Stochastic Differential Equations Cheatsheet

All important topics from SDEs (at least in my opinion)

This is a topic I've "approached from the side" by accident often when studying Measure Theory, Real Analysis, and Probability. SDEs have a ton of applications and they're super interesting so I wanted to work on a public cheatsheet!

All my side comments are italicized.

Important Problem Prototypes

Stochastic DEs

$\frac{dN}{dt} = \alpha(t)N(t)$ with $N(0) = N_0$

where $N_0$ is a constant.

Issue: $\alpha(t) = r(t) + \omega$ $\omega$ is undetermined noise, $r(t)$ nonrandom.

How do we solve this DE?

Sigma algebra definition

$\Omega$ is some set.

say... $\Omega = {1,2,3,4,5,6}$, the faces of a die

$\mathcal{F}$ is a family of subsets of $\Omega$ (See family vs. collection)

Then a $\sigma$-algebra is $\mathcal{F}$, a family of subsets of $\Omega$, such that these conditions hold:

A. $\emptyset \in \mathcal{F}$

B. The complement of any subset belonging to $\mathcal{F} \in \mathcal{F}$. So,

$$F \in \mathcal{F} \implies F^C \in \mathcal{F}$$ where $$F^c := \Omega / F$$

C. $$A_1, A_2, \dots \in \mathcal{F} \implies A:= \bigcup_{i=1}^{\infty} A_i \in \mathcal{F}$$ (the union of all subsets in $\mathcal{F}$ is in $\mathcal{F}$)

A sigma algebra of the subsets of any set seems to me to be an intuitive way of making sure your subsets, their complements, and their "closures" are all accounted for

If this doesn't make sense, I reccomend reading Section 2.4 (on discrete events) of Mathematical Statistics by Wackerly, Mendehall, and Scheaffer

Measurable Space

The pair $(\Omega, \mathcal{F})$ is a measurable space

rememeber that $\mathcal{F}$ is a $\sigma$-algebra on the subsets of $\Omega$

It can have a probability measure.

Probability Measure on Measurable Space

A probability measure on a measurable space is a function $P:\mathcal{F} \to [0,1]$) such that these conditions hold:

A. $P(\emptyset) = 0$, and $P(\mathcal{F}) = 1$

B. if $A_1, A_2, \dots \in \mathcal{F}$, and they are all disjoint, aka : $$A_i \cap A_j = \emptyset \text{ in } {A_i}_{i=1}^{\infty}$$

then

$$ P(\bigcup_{i = 1}^{\infty}A_i) = \sum_{i=1}^{\infty}{P(A_i)} $$

Probability Space

The 3-tuple $(\Omega, \mathcal{F}, P)$, obeying definitions above, is a probablity space.

Complete Probability Space and "P outer measure zero"

Recall that $P:\mathcal{F} \to [0,1]$ is a probablity measure on $(\Omega,\mathcal{F})$ and that $F \in \mathcal{F}$.

Say $G \subset F$.

Then $(\Omega, \mathcal{F}, P)$ is a complete probability space if $\mathcal{F}$ contains all subsets $G$ with "P-outer measure zero", or $P^*(G)$, where:

$$ P^*(G) := \text{inf}{P(F); f \in \mathcal{F}, G \subset F} = 0 $$

to me, this definition assures that our probability space has a general concept of a floor, or lower bound, at zero

also, remember from before that while $P(\emptyset) = 0$ is a requirement for a probability measure, it is not the only set that can be equal to zero under the probabiltiy measure, P. Adding all those other sets is what gives us a complete probability space

F-Measurable sets ... events!

$F$, the subsets of $\Omega$, belonging to $\mathcal{F}$, are called $\mathcal{F}$-measurable sets, or events (from probability theory)

this is where concepts from undergrad Probability courses really start to show their subtleties :)

Sigma algebra H generated by U

say $\mathcal{U}$ is a family of subsets of $\Omega$.

Then there always exists a smallest sigma $\mathcal{H_U}$ algebra generated by $\mathcal{U}$ (that is, containing the family $\mathcal{U}$). So,

$$ \mathcal{H_u} = \bigcap{\mathcal{H}: \mathcal{H} \text{ a sigma algebra of } \Omega; \mathcal{U} \subset \mathcal{H} } $$

I think about this one in terms of the family in question, we can always construct a "smallest one"

We can prove this one.

My proof of smallest sigma algebra

TODO

Borel Sets and Borel sigma algebras

Say $\mathcal{U}$ is the collection of open subsets of $\mathbb{R}^n$ (though it can be any $\Omega$ which is a topological space) (say U contians (0,1), (0,1000), (-pi, infinity)...)

Then the smallest $\sigma$-algebra generated by $\mathcal{U}$, ($\mathcal{H_U}$) is a Borel algebra. So,

$$ \mathcal{B}:=\mathcal{H_U} $$

and $B \in \mathcal{B}$ are Borel Sets. (subsets belonging to the sigma algebra are the borel sets)

Why Specify Borel sigma algebras?

This above is useful because..

F Measurable Functions on Borel Sets

Doob-Dynkin Lemma

TODO

Important for conditional expectation, conditioning on sigma algebras.

Stochastic Process

$$ {X_t}_{t \in T} $$

where $X$ is a random variable, the probability space is $(\Omega, \mathcal{F}, P)$ with range in euclidian space.