📂Stochastic Differential Equations

Linear, Homogeneous, Autonomous Stochastic Differential Equations

Definition ¹

Let’s assume we have a probability space $( \Omega , \mathcal{F} , P)$ and a filtration $\left\{ \mathcal{F}_{t} \right\}_{t \ge 0}$. Consider the following $n$-dimensional stochastic differential equation with respect to two functions $f$, $g$, and $\mathcal{F}_{t}$-adapted $m$-dimensional Wiener process $W_{t}$: $$ \begin{align*} d X_{t} =& f \left( t, X_{t} \right) dt + g \left( t, X_{t} \right) d W_{t} \\ f =& a(t) + A(t) X_{t} \\ g =& b(t) + B(t) X_{t} \\ a, b &: [0,T] \to \mathbb{R}^{n} \\ A, B &: [0,T] \to \mathbb{R}^{n \times m} \end{align*} $$

1. An SDE that is expressed as $f, g$ is called Linear. $$ d X_{t} = \left( a(t) + A(t) X_{t} \right) dt + \left( b(t) + B(t) X_{t} \right) dW_{t} $$
2. An SDE that is $a(t) = b(t) = 0$ is called Homogeneous. $$ d X_{t} = X_{t} \left[ A(t) dt + B(t) d W_{t} \right] $$
3. An SDE that is $B(t) = 0$ is referred to as Linear in the Narrow Sense. $$ d X_{t} = a(t) dt + A(t) X_{t} dt + b(t) d W_{t} $$
4. An SDE is said to be Autonomous Linear when $a,A,b,B$ is independent of time $t$. $$ d X_{t} = \left( a + A X_{t} \right) dt + \left( b + B X_{t} \right) d W_{t} $$

Examples

These are among the simpler types of SDEs, and their solutions are well-known.

Homogeneous SDE

The most famous example is the SDE defining Geometric Brownian Motion (GBM, Geometric Brownian Motion). $$ d X_{t} = \mu X_{t} dt + \sigma X_{t} d W_{t} $$

Linear SDE in the Narrow Sense

The following is known as the Ornstein–Uhlenbeck Equation or the Langevin Equation.

$$ d X_{t} = \mu X_{t} dt + \sigma d W_{t} $$