STOCHASTIC CALCULUS 07

Moments of Stochastic Processes

Moments summarise the distribution of a stochastic process through quantities such as its mean, second moment and variance.

The main idea

For a stochastic process \(X_t\), the value at a fixed time \(t\) is a random variable.

Different realisations may therefore give different values:

\[ X_t^{(1)},\quad X_t^{(2)},\quad X_t^{(3)},\quad \ldots \]

Moments describe important features of the distribution of these possible values.

\[ \boxed{ \text{distribution of }X_t \longrightarrow \text{moments} } \]

1. The first moment

The first moment is the expected value:

\[ \boxed{ m_1(t)=E[X_t] } \]

It gives the average value of \(X_t\) across many hypothetical realisations.

2. Expectation is not one trajectory

A single stochastic trajectory gives one possible value of \(X_t\).

The expectation instead averages over all possible outcomes according to their probabilities.

\[ \boxed{ X_t = \text{random outcome} } \] \[ \boxed{ E[X_t] = \text{average over possible outcomes} } \]

3. Monte Carlo estimate of the mean

Suppose \(M\) independent trajectories are simulated.

At time \(t\), we obtain

\[ X_t^{(1)}, \ldots, X_t^{(M)}. \]

Then

\[ \boxed{ E[X_t] \approx \frac{1}{M} \sum_{j=1}^{M} X_t^{(j)} } \]

This is the sample mean.

4. The second moment

The second moment is

\[ \boxed{ m_2(t)=E[X_t^2] } \]

It is not the same as

\[ (E[X_t])^2. \]

The distinction is fundamental.

5. Why the second moment matters

The second moment contains information about both the location and spread of the distribution.

It is used to calculate the variance:

\[ \boxed{ \operatorname{Var}(X_t) = E[X_t^2] - (E[X_t])^2 } \]

6. Variance

Variance measures the spread of possible values around the mean.

Equivalent notation is

\[ \operatorname{Var}(X_t) = E\left[ \left( X_t-E[X_t] \right)^2 \right]. \]

A small variance means trajectories tend to remain close to the mean.

A large variance means outcomes are more widely spread.

7. Standard deviation

The standard deviation is

\[ \boxed{ \operatorname{SD}(X_t) = \sqrt{\operatorname{Var}(X_t)} } \]

It has the same units as \(X_t\), which often makes it easier to interpret than variance.

8. A simple discrete example

Suppose \(X\) takes the values

\[ 1,\quad 2,\quad 3 \]

with equal probabilities.

Then

\[ E[X] = \frac{1+2+3}{3} = 2. \]

The second moment is

\[ E[X^2] = \frac{1^2+2^2+3^2}{3} = \frac{14}{3}. \]

Therefore,

\[ \operatorname{Var}(X) = \frac{14}{3}-4 = \frac{2}{3}. \]

9. Why \(E[X^2]\neq(E[X])^2\)

In the previous example,

\[ E[X^2] = \frac{14}{3}, \]

while

\[ (E[X])^2=4. \]

They differ because expectation is taken after squaring in the first expression.

\[ \boxed{ E[X^2] \neq (E[X])^2 \text{ in general} } \]

10. Higher moments

More generally, the \(n\)-th raw moment is

\[ \boxed{ m_n(t)=E[X_t^n] } \]

Higher moments can describe additional features of a distribution.

For example, third and fourth moments are related to skewness and kurtosis.

11. Moments change with time

For a stochastic process, moments usually depend on time.

Therefore we may study

\[ E[X_t], \] \[ E[X_t^2], \]

and

\[ \operatorname{Var}(X_t) \]

as functions of \(t\).

12. Example: Brownian motion

For Brownian motion,

\[ W_t\sim N(0,t). \]

Therefore,

\[ \boxed{ E[W_t]=0 } \]

and

\[ \boxed{ E[W_t^2]=t } \]

\[ \boxed{ \operatorname{Var}(W_t)=t. } \]

13. Example: drifted Brownian motion

Consider

\[ dX_t = \mu\,dt + \sigma\,dW_t, \qquad X_0=x_0. \]

Its solution is

\[ X_t = x_0+\mu t+\sigma W_t. \]

Taking expectation,

\[ E[X_t] = x_0+\mu t. \]

Therefore,

\[ \boxed{ E[X_t] = x_0+\mu t } \]

14. Variance of drifted Brownian motion

The deterministic terms

\[ x_0+\mu t \]

do not contribute to variance.

Therefore,

\[ \operatorname{Var}(X_t) = \sigma^2\operatorname{Var}(W_t). \]

Since

\[ \operatorname{Var}(W_t)=t, \]

we obtain

\[ \boxed{ \operatorname{Var}(X_t) = \sigma^2t } \]

15. Mean and spread can behave differently

For the previous model,

\[ E[X_t]=x_0+\mu t \]

describes systematic movement.

Meanwhile,

\[ \operatorname{SD}(X_t)=\sigma\sqrt{t} \]

describes increasing uncertainty.

\[ \boxed{ \text{mean} \rightarrow \text{central behaviour} } \] \[ \boxed{ \text{variance} \rightarrow \text{uncertainty around it} } \]

16. Diagram: mean and variability

17. Deriving moment equations from an SDE

Suppose

\[ dX_t = a(X_t)\,dt + b(X_t)\,dW_t. \]

Taking expectation formally gives

\[ E[dX_t] = E[a(X_t)]dt + E[b(X_t)dW_t]. \]

Under suitable conditions,

\[ E[b(X_t)dW_t]=0. \]

Therefore,

\[ \boxed{ \frac{d}{dt}E[X_t] = E[a(X_t)] } \]

18. Why expectation may not close

Suppose the drift is nonlinear:

\[ a(X)=X^2. \]

Then

\[ \frac{d}{dt}E[X_t] = E[X_t^2]. \]

The equation for the first moment depends on the second moment.

