The Markov and Martingale Properties

In order to formally define the concept of Brownian motion and utilise it as a basis for an asset price model, it is necessary to define the Markov and Martingale properties. These provide an intuition as to how an asset price will behave over time.

The Markov property states that a stochastic process essentially has "no memory". This means that the conditional probability distribution of the future states of the process are independent of any previous state, with the exception of the current state. The Martingale property states that the future expectation of a stochastic process is equal to the current value, given all known information about the prior events.

Both of these properties are extremely important in modeling asset price movements.

## The Markov Property

A sensible way to introduce the Markov property is through a sequence of random variables $Z_i$, which can take one of two values from the set $\{1,-1\}$. This is known as a coin toss. We can calculate the expectations of $Z_i$:

\begin{eqnarray*} \mathbb{E}(Z_i) = 0, \mathbb{E}(Z_i^2) = 1, \mathbb{E}(Z_i Z_k) = 0 \end{eqnarray*}

The key point is that the expectation of $Z_i$ has no dependence on any previous values within the sequence. Let us take the partial sums of our random variables within our coin toss, which we will denote by $S_i$:

\begin{eqnarray*} S_i = \sum^i_{k=1} Z_i \end{eqnarray*}

We can now calculate the expectations of our partial sums, using the linearity of the expectation operator:

\begin{eqnarray*} \mathbb{E}(S_i) = 0, \mathbb{E}(S_i^2)= \mathbb{E}(Z_1^2 + 2 Z_1 Z_2 + ...) = i \end{eqnarray*}

We see that, again, there is no dependence on the expectation of $S_i$ of any previous value within the sequence of partial sums. We can extend this to discuss conditional expectation. Conditional expectation is the expectation of a random variable with respect to some conditional probability distribution. Hence, we can ask that if $i=4$ (i.e. we carry out four coin tosses), what does this mean for the expectation of $S_5$?

\begin{eqnarray*} \mathbb{E}(S_5 | Z_1, Z_2, Z_3 , Z_4) = S_4 \end{eqnarray*}

That is, the expected value of $S_i$ is only dependent upon the previous value $S_{i-1}$, not on any values prior to that. This is known as the Markov Property. Essentially, there is no memory of past events beyond the point our variable is currently at within the sequence. Nearly all financial models discussed in these articles will possess the Markov property.

## The Martingale Property

An additional property that holds for our sequence of partial sums is the Martingale property. It states that the conditional expectation of the sequence of partial sums, $S_i$ is simply the current value:

\begin{eqnarray*} \mathbb{E}(S_i | S_k, k<i) = S_k \end{eqnarray*}

Essentially, the martingale property ensures that in a "fair game", knowledge of the past will be of no use in predicting future winnings.

These properties will be of fundamental importance in regard to defining Brownian motion, which will later be used as a model for an asset price path.

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