[5] Stochastic Integrals

1. Without randomness

(Notation)

• $X(t) =$ the position of the particle at time $t$.
• $\mu(t,x) =$ drift term $=$ the velocity of the particle when it is located at $x$ at time $t$.
• $\Delta t =$ a very small change in time.
• $\Delta X(t) = X(t+\Delta t)-X(t) =$ a very small change in location from time $t$ to time $t+\Delta t$

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When a particle moves on the real line without randomness, its trajectory can be described by $\Delta X(t) = \mu(t, X(t))\Delta t$

We can determine $X(t)$ by solving ordinary differential equation $\dot{X}(t)$.

$X(t+\Delta t) = X(t) + \mu(t, X(t)) \Delta t$

$\Rightarrow \ \frac{X(t+\Delta t)-X(t)}{\Delta t} = \mu(t, X(t))$

$\Rightarrow \ \frac{\Delta X(t)}{\Delta t} = \mu(t, X(t))$

$\Rightarrow \ \dot{X}(t) = \mu(t, X(t))$

2. With randomness

(Notation)

• $W(t)=W_t =$ the Wiener process.
• $\Delta W(t) = W(t+\Delta t) - W(t) =$ the displacement by the Brownian motion from time $t$ to time $t+\Delta t$
• $\sigma(t,x) =$ diffusion term measuring the strength of randomness.

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With randomness, its trajectory can be described by $\frac{\Delta X(t)}{\Delta t} = \mu(t, X(t))+\sigma(t,X(t))\frac{\Delta W(t)}{\Delta t}$

However we cannot solve $\dot{X}(t)=\mu(t, X(t))+\sigma(t,X(t))\dot{W}(t)$ because $W(t)$ is not differentiable.

Instead, we can rewrite its trajectory
$dX(t) = \mu(t, X(t))dt+\sigma(t,X(t))dW(t)$

Then, integrate both sides
$\int_o^t dX(s) = \int_0^t\mu(s, X(s))ds + \int_0^t\sigma(s,X(s))dW(s)$

$\Rightarrow \ X(t,w)=X_0+\int_0^t\mu(s, X(s,w))ds + \int_0^t\sigma(s,X(s,w))dW(s,w)$

3. Information

Let $X$ be a stochastic process.

• $\mathcal{F}_t^X =$ $\sigma$-algebra generated by $X$ on the time interval $[0,t]$. It is called the infromation generated by $X$ on $[0,t]$
• If a random variable $Z:\Omega\rightarrow\R$ which is probabilities of events in term of $Z$ can be calculated based on the $\sigma$-algebra $\mathcal{F}_t^X$, then we write $Z\in\mathcal{F}_t^X$. $Z$ is said to be $\mathcal{F}_t^X$-measurable
• If another stochastic process $Y(t)$ satisfies $Y(t) \in \mathcal{F}_t^X$ for all $t \geq 0$, $Y$ is said to be adapted to the filtration $F = \{\mathcal{F}_t^X\}_{t\geq 0}$