So knowing only

\[ E[X_t] \]

is not enough.

19. Deriving the second-moment equation

To find an equation for

\[ E[X_t^2], \]

apply Itô's lemma to

\[ f(X)=X^2. \]

Since

\[ f'(X)=2X \]

and

\[ f''(X)=2, \]

we obtain

\[ d(X_t^2) = \left[ 2X_ta(X_t) + b(X_t)^2 \right]dt + 2X_tb(X_t)\,dW_t. \]

Taking expectations gives

\[ \boxed{ \frac{d}{dt}E[X_t^2] = E\left[ 2X_ta(X_t)+b(X_t)^2 \right] } \]

20. Moment hierarchy

For nonlinear models, the second-moment equation may depend on the third moment.

The third-moment equation may depend on the fourth moment, and so on.

\[ \boxed{ m_1 \rightarrow m_2 \rightarrow m_3 \rightarrow \cdots } \]

This is called a moment hierarchy.

21. Example of a hierarchy

Suppose

\[ dX_t = X_t^2dt + \sigma dW_t. \]

Then

\[ \frac{d}{dt}E[X_t] = E[X_t^2]. \]

For the second moment,

\[ \frac{d}{dt}E[X_t^2] = 2E[X_t^3]+\sigma^2. \]

So the first two equations are not closed.

22. Moment closure

To obtain a finite system, higher moments may sometimes be approximated using lower moments.

This is called moment closure.

\[ \boxed{ \text{higher moment} \approx \text{function of lower moments} } \]

Different closure assumptions lead to different approximations.

23. A simple closure idea

One crude approximation is

\[ E[X^2] \approx (E[X])^2. \]

But this ignores variance because

\[ E[X^2] = \operatorname{Var}(X) + (E[X])^2. \]

So such an approximation is reasonable only when variability is sufficiently small.

24. Covariance

For two random variables \(X_t\) and \(Y_t\), covariance is

\[ \boxed{ \operatorname{Cov}(X_t,Y_t) = E[X_tY_t] - E[X_t]E[Y_t] } \]

It measures how the two quantities vary together.

25. Why covariance appears in multivariable models

Suppose a model contains a term

\[ X_tY_t. \]

Then expectation gives

\[ E[X_tY_t]. \]

This can be written as

\[ E[X_tY_t] = \operatorname{Cov}(X_t,Y_t) + E[X_t]E[Y_t]. \]

So interactions naturally introduce covariance into moment equations.

26. Epidemic example

Suppose a stochastic epidemic model contains the infection term

\[ \beta\frac{S_tI_t}{N}. \]

Taking expectations gives

\[ \frac{\beta}{N} E[S_tI_t]. \]

In general,

\[ E[S_tI_t] \neq E[S_t]E[I_t]. \]

Instead,

\[ \boxed{ E[S_tI_t] = \operatorname{Cov}(S_t,I_t) + E[S_t]E[I_t] } \]

27. Why this matters

A deterministic epidemic model uses

\[ SI. \]

A mean equation for a stochastic model naturally involves

\[ E[SI]. \]

These are not generally the same as multiplying the means:

\[ \boxed{ E[SI] \neq E[S]E[I] } \]

unless covariance is zero.

28. Moments versus full distributions

Moments do not usually describe the complete distribution.

Two different distributions can sometimes have the same mean and variance.

But moments can still provide compact and useful summaries when solving for the full probability distribution is difficult.

29. Moments and simulation

There are two main ways to obtain moment information.

One is to simulate many trajectories and estimate the moments numerically.

The other is to derive differential equations for the moments directly.

\[ \boxed{ \text{Monte Carlo trajectories} \rightarrow \text{estimated moments} } \] \[ \boxed{ \text{SDE + Itô calculus} \rightarrow \text{moment equations} } \]

30. Why moment equations can be useful

If the moment equations close or can be approximated accurately, they may be much cheaper to solve than simulating thousands of stochastic trajectories.

They can directly provide quantities such as

\[ E[X_t] \]

and

\[ \operatorname{Var}(X_t). \]

31. The structure to remember

\[ \boxed{ m_1(t)=E[X_t] } \] \[ \boxed{ m_2(t)=E[X_t^2] } \] \[ \boxed{ \operatorname{Var}(X_t) = m_2(t)-m_1(t)^2 } \] \[ \boxed{ \operatorname{Cov}(X_t,Y_t) = E[X_tY_t]-E[X_t]E[Y_t] } \]

32. The central idea

A stochastic model produces a distribution of possible outcomes rather than one deterministic trajectory.

Moments summarise that distribution.

\[ \boxed{ \text{mean} = \text{central behaviour} } \] \[ \boxed{ \text{variance} = \text{spread} } \] \[ \boxed{ \text{covariance} = \text{joint variation} } \]

Moment equations provide a bridge between stochastic differential equations and practical quantities used to describe uncertainty.