STOCHASTIC CALCULUS 11

From Jump Processes to Diffusion Approximations

A diffusion approximation replaces many small random jumps by a continuous stochastic process with the same local mean and variance.

The main idea

Some stochastic systems change through discrete events.

For example, in an epidemic,

\[ I\rightarrow I+1 \]

may represent one new infection, while

\[ I\rightarrow I-1 \]

may represent one recovery.

A continuous-time Markov chain represents these individual jumps directly.

When many such events occur, the jump process can sometimes be approximated by an SDE.

\[ \boxed{ \text{discrete random jumps} \longrightarrow \text{continuous random diffusion} } \]

1. A simple birth-death process

Suppose a population \(X(t)\) can increase by one at rate

\[ b(X) \]

and decrease by one at rate

\[ d(X). \]

The possible transitions are

\[ X\rightarrow X+1 \]

and

\[ X\rightarrow X-1. \]

2. Probabilities over a short interval

For a sufficiently small time interval \(\Delta t\),

\[ P(X\rightarrow X+1) \approx b(X)\Delta t, \] \[ P(X\rightarrow X-1) \approx d(X)\Delta t. \]

The probability of no event is approximately

\[ 1-[b(X)+d(X)]\Delta t. \]

3. Mean change over a short interval

The change \(\Delta X\) can approximately be

\[ +1, \qquad -1, \qquad 0. \]

Therefore,

\[ E[\Delta X] \approx (+1)b(X)\Delta t + (-1)d(X)\Delta t. \]

\[ \boxed{ E[\Delta X] \approx [b(X)-d(X)]\Delta t } \]

4. The drift appears naturally

The quantity

\[ b(X)-d(X) \]

is the net expected rate of change.

Therefore the deterministic drift is

\[ \boxed{ a(X) = b(X)-d(X) } \]

5. Variance of the short-time change

Now consider

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

Since

\[ (+1)^2=1 \]

and

\[ (-1)^2=1, \]

we have

\[ E[(\Delta X)^2] \approx b(X)\Delta t + d(X)\Delta t. \]

To first order in \(\Delta t\),

\[ \boxed{ \operatorname{Var}(\Delta X) \approx [b(X)+d(X)]\Delta t } \]

6. The diffusion coefficient

For an SDE,

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

the variance of the random part over a small interval is approximately

\[ \sigma(X)^2\Delta t. \]

Matching this with the jump-process variance gives

\[ \sigma(X)^2 = b(X)+d(X). \]

Therefore,

\[ \boxed{ \sigma(X) = \sqrt{b(X)+d(X)} } \]

7. Diffusion approximation

The birth-death process can therefore be approximated by

\[ \boxed{ dX_t = [b(X_t)-d(X_t)]\,dt + \sqrt{b(X_t)+d(X_t)}\,dW_t } \]

The drift comes from the difference of the event rates.

The diffusion comes from their combined random variation.

8. Why the rates are added inside the variance

Births and deaths affect the mean in opposite directions:

\[ +1 \quad\text{and}\quad -1. \]

But both contribute positively to variability because

\[ (+1)^2=(-1)^2=1. \]

Therefore,

\[ \boxed{ \text{drift} = \text{difference of rates} } \] \[ \boxed{ \text{variance rate} = \text{sum of rates} } \]

9. Epidemic SIS example

Suppose \(I(t)\) is the number of infectious individuals in an SIS epidemic.

An infection increases \(I\) by one:

\[ I\rightarrow I+1. \]

A recovery decreases \(I\) by one:

\[ I\rightarrow I-1. \]

Let the infection rate be

\[ b(I) = \beta\frac{I(N-I)}{N}, \]

and recovery rate be

\[ d(I)=\gamma I. \]

10. Deterministic SIS drift

The net rate is

\[ b(I)-d(I). \]

Therefore,

\[ \boxed{ a(I) = \beta\frac{I(N-I)}{N} - \gamma I } \]

which is exactly the usual deterministic SIS rate equation.

11. Stochastic SIS diffusion

The local variance rate is

\[ b(I)+d(I). \]

Therefore the diffusion approximation becomes

\[ \boxed{ dI_t = \left[ \beta\frac{I_t(N-I_t)}{N} - \gamma I_t \right]dt + \sqrt{ \beta\frac{I_t(N-I_t)}{N} + \gamma I_t }\,dW_t } \]

12. Why this is better than choosing arbitrary noise

The diffusion coefficient has been derived from the actual random event structure.

So the noise strength changes automatically with the infection and recovery rates.

\[ \boxed{ \text{event mechanism} \rightarrow \text{drift and diffusion} } \]

13. Mean and variance matching

The diffusion approximation is constructed so that over a short interval,

\[ E[\Delta X] \]

and

\[ \operatorname{Var}(\Delta X) \]

match those of the underlying jump process to leading order.

This is the key idea behind the approximation.

