# Stochastic processes

```
graph BT;
SP[Stochastic Process];
GSP[Gaussian Stochastic Process]-->SP;
PP[Point Process]-->SP;
RF[Random Field]-->SP;
MRF[Markov Random Field]-->RF;
IM[Ising Model]-->MRF;
CTP[Continuous Time Process]-->RF;
DTP[Discrete Time Process]-->SP;
MP[Markov Process]-->CTP;
MP[Markov Process]-->MRF;
MC[Markov Chain]-->DTP;
GMP[Gauss-Markov Process]-->GSP;
GMP-->MP;
OUP[Ornstein-Uhlenbeck Process]-->GMP;
PPP[Poisson Point Process]-->PP;
PPP-->MP;
LP[Lévy process]-->MP;
WP[Wiener Process]-->LP;
HP[Hawkes Process]-->PP;
```

For the length of this chapter, assume we are give a probability space $(Ω, Σ, \operatorname{Pr})$.

**Definition.** Given a probability space $(Ω, Σ, \operatorname{Pr})$, a measurable space $(Ω_X, Σ_X)$ and a set $T$,
a *stochastic process* is a function $X: T \times Ω → Ω_X$ such that for each $t ∈ T$ the restricted function $X\p{t, \dummyarg}: Ω → Ω_X$ is a random variable.

**Note.** The set $T$ is called the *index set* or *parameter set* of the stochastic process. The set $Ω_X$ is called the *state space*. Given $ω ∈ Ω$ the restricted function $X\p{\dummyarg, ω}: T → Ω_X$ is called a *sample function*, *realization*, *sample path*, *trajectory*, *path function*, or *path*.

Depending on the nature of the index set $T$, a stochastic process can be called many things:

**Definition.** If the index set $T$ is a topological space then $X$ is a *random field*.

**Note.** Nearly all stochastic processes are random fields by definition, but the term is generally reserved for when $T$ has more than one dimension.

**Definition.** If the index set $T$ is a countable ordered set then $X$ is a *discrete-time process*.

**Definition.** If the index set $T$ is an interval of $\R$ then $X$ is a *continuous-time process*.

**Definition.** If the index set $T$ are the vertices of a graph and satisfies **To do.** properties, then $X$ is a *Markov random field*.

**Note.** Similarly to the above definitions, a stochastic process can also be named based on the state space $Ω_X$. *discrete stochastic process*, *integer-valued stochastic process*, *real-valued stochastic process*, $n$-*dimensional vector process*.

**Definition.** *Stationary process*.

**Definition.** *Markov process*.

**Definition.** *Martingale*.

**Definition.** *Lévy process*.

**Definition.** *Bernoulli process*.

**Definition.** *Random walk*.

**Definition.** *Wiener process*.

**To do.** **Theorem.** Every continuous-time independent-increment process is a Gaussian process. Proof using central limit theorem.

https://www.whoi.edu/cms/files/lecture06_21268.pdf

## Point processes

**Definition.** *Point process*.

**Definition.** *Poisson point process*.

**Definition.** *Cox process*.

**Definition.** *Hawkes process*.

https://arxiv.org/abs/1507.02822

## Gaussian processes

**Definition.** *Gaussian process*.

*Kernel*.

https://en.wikipedia.org/wiki/Covariance_function

https://www.cs.toronto.edu/~duvenaud/cookbook/

$$ \Prc{\vec f_*}{\vec x_*, \vec x, \vec y} $$

**To do.** Kernel trick.

http://www.gaussianprocess.org/gpml/chapters/RW4.pdf

**Definition.** *Gauss-Markov process*.

**Definition.** *Matérn kernel*.

**Definition.** *Squared exponential kernel* aka *RBF kernel*. Special case of Matérn $\nu = \infty$.

**Definition.** *Rational quadratic kernel*.

**Definition.** *Ornstein–Uhlenbeck process*. Aka absolute exponential. Special case of Matérn with $\nu = \frac 12$.

### Gaussian Process Regression

**Note.** (Gaussian process regression)[https://en.wikipedia.org/wiki/Gaussian_process_regression] is also known as *Kriging* and *Wiener–Kolmogorov prediction*.