6.3 Interpretation of stochastic integrals

Let us assume a one dimensional stochastic differential equation with multiplicative noise [41]

$$\frac{d X(t)}{dt}=a(X(t),t)+b(X(t),t) \, \eta(t) \quad.$$ (6.9)

The increment $\,d{X}\,$ during a short time interval $\,d{t}\,$ is given by

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

The second term, which is a stochastic integral, has to be investigated in more detail. We can evaluate the integrand at the beginning of the interval $[t,t+\,d{t}\,]$, multiply it by the length of the interval and use the result as the increment for small $\,d{t}\,$. Thus, we obtain

$$\,d{X(t)}\,=a\Bigl(X(t),t\Bigr) \,d{t}\, + b\Bigl(X(t),t\Bigr)\eta(t) \sqrt{dt} \quad,$$

where $\eta(t)$ is a standard Gaussian random variable at each discrete time step with

$$\langle {\eta(t)\eta(t')} \rangle =\delta(t,t') \quad.$$

For $b(X(t),t)=\sigma=\mathrm{const.}$, that is the case for additive noise, it is equivalent to equation ().

However, we could also evaluate the integrand $b$ at any other time $t'$ in the interval $[t,t+\,d{t}\,]$ and at

$$\bar X(t) = (1-\alpha)X(t)+\alpha X(t+\,d{t}\,)=$$

$$= (1-\alpha)X(t)+\alpha \Bigl(X(t)+ \,d{X(t)}\,\Bigr)=$$

$$= X(t)+\alpha \,d{X(t)}\,$$ (6.10)

In this general case we get for the increment $\,d{X(t)}\,$ an implicit expression

$$\,d{X(t)}\,=a\Bigl(\bar X(t),t'\Bigr) \,d{t}\, + b\Bigl(X(t)+\alpha \,d{X(t)}\,,t'\Bigr)\eta(t) \sqrt{dt} \quad.$$

With the abbreviation $b'=\partial b(X,t)/\partial X$ we get

$$b\Bigl(X(t)+\alpha \,d{X(t)}\,,t'\Bigr)\eta(t) \sqrt{dt} = b\Bigl(X(t),t\Bigr)\eta(t) \sqrt{dt} +$$

$$\alpha b'\Bigl(X(t),t'\Bigr) \,d{X(t)}\,\eta(t) \sqrt{dt}+ \cdots$$

$$= b\Bigl(X(t),t\Bigr)\eta(t) \sqrt{dt} +$$

$$\alpha b'\Bigl(X(t),t'\Bigr) b\Bigl(X(t),t'\Bigr) \eta^2(t) \sqrt{dt}+$$

$$O(\,d{t^{3/2}}\,)$$ (6.11)

Finally, we get for the increment $\,d{X(t)}\,$

$$\,d{X(t)}\, = \left[ a\Bigl(X(t),t\Bigr) + \alpha b'\Bigl(X(t),t\Bigr)b\Bigl(X(t),t\Bigr) \eta^2(t) \right] \,d{t}\, +$$

$$b\Bigl(X(t),t\Bigr)\eta(t) \sqrt{dt} \,.$$ (6.12)

In this equation we find an additional drift term, which contains $\alpha$ and $\eta^2(t)$. The latter can be replaced by 1 for terms up to the order of $\,d{t}\,$. Depending on the choice of $\alpha$ and the interpretation of the integral, we get different drift terms.

If we set $\alpha=0$, we get

$$\,d{X(t)}\, = a(X(t),t) \,d{t}\, + b(X(t),t)\eta(t) \sqrt{dt}$$ (6.13)

and we call it the Itô interpretation of the stochastic differential equation

$$\dot X(t)=a(X(t),t)+b(X(t),t)\eta(t) \quad.$$ (6.14)

For $\alpha=1/2$, we get

$$\,d{X(t)}\, = \left[ a(X(t),t) + \frac{1}{2} b'(X(t),t)b(X(t),t) \right] \,d{t}\, +$$

$$b(X(t),t)\eta(t) \sqrt{dt}$$ (6.15)

and we call it the Stratonovich interpretation, which is indicated by writing

$$\dot X(t)=a(X(t),t)+b(X(t),t) \circ \eta(t) \quad.$$ (6.16)

Thus, we have to distinguish between the interpretation of a stochastic differential equation and the version, in which it is written. The stochastic differential equation () can be written in an Itô version using () as

$$\dot X(t)=a(X(t),t)+\frac{1}{2}b(X(t),t)b'(X(t),t)+b(X(t),t) \eta(t)$$ (6.17)

where we find the noise induced drift term

$$\frac{1}{2}b(X(t),t)b'(X(t),t) \quad.$$ (6.18)

Reversely, () can be written in a Stratonovich version as

$$\dot X(t) = a(X(t),t)- \frac{1}{2}b(X(t),t)b'(X(t),t)+b(X(t),t)\circ\eta(t)$$

$$= \bar a(X(t),t)+b(X(t),t)\circ\eta(t) \quad.$$ (6.19)

Due to the different drift terms, the two interpretations yield different dynamical properties [41]. Itô calculus is commonly chosen on certain mathematical grounds, since rather general results of probability theory can then be employed. On the other hand, white noise is usually an idealization of physical (coloured) noise with short autocorrelation time, in which case the two time covariance function is given by

$$\langle {\eta(t)\eta(t+\tau)} \rangle = \frac{\sigma^2}{2m} \mathrm{e}^{-m\vert\tau\vert}$$

with a short time constant $m^{-1}$.

The Wong-Zakai-Theorem [42] then says, that in the formal zero-correlation-time limit

$$\sigma \to \sigma m \quad, \quad m \to \infty$$

the coloured noise becomes white noise and we obtain the Stratonovich-Interpretation for the stochastic differential equation. The results coincide with those obtained in the limit of fluctuations with finite autocorrelation time. Therefore, Stratonovich calculus is usually preferred in physical applications.