## Another Stochastics Question

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### Another Stochastics Question

X_t = int_0_t (g(u) dW(u))

g is a cont function.

What distribution will X_t follow?
f0X_in_s0X

Posts: 86
Joined: Tue Dec 04, 2007 5:47 pm

I guess you mean that g is continuous and a deterministic function of time?

Then you have an Ito integral of a deterministic integrand, so X_t is normally distributed with mean 0 and variance \int_{0}^{t} g^2(u)du.

The mean is zero because at each infinitesimal timestep you're just multiplying a Gaussian-distributed, zero-mean variable (dW_u) by a constant -- which doesn't change its mean -- then summing over all infinitesimal timesteps, which also doesn't change the mean.

Derive the variance by using Ito's isometry or by deriving the moment-generating function (which also proves mean is 0).

The result also holds if g is discts.
INFIDEL

Posts: 62
Joined: Sun Aug 26, 2007 5:57 am

Thanks Infidel.

Is this a standard result? Like, given in maybe Shreve. Or does one have to expicitily derive the Moment generation function to show that X_t is normally distributed?
f0X_in_s0X

Posts: 86
Joined: Tue Dec 04, 2007 5:47 pm

the result is pretty obvious when g is piecewise constant. And you can approximate the general case with that case.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

mj's hit the nail on the head... Approximating by pcewise const fns is very useful: e.g. if your g(u) was more general -- an adapted stochastic process -- this is how you'd construct the proof that your X_t is a martingale, and of the Ito isometry.

Yes look at Shreve, esp. Chapt. 4 for this. I think it's a pretty good book. I'd say it's quite applied, even though it might look pure. I mean Shreve, not Karatzas and Shreve which is pure pure and might be hard going if you don't have a maths/hard-core physics background.
INFIDEL

Posts: 62
Joined: Sun Aug 26, 2007 5:57 am