## Lecture 1. Introduction to the Brownian motion

The first stochastic process that has been extensively studied is the Brownian motion, named in honor of the botanist Robert Brown (1773-1858), who observed and described in 1828 the random movement of particles suspended in a liquid or gas. One of the first mathematical studies of this process goes back to the mathematician Louis Bachelier (1870-1946), in 1900, who presented a stochastic modelling of the stock and option markets. But, mainly due to the lack of rigorous foundations of probability theory at that time, the seminal work of Bachelier has been ignored for a long time by mathematicians. However, in his 1905 paper, Albert Einstein (1879-1955) brought this stochastic process to the attention of physicists by presenting it as a way to indirectly confirm the existence of atoms and molecules. The rigorous mathematical study of stochastic processes really began with the mathematician Andrei Kolmogorov (1903-1987). His monograph published in Russian in 1933 built up probability theory in a rigorous way from fundamental axioms in a way comparable to Euclid’s treatment of geometry. From this axiomatic, Kolmogorov gives a precise definition of stochastic processes. His point of view stresses the fact that a stochastic process is nothing else but a random variable valued in a space of functions (or a space of curves). For instance, if an economist reads a financial newspaper because he is interested in the prices of barrel of oil for last year, then he will focus on the curve of these prices. According to Kolmogorov’s point of view, saying that these prices form a stochastic process is then equivalent to saying that the curve that is seen is the realization of a random variable defined on a suitable probability space. This point of view is mathematically quite deep and provides existence results for stochastic processes (Daniell-Kolmogorov existence result, as well as pathwise regularity results (Kolmogorov continuity theorem).

Joseph Doob (1910-2004) writes in his introduction to his famous book “Stochastic processes”:

[a stochastic process is] any process running along in time and controlled by probability laws…[more precisely] any family of random variables $X_t$ [where] a random variable … is simply a measurable function…

Doob’s point of view, which is consistent with Kolmogorov’s and built on the work by Paul Levy (1886-1971), is nowadays commonly given as a definition of a stochastic process. Relying on this point of view that emphasizes the role of time, Doob’s work, developed during the 1940’s and the 1950’s has quickly become one of the most powerful tools available to study stochastic processes.

Let us now describe the seminal considerations of Bachelier. Let $X_t$ denote the price at time $t$ of a given asset on a financial market. We will assume that $X_0=0$ (otherwise, we work with $X_t-X_0$). The first observation is that the price $X_t$ can not be predicted with absolute certainty. It seems therefore reasonable to assume that $X_t$ is a random variable defined on some probability space. One of the initial problems of Bachelier was to understand the distribution of prices at given times, that is the distribution of the random variable $(X_{t_1},...,X_{t_n})$, where $t_1,...,t_n$ are fixed.

The two following fundamental observations of Bachelier were based on empirical observations:

1. If $\tau$ is very small then, in absolute value, the price variation $X_{t+\tau}-X_t$ is of order $\sigma \sqrt{\tau}$ where $\sigma$ is a positive parameter (nowadays called the volatility of the asset);
2. The expectation of a speculator is always zero. Quoted and translated from Bachelier: I seems that the market, the aggregate of speculators, can believe in neither a market rise nor a market fall, since, for each
quoted price, there are as many buyers as sellers.. (nowadays, a generalization of this principle is called the absence of arbitrage).

Next, Bachelier assumes that for every $t>0$, $X_t$ has a density with respect to the Lebesgue measure, let us say $p(t,x)$. The two above observations imply that for $\tau$ small,

$p(t+\tau,x)=\frac{1}{2} p(t,x-\sigma \sqrt{\tau})+\frac{1}{2} p(t,x+\sigma \sqrt{\tau}).$

Indeed, due to the first observation, if the price is $x$ at time $t+\tau$, it means that at time $t$ the price was equal to $x-\sigma \sqrt{\tau}$ or to $x+\sigma \sqrt{\tau}$. According to the second observation, each of this case produces with probability $\frac{1}{2}$.

Now Bachelier assumes that $p(t,x)$ is regular enough and uses the following approximations coming from a Taylor expansion:

$p(t+\tau,x) \simeq p(t,x)+\tau \frac{\partial p}{\partial t} (t,x)$
$p(t,x-\sigma \sqrt{\tau}) \simeq p(t,x)-\sigma \sqrt{\tau} \frac{\partial p}{\partial x} (t,x)+\frac{1}{2} \sigma^2 \tau \frac{\partial^2 p}{\partial x^2} (t,x)$
$p(t,x+\sigma \sqrt{\tau}) \simeq p(t,x)+\sigma \sqrt{\tau} \frac{\partial p}{\partial x} (t,x)+\frac{1}{2} \sigma^2 \tau \frac{\partial^2 p}{\partial x^2} (t,x).$

This gives

$\frac{\partial p}{\partial t} =\frac{1}{2} \sigma^2 \frac{\partial^2 p}{\partial x^2} (t,x).$

This is the so-called heat equation, which is the primary example of a diffusion equation. Explicit solutions to this equation are known, and by using the fact that at time $0$, $p$ is the Dirac distribution at $0$, it is obtained that:

$p(t,x)=\frac{e^{-\frac{x^2}{2\sigma^2 t} } }{\sigma \sqrt{2 \pi t}}.$

It means that $X_t$ has a Gaussian distribution with mean $0$ and variance $\sigma^2$. Let now $0 be fixed times and $I_1,...,I_n$ be fixed intervals. In order to compute

$\mathbb{P} (X_{t_1} \in I_1,...,X_{t_n} \in I_n)$

the next step is to assume that the above analysis did not depend on the origin of time, or more precisely that the best information available at time $t$ is given by the price $X_t$. That leads first to the following computation

$\mathbb{P} (X_{t_1} \in I_1,X_{t_2} \in I_2)$
$=\int_{I_1} \mathbb{P}(X_{t_2} \in I_2 \mid X_{t_1}=x_1) p(t_1,x_1) dx_1$
$=\int_{I_1} \mathbb{P}(X_{t_2-t_1} +x_1 \in I_2 \mid X_{t_1}=x_1) p(t_1,x_1) dx_1$
$=\int_{I_1 \times I_2} p(t_2-t_1,x_2-x_1) p(t_1,x_1) dx_1dx_2$

which is easily generalized to

$\mathbb{P} (X_{t_1} \in I_1,...,X_{t_n} \in I_n)$
$= \int_{I_1 \times \cdots \times I_n} p(t_n-t_{n-1},x_n-x_{n-1}) \cdots p(t_2-t_1,x_2-x_1) p(t_1,x_1) dx_1dx_2 \cdots dx_n$

In many regards, the previous computations were not rigorous but heuristic. One of the main issues here, is that the sequence of random variables $X_t$ is not well defined from a mathematical point of view. In the next posts, we will provide a rigorous construction of this object $X_t$ on which worked Bachelier and that is called a Brownian motion.

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