6.2 Stochastic differential equations

For simplicity let us assume a one dimensional stochastic differential equation with additive noise [41]

$$\frac{d X(t)}{dt}=a(X(t),t)+\eta(t) \quad.$$

$a(X(t),t)$ in this Langevin equation can be interpreted as a deterministic or averaged drift term perturbed by a noisy diffusive term $\eta(t)$ which is a Gaussian random variable.

For the increase $\,d{X}\,$ during a time step $\,d{t}\,$ we get (to first order)

$$\,d{X(t)}\,=a(X(t),t) \,d{t}\, + \,d{W(t)}\,$$ (6.5)

with

$$\,d{W(t)}\,=\int_t^{t+\,d{t}\,}\eta(t')\,d{t'}\, \quad.$$

If we interpret the above integral as a limit of a sum, then $\,d{W}\,$ is a Gaussian random variable, because it is the sum of Gaussian random variables. Thus,

$$\langle {dW(t)} \rangle =0$$

and (cf. eqn. ())

$$\langle {(\,d{W(t)}\,)^2} \rangle = \int_t^{t+\,d{t}\,} \,d{t_1}\, \int_t^{t+\,d{t}\,} \,d{t_2}\, \langle {\eta(t_1) \eta(t_2)} \rangle$$ (6.6)

$$= \int_t^{t+\,d{t}\,} \,d{t_1}\, \int_t^{t+\,d{t}\,} \,d{t_2}\, \sigma^2 \delta(t_1-t_2)$$ (6.7)

$$= \sigma^2 \,d{t}\, \quad.$$ (6.8)

As long as the intervals $[t,t+\,d{t}\,]$ and $[t',t'+\,d{t}\,]$ do not overlap, which is true for successive time steps, we get

$$\langle {\,d{W(t)}\, \,d{W(t')}\,} \rangle = 0 \quad.$$

It should be emphasized, that only the second moment of $\,d{W(t)}\,$ is linear in $\,d{t}\,$. $\,d{W(t)}\,$ is only of the order of $\sqrt{dt}$. This important aspect is made clear by writing

$$dW(t)=\sigma \eta(t) \sqrt{dt}$$

where $\eta(t)$ denotes a Gaussian random variable.