Intro to Random Walk and the Brownian motion (StoCal-1)

In this post, I am going to talk about the so called symmetric random walk, scaled symmetric random walk and the Brownian motion as its limiting distribution which is a direct consequence of the Central limit theorem.

Random Walk

Consider a game where you walk one step up if the result on the coin toss is head with probability $p$ otherwise one step down with probability $1-p$ . Now the question is where would you be at $n$ th step? Let’s define a random variable,

where $\omega_i$ ‘s are sequence of coin tosses with probability as mentioned above. If we calculate the expectation of this rv, that is, $E[X_j]=1*p+(-1)*(1-p)=2p-1$ . So when $p>0.5$ we have positive expectation and when $p<0.5$ we have negative expectation. But mathematically the most interesting case arises when $p=0.5$ , in this case the expectation is zero for the given rv. A random or stochastic process is the collection of random variables with the same probability space and takes values depending on whether they are discrete or continuous. Generally it is denoted by $\{X_t\}$ for continuous time data where $t$ represents time or by $\{X_k\}$ where $k$ represents the steps of discrete data.

Suppose $M_0=0, M_k=\sum_{j=1}^k X_j$ , $k=1,2,...$

The random process $M_k,k=0,1,2,...$ defined above is called as the one dimensional random walk (starting at zero, one can change the starting position and add it to the summation). When $p=0.5$ , this process is called as the symmetric random walk which is the topic we are going to discuss, otherwise it is called as the asymmetric random walk, which is also used in many cases. Now for the question we started with, at $n$ th step our position will be given by $M_n=\sum_{j=1}^n X_j$ . The expectation, which would be our best guess given the information, is zero for the symmetric random walk, and for the asymmetric case it is, $n(2p-1)$ , since each trial is assumed to be independent so that the random walk is the sum of independent rvs. For example, when $n=1000$ and $p=0.46$ the expectation is $-80$ and when $n=1000$ and $p=0.54$ the expectation is $80$ , while for the symmetric case it is always $0$ . The image you see on the top of the blog is a simulation of $10$ symmetric random walks.

A note for Real life: Suppose there are two persons working or competing in the same field, one has probability of success as 0.46 while other one has probability of success as 0.54. Clearly if both of them make random choices the one with higher probability would probabilistically do far better in future. There is a first catch that very few people actually know their probability of success in the given field, some overestimate it and some underestimate it. It is generally better to underestimate it and work toward improving it (I call such people as an under-optimist) than overestimating it and getting carried away (an over-optimist). The second catch is that, this probability changes. For example, if the person with low success probability starts making their life disciplined and their decisions less random, then they are going to increase their probability of success, and in long run would win. That’s why every successful person tells you to fail early and learn from it.

Some properties of the Random walk

(1) A random walk has independent increments i.e. for non-overlapping steps increments are independent of each other. Mathematically, suppose we choose $0=k_0<k_1<...<k_m$ , the random variables defined as $M_{k_1}=(M_{k_1}-M_{k_0}),(M_{k_2}-M_{k_1}),...,(M_{k_m}-M_{k_{m-1}})$ are independent. For any $i$ , $M_{k_{i+1}}-M_{k_i}=\sum_{j=k_i+1}^{k_{i+1}} X_j$ ,

is called an increment of the random walk.

(2) We can easily check that expectation for any increment is zero while $Var(M_{k_{i+1}}-M_{k_i})=k_{i+1}-k_i+1$ , which means if $T=k_{i+1}$ and $k_i+1=0$ then variance is $T$ , same as the time step between the two times. In general, for non-negative $k$ and $l$ where $k<l$ , the variance in this interval is $l-k$ .

(3) Quadratic Variation: This is one of the unique properties of the stochastic processes because of this we need stochastic calculus instead of ‘ordinary’ calculus. We would discuss this further when we talk about stochastic calculus. For the symmetric random walk, the quadratic variation up to time $k$ is defined to be

$[M,M]_k=\sum_{j=1}^k(M_j-M_{j-1})^2=k$ .

Scaled Symmetric Random Walk

As we have seen that the symmetric random walk takes discrete values so it is not a very good model for much finer values (as in the case of stock prices) but that’s not an issue as we can modify the model to make it finer. For this reason we define $W^{(n)}(t)=\frac{1}{\sqrt{n}}M_{nt}$ where $n$ and $t$ are chosen in such a way that $nt$ is an integer. Here we are speeding up the random walk but also scaling down the values so that we will have more oscillation within $0\leq t\leq1$ . This $t$ is sometimes called as the tick size, for example we can choose $t$ to be up to two decimal places and $n$ as $100$ and for in between values of $t$ when $nt$ is not an integer that can be obtained using interpolation from the nearest integer valued $ns$ and $nr$ where $s<t<r$ . It all looks complicated but in essence the value of let’s say $W^{(100)}(0.25)=\frac{1}{10}M_{25}$ depends on the $25$ coin tosses, by definition of the symmetric random walk this value will always lie between $-2.5$ and $2.5$ moving by $0.1$ . Now we can draw histogram of distribution of scaled symmetric random walk, further we will show that the normal distribution is a good approximation to it, by plotting normal density over the histogram of $W^{(100)}(0.25)$ .

**Normal distribution approximation to the distribution of $W^{(100)}(0.25)$**

**Normal distribution approximation to the distribution of $W^{(100)}(0.25)$**

Brownian Motion as the limit of the Scaled Symmetric Random Walk

We define Brownian motion, $W(t)$ , as the limit of scaled symmetric random walks $W^{(n)}(t)$ as $n \to \infty$ . The point to keep in mind here is that $W(t)$ is also a random process, it takes values depending on the path randomly obtained by $M_{nt}$ . So for the same $t$ , it will have different values depending on the path of the random walk, that is the whole point of defining it. By central limit theorem, for a fixed $t$ , the distribution of $W^{(n)}(t)$ converges to a normal distribution with mean zero and variance $t$ which by definition becomes the distribution of $W(t)$ for fixed $t$ . Also $W(t)$ inherits properties from these random walks such as independent increments, expected value of increments is zero and variance of the increments is the length of the time interval.

Here we are not going into too much details of the one dimensional Brownian motion, which is sometimes called as the Wiener process named after Norbert Wiener, a mathematician who studied it and proved many of its properties. The name Brownian motion comes from the botanist Robert Brown who observed the random movement of the pollen grains, he did not provide the theory behind the Brownian motion but his name got stuck to it anyway. Albert Einstein also studied this phenomena for molecules in a gas which ultimately made the way for the existence of atoms. But the first person to develop mathematics of Brownian motion as an stochastic process and its application to finance was Louis Bachelier in his PhD thesis popularly known as Bachelier’s Thesis.

References:

Steven Shreve, Stochastic Calculus for Finance II

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Intro to Random Walk and the Brownian motion (StoCal-1)

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