STOCHASTIC CALCULUS 09

Stochastic Modelling and Interpretation

A stochastic model describes both systematic change and uncertainty, allowing us to study distributions, probabilities and the range of possible outcomes.

The main idea

A deterministic model produces one trajectory when the initial conditions and parameters are fixed.

A stochastic model instead produces a distribution of possible trajectories.

\[ \boxed{ \text{deterministic model} \rightarrow \text{one trajectory} } \] \[ \boxed{ \text{stochastic model} \rightarrow \text{many possible trajectories} } \]

1. Why introduce randomness?

Randomness should not be added simply because a real system looks irregular.

A stochastic term should represent a genuine source of uncertainty or random variation in the mechanism being modelled.

Examples include random transmission events, environmental fluctuations, demographic variation and measurement uncertainty.

2. Mechanism comes first

Suppose a deterministic model is

\[ \frac{dX}{dt}=a(X,t). \]

A stochastic version may be written as

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

The drift should retain the underlying mechanism of the model.

The diffusion term should represent the source and scale of random variation.

3. The diffusion coefficient is not arbitrary

The coefficient

\[ b(X_t,t) \]

determines how strongly randomness affects the state.

Its form should therefore be justified by the modelling assumptions.

\[ \boxed{ \text{diffusion coefficient} = \text{mathematical representation of uncertainty strength} } \]

4. Additive and multiplicative noise

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

the noise strength is independent of the state.

This is additive noise.

If instead

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

the fluctuation size changes with the state.

This is multiplicative noise.

5. Interpretation of a sample path

A sample path is one possible realisation of the stochastic model.

It should not automatically be interpreted as the prediction of the model.

\[ \boxed{ \text{one path} = \text{one possible outcome} } \]

The full stochastic prediction is represented by the distribution of possible outcomes.

6. Why many trajectories are useful

Simulating many independent paths allows us to estimate:

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

quantiles, threshold probabilities and distributions of important outcomes.

This gives more information than a single trajectory.

7. Mean trajectory

The mean

\[ E[X_t] \]

describes the average across possible realisations at time \(t\).

It is useful, but it does not describe the full uncertainty.

Two models can have the same mean while having very different variances and tail probabilities.

8. Variance and uncertainty

The variance

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

describes how widely the possible values are spread.

\[ \boxed{ \text{mean} = \text{central behaviour} } \] \[ \boxed{ \text{variance} = \text{uncertainty around that behaviour} } \]

9. Quantiles

A quantile gives a value below which a specified proportion of the distribution lies.

For example, the \(95\%\) quantile \(q_{0.95}\) satisfies approximately

\[ P(X_t\le q_{0.95})=0.95. \]

Quantiles are useful when the full distribution is asymmetric or when extreme outcomes matter.

10. Threshold probabilities

Suppose \(L\) is an important threshold.

A stochastic model allows us to investigate

\[ \boxed{ P(X_t>L) } \]

rather than only asking whether the mean is above or below \(L\).

11. Mean below a threshold does not imply safety

Suppose

\[ E[X_t] There may still be a substantial probability that

\[ X_t>L. \]

Therefore,

\[ \boxed{ E[X_t]L)=0 } \]

This distinction is important in risk-based decision making.

12. Diagram: same mean, different uncertainty

13. Deterministic and stochastic models answer different questions

A deterministic model is often useful for questions such as

\[ \boxed{ \text{What is the expected or typical direction of the system?} } \]

A stochastic model can additionally address

\[ \boxed{ \text{How uncertain is the outcome?} } \] \[ \boxed{ \text{What is the probability of an extreme event?} } \] \[ \boxed{ \text{How variable is the timing of an event?} } \]

14. Stochastic does not automatically mean better

A stochastic model is not automatically superior to a deterministic model.

The appropriate model depends on the question, scale, available data and importance of uncertainty.

If only the average trend is required, a deterministic model may be sufficient.

If probabilities and variability matter, stochastic modelling may provide essential additional information.

15. Randomness at different scales

Different sources of randomness can matter at different scales.

In a very large population, some individual-level fluctuations may average out.

In a small population, individual random events may strongly affect the outcome.

The model should reflect the scale of the system.

16. Intrinsic uncertainty

Intrinsic uncertainty arises from randomness inside the process itself.

For example, infection and recovery may occur as random individual events.

Even if all parameters were known exactly, different epidemic realisations could still occur.

17. Parameter uncertainty

A different source of uncertainty comes from not knowing parameter values exactly.

