Another Stochastics Question

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Another Stochastics Question

Postby f0X_in_s0X » Fri Apr 25, 2008 11:11 am

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

g is a cont function.

What distribution will X_t follow?
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Postby INFIDEL » Fri Apr 25, 2008 3:34 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.
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Postby f0X_in_s0X » Fri Apr 25, 2008 6:53 pm

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?
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Postby mj » Fri Apr 25, 2008 9:38 pm

the result is pretty obvious when g is piecewise constant. And you can approximate the general case with that case.
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Postby INFIDEL » Sat Apr 26, 2008 2:40 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.
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