Stochastic Process

def. Time Series is a stochastic process indexed by discrete (integer) time points.

• See discussion: Is a time series the same as a stochastic process? - Cross Validated

def. Stochastic Process is simply the sum of the sequence of random variables: $X_1,\dots ,X_n$:

$$W_k=\{\begin{array}{ll}{\sum }_{i=1}^k{R}_i& \text{if }k>0\\ w_0& \text{if }k=0\end{array}$$

• $R_i$ are identically distributed.: Stationary

• $E(W_{k+1}|W_1\dots W_k)=W_k$ then: Martingale

• $P(W_{k+1}=w_{k+1}|W_1=w_1,\dots W_k=w_k)=P(W_{k+1}=w_{k+1}|W_k=w_k)$: Markov

  random variable - Why do stochastic processes involve time? - Cross Validated

Weiner Process §

Standard Weiner Process §

Motivation. Assume there are random variables $X_1\dots$ as:

$$X_i=\{\begin{array}{ll}\frac{1}{\sqrt{n}}& 0.5\text{ probability}\\ -\frac{1}{\sqrt{n}}& 0.5\text{ probability}\end{array}$$

1. let $W_k$ be a stochastic process such that:

$$W^{(n)}(t)=\{\begin{array}{ll}0& \text{if }t=0\\ R_1+\cdots +R_t& \text{if}t\in \mathbb{N}\\ \text{affine extension on closest interval }& \text{else}\end{array}$$

Then we get the Weiner process as:

$$\mathfrak{B}(t)=\underset{n\to \infty }{\lim }{W}^{(n)}(t)$$

def. Brownian Motion. (=Weiner process) $\mathfrak{B}(t)$ (written as ${\mathfrak{B}}_t$) is a set of random variables continuous-time indexed and has the following properties:

1. Is a continuous process
2. ${\mathfrak{B}}_0=0$
3. ${\mathfrak{B}}_t-{\mathfrak{B}}_s{\sim }_{d}N(0,t-s)$
   1. Thus, ${\mathfrak{B}}_t{=}_d\mathcal{N}(0,t)$
4. Any interval ${\mathfrak{B}}_a-{\mathfrak{B}}_b$ and ${\mathfrak{B}}_c-{\mathfrak{B}}_d$ where $[a,b]$ and $[c,d]$ do not overlap is independent. Notation wise, we think of $\mathfrak{B}=\{{\mathfrak{B}}_t{\}}_{t\ge 0}$, i.e. a set of random variables indexed by $t$.

Remark. Brownian Motion can also be defined as an Ito Process that satisfies the following stochastic differential equation:

$$d\mathfrak{B}=\mu \cdot dt+{\sigma }^{2}d\mathfrak{B}$$

Trivially, in the case of standard brownian motion, $\mu =0,{\sigma }^2=1$ Thus $d\mathfrak{B}=0\cdot dt+1\cdot d\mathfrak{B}$.

Properties. Brownian Motion satisfies the following properties:

• Martingale Property: Brownian motion is martingale: $\forall s\le t,\mathbb{E}({\mathfrak{B}}_t|{\mathcal{F}}_s)={\mathfrak{B}}_s$ where ${\mathcal{F}}_s=\sigma ({\mathfrak{B}}_s)$
• Markov Property:
  • $\forall s\ge 0,\{{\mathfrak{B}}_{s+t},t\ge 0\}$ is independent of ${\mathcal{F}}_s$ and…
  • $\{{\mathfrak{B}}_{s+t},t\ge 0\}{=}_d\{{\mathfrak{B}}_t\}$
  • Alternatively: $\mathbb{P}(X_u\in A\mid \sigma (X_s))=\mathbb{P}(X_u\in A\mid X_t)$ where any combination $s\le t\le u$
• Scaling Invariance: If ${\mathfrak{B}}_t$ is a brownian motion, then $\frac{\mathfrak{B}(a^{2}t)}{a}$ is also a brownian motion $\forall a>0$
• Quadratic Variation property: $[\mathfrak{B}{]}_t:={\sum }_{i=1}^n({\mathfrak{B}}_{t_i}-{\mathfrak{B}}_{t_{i-1}}{)}^2=t$

With Scale and Drift §

def. Weiner Process with Drift and Scaling (WPDS). Such is a Weiner process $X(t)$ that has the following properties:

1. Is a continuous process
2. $X(0)=x_0$
3. $X(t+\Delta t)-X(t){=}_{d}N(\mu \Delta t,{\sigma }^2\Delta t)$
4. Any interval $X(a)-X(b)$ and $X(c)-X(d)$ where $[a,b]$ and $[c,d]$ do not overlap is independent.

