Summary of some basic stochastic calculus results

[Quasar Chunawala]

Hi guys,

I had taken down some quick summary notes when reviewing stochastic calculus recently, deriving things from first principles and then solving lots of exercise problems from standard sources. Might be helpful for a quick refresher on sell-side math. So, sharing it here.

Cheers.

 

[Daniel Duffy]

p. 26 A good test: compute Lebesgue integral numerically + manually using N subdivisions of y axis

e.g. f(x) = x^2

N = 2,4,8 etc.

// figures not numbered.

 

[Quasar Chunawala]

That's a good sense check exercise. Here's my solution for [imath]\int_{0}^{1} f d\mu [/imath].

For [imath]N=2[/imath] subdivisions, we can approximate the [imath]f(x)[/imath] as the step function:

[math]0 \cdot 1_{0 \leq x \leq 1/\sqrt{2}} + (1/2) \cdot 1_{1/\sqrt{2} \leq x \leq 1}[/math]
So, I get:

[math]\int_0^1 f d\mu \approx (0.5)(0.29289) = 0.14644[/math]
For [imath]N=4[/imath] subdivisions, we can approximate the [imath]f(x)[/imath] as the step function:

[math]0 \cdot 1_{0 \leq x \leq 1/\sqrt{4}} + (1/4) \cdot 1_{1/\sqrt{4} \leq x \leq 1/\sqrt{2}} + (2/4) \cdot 1_{1/\sqrt{2} \leq x \leq \sqrt{3}/2} + (3/4) \cdot 1_{\sqrt{3}/2 \leq x \leq 1} [/math]
So, I get:

[math]\int_0^1 f d\mu \approx (0.25)(0.207106) + (0.50)(0.15891) + (0.75)(0.133974) = 0.231712[/math]
I think I should expect the numerical answer to be much closer to [imath]0.3333[/imath] for [imath]N=8[/imath]. Will code it up after work tonight!

 

[Daniel Duffy]

Another way, from 1st principles from definition of upper and lower limits U and L, value always in (L,U), let N -> 2,4, ...
The way Archimedes wudda done it.

 

[Quasar Chunawala]

Another way, from 1st principles from definition of upper and lower limits U and L, value always in (L,U), let N -> 2,4, ...
The way Archimedes wudda done it.

Oh yeap, basically the only requirement is that the pre-images; [imath]E_j[/imath]'s should be nice sets, right, (and not some pathological set), which is what we mean by [imath]f[/imath] being measurable, or the random variable [imath]X[/imath] being (lebesgue) measurable in the first place.

Even the properties of expectation are easily gotten. 1) Prove something for a step random variable 2) Expectation of a non-negative random variable is simply [imath]\mathbb{E}[X] = \sup \{\mathbb{E}Z: \text{Z is step variable }, Z\leq X \}[/imath], so pass to the limit.

With little work, you get some very powerful properties.