---
title: Brownian Motion
---

```{mermaid}
flowchart LR
    markdown["`Random walk"]
    newLines["`Line1
    Line 2
    Line 3`"]
    markdown --> newLines
```

## Brownian motion as a markov process
${X(t), t \geq 0}$. 

- Markov property: Prediction of process $X(s+t) with known information about $X(r), r \leq s$ depends only on X(s)
- 


## Filtration

Describes the past of a process $X$. 

$$
F_t = \sigma {X(s), s \in [0,t]}
$$


## Resources 

- Oliver Ibe's Markov processes for stochastic modeling