Moment equations
A stochastic differential equation describes individual random trajectories. Moment equations ask a different question: how do summary quantities of the entire distribution—such as its mean, variance and higher moments—change through time?
What is a moment?
For a random variable \(X_t\), the raw moments are
\[m_k(t)=E[X_t^k].\]| Moment | Expression | What it helps describe |
|---|---|---|
| first | \(E[X_t]\) | mean or centre |
| second | \(E[X_t^2]\) | used to obtain variance |
| third | \(E[X_t^3]\) | related to asymmetry |
| fourth | \(E[X_t^4]\) | related to tail shape and peakedness |
The variance is not itself a raw moment, but is obtained from the first two:
\[\boxed{\operatorname{Var}(X_t)=E[X_t^2]-E[X_t]^2}.\]Start with a general Itô SDE
Consider
\[\boxed{dX_t=a(X_t,t)dt+b(X_t,t)dW_t}.\]Here \(a\) is the drift and \(b\) is the diffusion coefficient.
First moment: derive the mean equation
Write the SDE in integral form:
\[X_t=X_0+\int_0^t a(X_s,s)ds+\int_0^t b(X_s,s)dW_s.\]Take expectations:
\[E[X_t]=E[X_0]+E\!\left[\int_0^t a(X_s,s)ds\right]+E\!\left[\int_0^t b(X_s,s)dW_s\right].\]Under the usual integrability and adaptedness conditions, the Itô integral has expectation zero:
\[E\!\left[\int_0^t b(X_s,s)dW_s\right]=0.\]Therefore
\[E[X_t]=E[X_0]+\int_0^t E[a(X_s,s)]ds,\]and differentiating gives
\[\boxed{\frac{d}{dt}E[X_t]=E[a(X_t,t)]}.\]A simple linear example
Suppose
\[dX_t=rX_tdt+\sigma X_tdW_t.\]Since \(a(X)=rX\),
\[\frac{d}{dt}E[X_t]=rE[X_t].\]If \(X_0=x_0\),
\[\boxed{E[X_t]=x_0e^{rt}}.\]In this case the first-moment equation closes by itself because the drift is linear.
Second moment: why Itô's lemma is needed
To obtain \(E[X_t^2]\), apply Itô's lemma to
\[f(X)=X^2.\]Its derivatives are
\[f'(X)=2X,\qquad f''(X)=2.\]Itô's lemma gives
\[d(X_t^2)=2X_t\,dX_t+(dX_t)^2.\]Substituting the SDE and using \((dW_t)^2=dt\) gives
\[d(X_t^2)=\left[2X_ta(X_t,t)+b(X_t,t)^2\right]dt+2X_tb(X_t,t)dW_t.\]Taking expectations removes the Itô-integral term:
\[\boxed{\frac{d}{dt}E[X_t^2]=E\!\left[2X_ta(X_t,t)+b(X_t,t)^2\right]}.\]Why the diffusion appears in the second moment
The diffusion term does not appear explicitly in the general first-moment equation, but \(b^2\) appears directly in the second-moment equation.
This happens because Brownian quadratic variation produces the Itô correction.
Complete second-moment calculation for multiplicative noise
Return to
\[dX_t=rX_tdt+\sigma X_tdW_t.\]Here
\[a(X)=rX,\qquad b(X)=\sigma X.\]Therefore
\[2Xa(X)+b(X)^2=2rX^2+\sigma^2X^2.\]Hence
\[\frac{d}{dt}E[X_t^2]=(2r+\sigma^2)E[X_t^2].\]So
\[\boxed{E[X_t^2]=x_0^2e^{(2r+\sigma^2)t}}.\]Obtain the variance
Using
\[\operatorname{Var}(X_t)=E[X_t^2]-E[X_t]^2,\]we obtain
\[\boxed{\operatorname{Var}(X_t)=x_0^2e^{2rt}\left(e^{\sigma^2t}-1\right)}.\]This shows explicitly that increasing \(\sigma\) increases the spread of possible trajectories.
Mean and variance can behave very differently
The general equation for the \(k\)-th moment
Choose
\[f(X)=X^k.\]Then
\[f'(X)=kX^{k-1},\qquad f''(X)=k(k-1)X^{k-2}.\]Itô's lemma gives
\[d(X_t^k)=\left[kX_t^{k-1}a(X_t,t)+\frac12k(k-1)X_t^{k-2}b(X_t,t)^2\right]dt+kX_t^{k-1}b(X_t,t)dW_t.\]Taking expectations gives
\[\boxed{\frac{d}{dt}E[X_t^k]=E\!\left[kX_t^{k-1}a(X_t,t)+\frac12k(k-1)X_t^{k-2}b(X_t,t)^2\right]}.\]When do moment equations close?
A set of moment equations is closed when the equations for the moments we want involve only those same moments.
For the multiplicative linear SDE above, the equations for \(E[X]\) and \(E[X^2]\) involve only \(E[X]\) and \(E[X^2]\). They can therefore be solved directly.
What goes wrong with nonlinear models?
Consider stochastic logistic growth:
\[dX_t=rX_t\left(1-\frac{X_t}{K}\right)dt+\sigma X_tdW_t.\]The mean equation is
\[\frac{d}{dt}E[X_t]=rE[X_t]-\frac{r}{K}E[X_t^2].\]So the first moment already depends on the second moment.
For the second moment,
\[\frac{d}{dt}E[X_t^2]=(2r+\sigma^2)E[X_t^2]-\frac{2r}{K}E[X_t^3].\]Now the second moment depends on the third.
The moment hierarchy
This continuing dependence on higher moments is called a moment hierarchy. In many nonlinear stochastic biological models, it does not terminate naturally.
Moment closure
A moment-closure approximation replaces a higher-order moment by an expression involving lower-order moments.
For example, a very simple approximation might use
\[E[X^2]\approx E[X]^2,\]which corresponds to neglecting variance. More informative closures retain variance and make assumptions about the shape of the distribution.
Different closure assumptions can produce different approximations, so the choice should be justified rather than applied automatically.
Why not just simulate?
Monte Carlo simulation and moment equations answer related questions in different ways.
| Moment equations | Monte Carlo simulation |
|---|---|
| directly describe selected distribution summaries | generate individual random trajectories |
| can be computationally efficient when closed | flexible even when analytic equations are difficult |
| may require closure approximations | requires many simulations for accurate statistics |
| can reveal mathematical structure | can estimate complex probabilities and distributions |
Biological interpretation
If \(X_t\) is a population size, the first moment tells us the expected population and the second moment helps determine its uncertainty. If \(X_t=I_t\) is an infectious population, moment equations can describe the expected infectious burden and its variability across possible epidemics.
However, moments do not always capture the entire distribution. Two distributions can share the same mean and variance while having different tail probabilities or extinction behaviour.
For several biological variables
With compartments such as \(S_t,I_t,R_t\), we also need mixed moments such as
\[E[S_tI_t].\]These describe dependence between variables and naturally appear when nonlinear interaction terms such as \(S_tI_t\) occur in the model.
Covariance is
\[\operatorname{Cov}(S_t,I_t)=E[S_tI_t]-E[S_t]E[I_t].\]Thus multi-compartment moment systems contain means, variances and covariances.