14. A more general event-based system

Suppose a system has events indexed by \(j\).

Each event changes the state by a vector

\[ \boldsymbol{\nu}_j \]

and occurs at rate

\[ \lambda_j(\mathbf{X}). \]

The mean rate of change is

\[ \boxed{ \mathbf{a}(\mathbf{X}) = \sum_j \boldsymbol{\nu}_j \lambda_j(\mathbf{X}) } \]

15. Covariance structure

The local covariance matrix is

\[ \boxed{ D(\mathbf{X}) = \sum_j \boldsymbol{\nu}_j \boldsymbol{\nu}_j^{T} \lambda_j(\mathbf{X}) } \]

This describes how the random event structure generates variability and covariance between state variables.

16. Multidimensional diffusion approximation

Choose a matrix \(B(\mathbf{X})\) satisfying

\[ BB^T=D. \]

Then the diffusion approximation has the form

\[ \boxed{ d\mathbf{X}_t = \mathbf{a}(\mathbf{X}_t)\,dt + B(\mathbf{X}_t)\,d\mathbf{W}_t } \]

17. SIR event structure

In an SIR epidemic, infection changes the state by

\[ \begin{pmatrix} -1\\ +1\\ 0 \end{pmatrix}, \]

while recovery changes it by

\[ \begin{pmatrix} 0\\ -1\\ +1 \end{pmatrix}. \]

The corresponding event rates may be

\[ \lambda_1 = \beta\frac{SI}{N} \]

and

\[ \lambda_2 = \gamma I. \]

18. SIR drift from event vectors

The drift is

\[ \boldsymbol{\nu}_1\lambda_1 + \boldsymbol{\nu}_2\lambda_2. \]

Therefore,

\[ \boxed{ \frac{dS}{dt} = -\beta\frac{SI}{N} } \] \[ \boxed{ \frac{dI}{dt} = \beta\frac{SI}{N} - \gamma I } \] \[ \boxed{ \frac{dR}{dt} = \gamma I } \]

So the deterministic compartment equations arise directly from expected event changes.

19. Random changes are correlated

In an infection event, \(S\) decreases at exactly the same time that \(I\) increases.

Therefore stochastic changes in compartments are not generally independent.

This dependence appears in the covariance matrix of the diffusion approximation.

20. Why one Brownian motion may be insufficient

A model with several independent event mechanisms may require several Brownian drivers.

For example, infection and recovery can be represented by separate Brownian components:

\[ dW_1, \qquad dW_2. \]

This preserves the structure of the different random event channels.

21. Diffusion approximation diagram

22. When is the approximation reasonable?

Diffusion approximations are most natural when event counts are sufficiently large and individual jumps are small relative to the scale of the system.

Then many discrete events combine into approximately continuous random fluctuations.

23. When can it be poor?

The approximation may be less reliable when population numbers are very small.

For example, near

\[ I=0, \]

the difference between zero infections and one infection is important.

A continuous diffusion can also produce non-integer or even negative numerical states unless handled carefully.

24. Extinction is especially sensitive

In a jump process,

\[ I=0 \]

can be an exact absorbing state.

A simple diffusion approximation may not preserve this boundary correctly without additional treatment.

Therefore jump-process models can be preferable when extinction probabilities are central.

25. CTMC and SDE answer related questions

A CTMC represents discrete events exactly at the model level.

An SDE provides a continuous approximation that may be easier to analyse or simulate for large systems.

\[ \boxed{ \text{CTMC} = \text{event-level stochastic model} } \] \[ \boxed{ \text{SDE} = \text{continuous diffusion approximation} } \]

26. Connection with the deterministic model

The three descriptions can be viewed as related:

\[ \boxed{ \text{jump process} } \] \[ \downarrow \] \[ \boxed{ \text{mean event rates} \rightarrow \text{deterministic ODE} } \] \[ \boxed{ \text{mean + event variance} \rightarrow \text{SDE} } \]

27. Why this connection matters

It explains where the deterministic drift comes from and why the stochastic diffusion coefficient should reflect the underlying event rates.

The deterministic and stochastic models are therefore connected through the same event mechanism.

28. General modelling workflow

\[ \boxed{ 1.\; \text{identify all event types} } \] \[ \boxed{ 2.\; \text{write each state-change vector} } \] \[ \boxed{ 3.\; \text{write each event rate} } \] \[ \boxed{ 4.\; \text{sum expected event changes to obtain drift} } \] \[ \boxed{ 5.\; \text{sum event covariance contributions} } \] \[ \boxed{ 6.\; \text{construct the diffusion matrix} } \]

29. The central idea

\[ \boxed{ \text{mean event effect} \rightarrow \text{drift} } \] \[ \boxed{ \text{event variability} \rightarrow \text{diffusion} } \]

A diffusion approximation is therefore not just a deterministic equation with arbitrary noise added.

It can be derived systematically from the local mean and variance of an underlying random event process.