For example,

\[ \beta \]

\[ \gamma \]

may be estimated from data and therefore uncertain.

This is different from stochastic process noise.

18. Observation uncertainty

Observed data can also contain measurement error, reporting delays or incomplete detection.

This is another distinct source of uncertainty.

\[ \boxed{ \text{process uncertainty} \neq \text{parameter uncertainty} \neq \text{observation uncertainty} } \]

19. Why the distinction matters

Different uncertainties should generally be represented differently.

A diffusion term in an SDE represents process-level random fluctuation.

It should not automatically be used as a substitute for uncertain parameters or measurement error.

20. Epidemic modelling example

A deterministic SIR model is

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

This describes one deterministic epidemic trajectory for fixed parameters and initial conditions.

21. A stochastic epidemic model

A diffusion approximation may be written schematically as

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

where

\[ \mathbf{X}_t = \begin{pmatrix} S_t\\ I_t\\ R_t \end{pmatrix}. \]

The drift describes the epidemic flows, while the diffusion matrix describes stochastic variation generated by those flows.

22. Peak size becomes random

In a deterministic epidemic model, the infectious peak

\[ I_{\max} = \max_t I(t) \]

is one fixed value.

In a stochastic model,

\[ I_{\max} \]

is itself a random variable.

\[ \boxed{ I_{\max}^{(1)}, \; I_{\max}^{(2)}, \; \ldots } \]

can differ across epidemic realisations.

23. Peak time is also random

Define

\[ T_{\max} = \operatorname*{arg\,max}_t I_t. \]

In a stochastic model, the time of the epidemic peak may vary between trajectories.

So we may study both

\[ E[I_{\max}] \]

and

\[ E[T_{\max}], \]

together with their distributions.

24. Capacity exceedance

Suppose \(C\) is a hospital-capacity threshold.

A deterministic model might compare

\[ I_{\max} \]

directly with \(C\).

A stochastic model can instead estimate

\[ \boxed{ P(I_{\max}>C) } \]

which may be more useful when decisions depend on risk.

25. Extinction probability

Another question that may be invisible in a deterministic model is whether an outbreak dies out early by chance.

For a stochastic epidemic process, we may study

\[ \boxed{ P(\text{early extinction}) } \]

\[ \boxed{ P(\text{major outbreak}) } \]

from a small number of initial infections.

26. Simulation workflow

\[ \boxed{ 1.\; \text{Specify the stochastic model} } \] \[ \boxed{ 2.\; \text{Choose parameters and initial conditions} } \] \[ \boxed{ 3.\; \text{Simulate many independent trajectories} } \] \[ \boxed{ 4.\; \text{Calculate the outcome of interest for each trajectory} } \] \[ \boxed{ 5.\; \text{Summarise the resulting distribution} } \]

27. Example of an outcome distribution

Suppose \(M\) epidemic trajectories are simulated.

For each trajectory, record

\[ I_{\max}^{(m)}. \]

Then the collection

\[ I_{\max}^{(1)}, \ldots, I_{\max}^{(M)} \]

approximates the distribution of possible epidemic peaks.

28. Estimating exceedance probability

If \(K\) of the \(M\) simulated peaks exceed capacity \(C\), then

\[ \boxed{ P(I_{\max}>C) \approx \frac{K}{M} } \]

This is a Monte Carlo estimate.

29. Confidence in Monte Carlo estimates

A Monte Carlo probability estimate contains sampling error because only finitely many trajectories are used.

For an estimated probability \(\hat p\), a common approximate standard error is

\[ \boxed{ \sqrt{ \frac{ \hat p(1-\hat p) }{M} } } \]

so increasing \(M\) improves sampling precision.

30. Validation remains essential

A stochastic model should still be compared with data and checked against the real process it is intended to describe.

Adding randomness does not compensate for poor assumptions, incorrect parameters or an inappropriate model structure.

31. Model complexity should serve the question

A useful model is not necessarily the most complicated one.

The model should contain enough structure to answer the scientific question while remaining interpretable and testable.

\[ \boxed{ \text{model complexity} \text{ should be justified by the modelling objective} } \]

32. The central idea

Stochastic modelling changes the focus from one predicted path to a distribution of possible outcomes.

\[ \boxed{ \text{mechanism} + \text{randomness} \rightarrow \text{distribution of trajectories} } \] \[ \boxed{ \text{distribution of trajectories} \rightarrow \text{means, variances and probabilities} } \]

This makes stochastic models particularly useful when uncertainty itself is part of the scientific or decision-making question.