Postscript version of these notes

STAT 380 Lecture 22

Properties of Brownian motion

• Suppose t> s. Then

Notice the use of independent increments and of .

• Again if t> s:

  

Suppose t< s. Then is a sum of two independent normal variables. Do following calculation:

, and independent. Z=X+Y.

Compute conditional distribution of X given Z:

Now Z is where so

for suitable choices of a and b. To find them compare coefficients of , x and 1.

Coefficient of :

so .

Coefficient of x:

so that

Finally you should check that

to make sure the coefficients of 1 work out as well.

Conclusion: given Z=z the conditional distribution of X is with a and b as above.

Application to Brownian motion:

• For t< s let X be X(t) and Y be X(s)-X(t) so Z=X+Y=X(s). Then , and . Thus

  

  and

  

  SO:

  

  and

  

The Reflection Principle

Tossing a fair coin:

Both sequences have the same probability.

So: for random walk starting at stopping time:

Any sequence with k more heads than tails in next m tosses is matched to sequence with k more tails than heads. Both sequences have same prob.

Suppose is a fair (p=1/2) random walk. Define

Compute ? Trick: Compute

First: if then

Second: if then

Fix y < x. Consider a sequence of H's and T's which leads to say T=k and .

Switch the results of tosses k+1 to n to get a sequence of H's and T's which has T=k and . This proves

This is true for each k so

Finally, sum over all y to get

Make the substitution k=2x-y in the second sum to get

Brownian motion version:

(called hitting time for level x). Then

Any path with and X(t)= y< x is matched to an equally likely path with and X(t)=2x-y> x.

So for y> x

while for y< x

Let to get

Adding these together gives

Hence has the distribution of |N(0,t)|.

On the other hand in view of

the density of is

Use the chain rule to compute this. First

where is the standard normal density

because P(N(0,1)> y) is 1 minus the standard normal cdf.

So

This density is called the Inverse Gaussian density. is called a first passage time

NOTE: the preceding is a density when viewed as a function of the variable t.

Martingales

A stochastic process M(t) indexed by either a discrete or continuous time parameter t is a martingale if:

whenever s< t.

Examples

• A fair random walk is a martingale.
• If N(t) is a Poisson Process with rate then is a martingale.
• Standard Brownian motion (defined above) is a martingale.

Note: Brownian motion with drift is a process of the form

where B is standard Brownian motion, introduced earlier. X is a martingale if . We call the drift

• If X(t) is a Brownian motion with drift then

  

  is a geometric Brownian motion. For suitable and we can make Y(t) a martingale.

• If a gambler makes a sequence of fair bets and is the amount of money s/he has after n bets then is a martingale - even if the bets made depend on the outcomes of previous bets, that is, even if the gambler plays a strategy.

Some evidence for some of the above:

Random walk: iid with

and with . Then

Things to notice:

treated as constant given .

Knowing is equivalent to knowing .

For j> k we have independent of so conditional expectation is unconditional expectation.

Since Standard Brownian Motion is limit of such random walks we get martingale property for standard Brownian motion.

Poisson Process: . Fix t> s.

Things to notice:

I used independent increments.

is shorthand for the conditioning event.

Similar to random walk calculation.