📂Stochastic Differential Equations

What is a Stochastic Differential Equation?

Definition ¹

$$ d X(t) = f \left( t, X(t) \right) dt + g \left( t, X(t) \right) d W_{t} \qquad , t \in \left[ t_{0} , T \right], T > 0 $$

Equations of the form above are called Stochastic Differential Equations, abbreviated as SDEs. Here, $f$ and $g$ are called the drift and diffusion coefficient functions, respectively. For the initial condition $X_{0} := X \left( t_{0} \right)$, the integral form is represented as follows.

$$ X(t) = X_{0} + \int_{t_{0}}^{t} f \left( s, X (s) \right) ds + \int_{t_{0}}^{t} g \left( s, X (s) \right) d W_{s} $$

Explanation

$$ d X_{t} = f \left( t, X_{t}\right) dt + g \left( t, X_{t} \right) d W_{t} $$

If this form doesn’t bother you, it’s either because you’ve studied Ito calculus very well or know very little about differential equations, one or the other. For someone familiar with differential equations but not with SDEs, naturally $g d W_{t}$ would be an eyesore. Unlike ODEs, SDEs incorporate this stochastic process, adding uncertainty to the model. Considering this term as $0$, that is, $g d W_{t} = 0$ as a nondeterministic system, we can see the following. $$ \begin{align*} d X(t) =& f \left( t, X(t) \right) dt + g \left( t, X(t) \right) d W_{t} \\ =& f \left( t, X(t) \right) dt + 0 \\ =& f \left( t, X(t) \right) dt \end{align*} $$ Dividing both sides by $dt$, we get $$ {{ d X (t) } \over { dt }} = f \left( t, X(t) \right) $$ Therefore, we can see that we have reclaimed the look of a well-known non-autonomous system.

Drift

In this explanation, recalling drift in time series analysis makes it quite natural to refer to the coefficient function $f$ as drift. This is because, regardless of the latter term, $f dt$ is the driving force that allows the system to be managed as a system itself.

Diffusion

Then, referring to $g$ as diffusion naturally suggests its role or characteristic of spreading or dispersing. In stochastic differential equations, this points to the concept of white noise, and unique results like Ito’s lemma arise due to such noise.