What is a stochastic differential equation?

Question

What is a stochastic differential equation?

Answer 1

The most obvious way to look at SDEs is in terms of the processes underlying stock markets.  If you look at the return on the stock at evenly spaced intervals, you will (supposedly) find that they follow something like a normal distribution. That is, if the share price at time t is S(t), then

(S(t+1) - S(t))/S(t) ~ N(μ,σ^2).

It's kinda useful to phrase that as

ΔS/S = μΔt + σ√(Δtε) where ε ~ N(0,1)

or

ΔS/S = μΔt + σΔW

Obviously, if you multiply both sides by S and sum over all the ΔS, you get the value of S at the end of the overall time period.

(ie Σ(SμΔt) + Σ(SσΔW) = Σ(ΔS) = S)

If we take the limit as Δt goes to 0, we end up with something like an integral.

S = ∫dS = ∫Sμdt + ∫SσdW.

I guess the ∫Sμdt is a typical integration, but the point is that W is a random variable, so you can't really integrate with respect to it.

Normally the above is written as dS = Sμdt + σSdW.  This latter expression is what is normally called a stochastic differential equation.  Note that it's not really a differential equation though, as the whole point is that W is random and so has no derivative.

I can't really say much more than this, mostly because I don't really understand the topic myself.