Stochastic Integration
Integration is one of those mathematical operations that should conform to common-sense: no matter what method of integration you’re using, no matter how high-falutin’, the area of a square should be the square of the length of its sides. And that’s the way its been up to now: Riemann, Lebesgue-Stieltjes, and Lebesgue integration… they’re all cut from the same cloth, all succeedingly more careful generalizations of the intuitive notion of measure.
Yesterday we were briefly introduced to a form of stochastic integration, namely that of integration of 
functions with respect to Brownian noise:
![\int_0^t f(s) dB_s = \sum_{i=1}^\infty \langle \chi_{[0,t]} f, \varphi_i \rangle \xi_i](/cz/latexrender/pictures/c3dd14a7b36ff1659acfa7694e345bee.png "\int_0^t f(s) dB_s = \sum_{i=1}^\infty \langle \chi_{[0,t]} f, \varphi_i \rangle \xi_i")
where 
is an orthonormal basis of 
and 
is a sequence of i.i.d. 
r.v.s, so the integral is a Gaussian r.v. for each 
. This definition does not seem to have much to do with the concept of measurement. Is this even well-defined? By the Riemann-Lebesgue lemma the sum converges, but apparently any o.n.b. can be used, so does changing it affect the integral? Despite the obscurity of its origins, I like the simple form of this integral.
Today, I found an online book which does a great job of motivating stochastic integrals. To abbreviate the example he gives, consider the problem of sending a probe to the moon. We’ll describe the state of the system as 
, where 
are the location, velocity, and momentum of the probe. In the ideal case, we know 
where 
is the force applied to the probe, so we can solve for
In the real world, there will be random effects we can’t account for deterministically, such as minor fluctuations in the gravitational field, collision with debris, etc., all of which we will call noise 
(note that this model doesn’t account for noise which is partially caused by the state 
, noise is a function of – I don’t know if this is a limitation of the theory, or just this example). Then we’d like to say that how much the noise affects the state of the system varies depending on the state of the system, giving the new equation
Since the noise is random, 
is in fact a stochastic process. Each elementary event 
determines a different sample path. Therefore if 
is such that the Lebesgue-Stieltjes integral is defined on each sample path (cadlag?), we can actually solve the above equation, and use it to calculate useful quantities. For instance, if 
represents the time when the probe reaches the moon (
if it misses), then we can calculate the expected distance from the target point 
, given a particular noise model:
![\mathbb{E}[\|s - X(T)\|].](/cz/latexrender/pictures/a0f324adcf4523d678e1e1d3e06c8f52.png "\mathbb{E}[\|s - X(T)\|].")
Pretty cool!
The problem is, apparently in general the sample paths of stochastic processes are not nice, so Lebesgue-Stieltjes integration cannot be done. Then you need to break out the power tools
Possibly relevant posts:
- Not a frame, but Bessel (5/26/2006)
- Exactness of differential forms (7/28/2007)
- Perfect Reconstruction, but not really (4/15/2005)
wow, alex, when reading your blog, your protriat shows up again in front of me, holding a library book stopping people in the hall way. i actually found some familiar concepts, like “gram schimidts” process, yes! ha, it’s not always easy to undertand what you are talking all about the math, but wish you every thing well in your grad study.
Hi, Alex
Really exciting formulation of stochastic integral. what book is it from ?
thanks
Cepreu
Hi Alex,
very nice and inspirational formulation of stochastic integral!
What book/lectures is it from ?
thanks,
Ceprey
Cepera, I posted a link above to the book I saw that explanation in. It’s also available as a volume of the Encyclopedia of Mathematics and Its Applications, as “Stochastic Integration with Jumps”. Despite the very nice motivation of the concept of a stochastic integral, it’s not a good book for an introduction to the subject. I like Introduction to Stochastic Integration by Kuo better for that.
Alex,
I meant first definition, where stochastic integral is represented by sum of function “f” projections onto basis multiplied by random Gaussian numbers.
i didn’t find it in Bichteler’s book.
thanks,
C
Ahh.. that was *given* to us in class, out of the blue, as our first working definition of a stochastic integral.