Year 12s: what will be the first way you research your future options? >>

Help with a Btownian motion Q

could someone guide or help me with the following please:

A Brownian motion (Wt)t≥0 is a stochastic process whose properties include:
(i) At time t ≥ 0 its value, Wt, is a normally distributed random variable with zeromean and variance t (which implies that W0 = 0);
(ii) If t ≥ 0 and s ≥ 0 then (Wt+s − Ws) is a normally distributed random variablewith zero mean and variance s (and is independent of Wt);
(iii) If 0 ≤ s < t ≤ u < v then (Wt − Ws) and (Wv − Wu) are independent.

The quadratic variation of (Wt)t≥0, denoted by [W], is defined as follows. For anyt > 0 divide the interval [0, t] into a partition π,
0 = t0 < t1 < t2 < · · · < tn = t,
and let |π| be the maximum distance between two consecutive points in the partition,
|π| = max0≤k=≤n−1(tk+1 − tk).
Set the value of [W] at t to be
[W]t = lim|π|→0n-1(summation sign) k=0 (Wtk+1 − Wtk )^2,
where it is implicit that as |π| → 0, n → ∞.
Show thatE( [W]t − t)= 0 and E( [W]t − t)^ 2= 0.[Hint: if φ is normally distributed with zero mean and E[φ^ 2] = ν then E[φ^ 4] = 3ν^2.]

Reply 1

Gregorius

OK, this is not my speciality, but it looks like a fairly standard bookwork crunch! Let's try and get you started. It looks like the question allows you to assume that the limits exist for any choice of partition satifying the conditions. So, what do you think would be the easiest choice of partition to work with? Now with that partition, can you think of a way of getting a standard limit theorem to apply?

Big hint: once you've chosen the partition correctly, a google search of quadratic variation brownian motion will give you the answers!