• Properties:
  • $\mu$: drift; the higher, the more it climbs
  • ${\sigma }^2$: scaling; the higher, the more volatile (See y-axis:)

thm. Standard WP to Scale and Drift. Given:

• $B(t)$ is a standard Weiner process
• $X(t)$ is a Weiner process with drift $\mu$ and scaling ${\sigma }^2$, with initial value $x_0$
• ⇒ Then the following relationship holds (two equivalent definitions)

$$\begin{array}{rl}X(t)& {=}_d{x}_0+\mu t+\sigma B(t)\\ dX(t)& =\mu \cdot dt+\sigma \cdot dB(t)\end{array}$$

Geometric Brownian Motion §

thm. Geometric Brownian Motion. Given $X(t)$ is a WPDS $\mu ,{\sigma }^2,x_0$ then the following is a geometric brownian motion with initial value $S(0)$: (two equivalent definitions)

$$\begin{array}{rl}G(t)& =G_0\exp [X(t)]=G_0\exp [x_0+\mu t+\sigma B(t)]\\ dG(t)& =\left(\mu +\frac{{\sigma }^2}{2}\right)G(t)\cdot dt+\sigma G(t)\cdot dB(t)\end{array}$$

Geometric to WPSD

The following is equivalent:

$$\begin{array}{rl}dS(t)& =\left(\mu +\frac{{\sigma }^2}{2}\right)S(t)\cdot dt+\sigma S(t)\cdot dB(t)\\ S(t)& =S_0\exp \left[\left(\mu -\frac{{\sigma }^2}{2}\right)t+\sigma B(t)\right]\end{array}$$

Properties of Stochastic Processes §

def. Adapted process. A stochastic process $\{X_t{\}}_{t\ge 0}$ is adapted to filtration set $\{{\mathcal{F}}_t{\}}_{t\ge 0}$ if $\forall i,X_i$ is a ${\mathcal{F}}_i$-measurable function.

Intuition. Recall that a sigma algebra $\mathcal{F}$ can be thought of as “resolution of information”. $X_i$ is realized and its information is ${\mathcal{F}}_i$. The series of filtrations ${\mathcal{F}}_1\subset {\mathcal{F}}_2\subset \cdots \subset {\mathcal{F}}_t\subset \cdots \subset \mathcal{F}$ correspond to the series of random variables $X_1,X_2,\dots$; for each timestep, the information gets higher and higher resolution.

Martingale Process §

Motivation. Many stochastic processes, including the standard brownian motion ${\mathfrak{B}}_t$ has the property that you can’t predict future trajectory based on information from its past trajectory. We formalize this as a martingale.

def. Martingale. A stochastic process $\{X_t{\}}_{t\ge 0}$ on a filtered probability space $(\Omega ,\mathcal{F},\{{\mathcal{F}}_t\},\mathbb{P})$ is a martingale w.r.t. its adapted filtration $\{{\mathcal{F}}_t\}$ if:

$$\forall s\le t,\mathbb{E}(X_t|{\mathcal{F}}_s)=X_s$$

This is equivalent to saying

• $\forall s\le t,\mathbb{E}(X_t)=\mathbb{E}(X_s)$ by taking expectation on both sides of above
• $\forall s\le t,\mathbb{E}(X_t-X_s)=0$
  • This is a simple proof that brownian motion is a martingale; $\mathbb{E}({\mathfrak{B}}_t-{\mathfrak{B}}_s{=}_d\mathcal{N}(0,t-s))=0$

Quadratic Variation §

Motivation. Quadratic variation is not about variation of probability distributions. It’s a way to measure how “shaky” a function (in this case, a Brownian motion) is in a given interval.

def. Quadratic Variation. For a stochastic process $\{X_t\}$, its quadratic variation on interval $[0,t]$, $[X{]}_t$ for $0=t_0<t_1<\cdots <t_n=t$, is:

$$\underset{n\to \infty }{\lim }\sum _{i=1}^n(X_{t_i}-X_{t_{i-1}}{)}^2$$