0.99, X is in [-2,58;2,58].

We say that X have N(μ, σ2) if (X – μ)/σ is standard

Normal distribution

Wiener’s construction

Let (ξn) be a sequence of independent standard Gaussian random variable. Wiener proved that the following series

Converges uniformly in [0,T], for almost ω.

Then t to Wt is a Wiener process.

Heat equation

Let

It is the density of Wt, that is

Then p satisfies the PDE (simple to be checked)

Martingales

Let f(t,x) be the solution to

Then Mt=f(t,Wt) enjoys the property of martingale. For example, Mt=Wt

2 – t is a martingale for f(t,x)=x2-t. So

In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings. A martingale is a sequence of random variables, at a particular time, the expectation of the next value is equal to the present observed value even given knowledge of all prior observed values.

Kiyoshi Itô (1915-2008)

Wolf Prize 1987Gauss Prize 2006

For his creation ofStochastic differentialAnd integral calculus

Itô stochastic calculus

Let f(t,x) be a good function and Wt the Wiener process. How to compute the term

f(t, Wt) ?

Itô formula:

Why the extra term

Which is bad or good ?

It’s extremly good

A first remark is that if f satisfies the condition

Then

is a martingale; this means that the last term gives A martingale. So for general f(x), the term

is a martingale.

Push-forward cases

Let (Wt1, …, Wt

m) be m independent Brownian motion, then

Or simply

is a martingale.

Brownian motions on a manifold ?

A m-dimensional manifold M is a topological space such that each point has a neighbourhood that is homeomorphic to Rm.

A covering of The circle.

How to construct

a Brownian motion

On M?

Examples of manifold

Here are two examples:

Brownian motion on M

The Laplace operator on Rm admits a counterpart on M.

The stochastic process is said to be the Brownian motion on M if

is a martingale, for any C2 real-valued function on the manifold M. The behaviour of Xt is related to geometry of M.

This means that many simple ideas could be pushed forward very far.

Definition of martingales

Let (Ω, Ft,F,P) be a probability space. A real valued process {Xt; t≥0} is said to be martingale with respect to (Ft, P) if

i) Xt is measurable w.r.t Ft,

ii) E(|Xt|) is finite,

iii) E(Xs|Ft)=Xt for s>t.

Roughly speaking, Ft is a collection of informations up to the time t, the condition i) means that Xt is only dependent of the past up to now, iii) means that knowing Ft, the mean of X in the future is equal to actual value.

Attention: when Ft changes or P changes, Xt is no more a martingale.

Good property of martinagle

Let {Xt;t≥0} be a martingale. If

Supt>0 E(|Xt|)<+∞,

Then for almost surely ω, as t tends to +∞,

Lim t->+∞ Xt(ω) exists.

This is a basic tool to prove the existence of limit in many situations in Analysis.

If Xt takes in a Banach space, such a result holds or not is dependent of the geometry of the Banach space.

Itô stochastic integral

In fact, Itô dealed with a large class of stochastic process {ut(ω);t>0}. Let {Wt;t>0} be the Wiener process, let Ft be generated by {Ws;0<s≤t} called Itô filtration. For {ut;t>0} adapted to Ft, the Itô stochastic integral

is defined as the limit of

Note that the time for u is taken for the left point of

Each sub-interval.

Energy identity

Let

Then {Mt;t≥0} is a martingale, and the identity of energy holds

Together with Itô formula, they are basic tools in stochastic calculus.

Stochastic differential equations(SDE)

The typical SDE is of the form

Which is understood as an integral equation

for t>s. There are more general SDE

where b and σ depend also on previous values of the process.

Such a SDE is useful in stochastic control theory.

Black-Scholes SDE

In the Black-Scholes model, the underlying stock price S(t) is assumed to evolve following SDE

Intuitively, W(t) is a process that "wiggles up and down" in such a random way that its expected change over any time interval is 0, r>0 is the interest rate and σ is the volatility of the market.

Wiener measure

Let B=C0([0,T],R) be the space of continuous curves in R. The Wiener process

defines a probability μ on B, which is called Wiener measure. Then (B,μ) becomes a probability space. Under (B,μ) , the w->w(t) is a Wiener process.

A. Weil showed that in infinite dimensional space, there does not exist any probability which is invariant under translations.

Cameron-Martin theorem

However

where Kh(w) is the density of translation:

where the dot denotes the derivative w.r.t the time t.

Paul Malliavin (1925-2010)

Malliavin Calculus

Malliavin calculus extends the calculus of variations from functions to stochastic processes, which leds to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equations. The calculus allows integration by parts with random variables. This operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.

Bundle of orthonormal frames

The difficulty for a manifold M of dimension m is there is not usually m vector fields {A1, …, Am} such that at each point they form a basis for the tangent space. There is a necessity to introduce the bundle of orthonormal frame O(M). An element r in O(M) if

where π(r) is the base of the frame. The Levi-Civita connection defines m canonical horizontal vector fields {H1, …, Hm} on O(M): π’(r) H1(r) = re1,

where {e1, …, em} is the canonical basis of Rm.

Intrinsic construction of Brownian motion

The SDE on O(M)

can be solved. Then ξt (w) = π(rt(w)) is a Brownian motion on M. This construction is much better than using martingale. Moreover the parallel transport along Brownian paths can be defined

While in Differential geometry, only along a smooth curve the parallel transport is defined.

Jean-Michel Bismut (1948-)

Girsanov transformation

It is a powful tool in stochastic calculus.

Let {ut(ω);t>0} be an adapted process w.r.t Ft. Consider

Then under a new probability dQ=KudP, {ηt;>0} is a Brownian motion. Here is the density Ku:

Bismut used it to get the integration by parts.

Integration by parts

The gradient of a functional f of Brownian motion (for example St in Black-Scholes) is seen as a process

{Dtf; t>0}. Here is the formula:

For the Brownian motion on a manifold M:

here Rict is Ricci curvature of M, geometry entering.

A book by J.M. Bismut

The last formula was proved in a fantastic book

«Large deviations and Mallavin Calculus »

In Progress in Math. 1984 (he was 36 years old). This book opened a new horizon for researchs in curved functional space, such as space of Riemannian paths, space of loops, loop groups: they are linked to Mathematical physic; loop groups are used in String theory. These spaces furnish also infinite dimensional examples in optimal transport theory developed by Y. Brenier, C. Villani and all.

It is a main topic on which I have been worked.

Navier-Stokes equations

A joint work with D.J. Luo