Prove that Y(t) = W(t)^2 - t is a martingale.

 

A process Y(t) is a martingale with respect to some filtration if:

1. The expected value of |Y(t)| is finite for all t.

2. Y(0) is integrable and its expected value is 0.

3. The expected value of Y(t) given the history up to time s is equal to Y(s) for all 0 <= s < t.

 

Let's prove each of these points for Y(t):

 

1. Both W(t)^2 and t are finite, so the expected value of Y(t) is finite.

 

2. Y(0) = W(0)^2 - 0 = 0, which is integrable, and its expected value is 0.

 

For the third point:

 

3. We need to show that the expected change in Y(t) given the history up to time s is zero for 0 <= s < t.

 

The increment of the Brownian motion over [s, t], i.e., W(t) - W(s), has mean 0 and variance t - s (*).

 

Now, the change in Y over [s, t] is:

Y(t) - Y(s) = W(t)^2 - t - W(s)^2 + s

This can be broken down as:

= (W(t) - W(s))^2 + 2*W(s)*(W(t) - W(s)) - (t - s)

 

Given the properties of Brownian motion, the expected value of (W(t) - W(s))^2 given the history up to time s is t - s, and the expected value of W(t) - W(s) given the history is 0.

 

So, the expected change in Y(t) given the history up to time s is:

= t - s + 0 - (t - s) = 0

 

Given this is true for any s < t, Y(t) is a martingale.

 

Thus, Y(t) = W(t)^2 - t is a martingale when W(t) is a standard Brownian motion.

 

 

(*)

 

To prove that W(t) - W(s), has mean 0 and variance t - s :

 

E(W(t) - W(s))= E(W(t)) - E(W(s))=t-s

 

Variance of W(t) - W(s):

 

To determine the variance of the increment W(t) - W(s) of a standard Brownian motion, remember that the increments are normally distributed with mean 0 and variance equal to the length of the increment.

 

The formula for variance is given by:

Var[X] = E[X^2] - E[X]^2

 

For the increment W(t) - W(s):

E[W(t) - W(s)] = 0 (since the expected value is 0)

 

Thus, E[W(t) - W(s)]^2 = 0

 

Our variance formula then simplifies to:

Var[W(t) - W(s)] = E[(W(t) - W(s))^2]

 

The term (W(t) - W(s))^2 represents the second moment of a normal distribution. 

 

E[(W(t) - W(s))^2]), represents the average or expected value of the squared increments over many possible realizations of the Brownian motion.

 

One of the fundamental properties of a standard Brownian motion is that the expected value of the squared increment over an interval [s, t] is equal to the length of that interval. In mathematical terms, this is expressed as:

 

E[(W(t) - W(s))^2] = t - s 

 

Substituting this into our variance formula, we get:

Var[W(t) - W(s)] = t - s