STOCHASTIC CALCULUS 10

Deterministic and Stochastic Models Compared

Deterministic and stochastic models describe the same kinds of systems from different mathematical viewpoints.

The main idea

A deterministic model produces one trajectory for fixed parameters and initial conditions.

A stochastic model produces a distribution of possible trajectories.

\[ \boxed{ \text{deterministic} \rightarrow \text{one path} } \] \[ \boxed{ \text{stochastic} \rightarrow \text{many possible paths} } \]

1. Deterministic model

Consider

\[ \frac{dX}{dt}=f(X,t), \qquad X(0)=X_0. \]

Once \(f\) and \(X_0\) are fixed, the model determines a particular trajectory \(X(t)\), assuming the usual conditions for uniqueness.

2. Stochastic model

A corresponding stochastic model may be

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

Even with the same initial value and parameters, different Brownian paths generate different realisations.

3. Same initial state, different outcomes

For a deterministic model,

\[ X_0 \]

leads to one model trajectory.

For a stochastic model,

\[ X_0 \]

can lead to

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

depending on the realised random fluctuations.

4. What does a deterministic trajectory represent?

A deterministic trajectory describes the evolution implied by the model's rate equations.

In many applications it may be interpreted as a mean-field or large-population approximation, but this interpretation depends on how the model was derived.

It should not automatically be assumed to equal the expectation of a related stochastic model.

5. Why the expectation need not equal the deterministic solution

Suppose

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

Taking expectations gives

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

In a nonlinear model,

\[ E[a(X_t)] \]

is generally not equal to

\[ a(E[X_t]). \]

\[ \boxed{ E[a(X_t)] \neq a(E[X_t]) \quad \text{in general} } \]

6. A simple nonlinear example

Suppose

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

Then

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

But

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

Therefore,

\[ \boxed{ \frac{d}{dt}E[X_t] = (E[X_t])^2 + \operatorname{Var}(X_t) } \]

so the variance affects the evolution of the mean.

7. Deterministic approximation

If variability is small, one may approximate

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

This produces a deterministic mean-field equation.

But it is an approximation, not an identity.

8. Linear models are simpler

For a linear SDE such as

\[ dX_t = aX_t\,dt + \sigma\,dW_t, \]

taking expectations gives

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

Here the mean obeys the same linear equation as the corresponding deterministic model.

This agreement does not generally extend to nonlinear systems.

9. A useful comparison

\[ \boxed{ \text{deterministic model} = \text{single state at each time} } \] \[ \boxed{ \text{stochastic model} = \text{probability distribution at each time} } \]

The distinction is not simply smooth versus noisy curves.

It is a difference in the mathematical object being modelled.

10. Distribution at a fixed time

In a deterministic model,

\[ X(t) \]

is one fixed value.

In a stochastic model,

\[ X_t \]

is a random variable with a probability distribution.

That distribution may be summarised by

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

quantiles and tail probabilities.

11. Diagram

12. Questions a deterministic model can answer

A deterministic model can be useful for questions such as

\[ \text{What is the predicted trajectory?} \] \[ \text{When does the peak occur?} \] \[ \text{How does changing a parameter alter the trajectory?} \]

These are often valuable modelling questions.

13. Questions a stochastic model adds

A stochastic model can additionally address

\[ \text{How variable is the peak?} \] \[ \text{What is the probability of exceeding a threshold?} \] \[ \text{What is the probability of early extinction?} \] \[ \text{How uncertain is the event timing?} \]

14. Why the average alone can be misleading

Suppose hospital capacity is \(C\).

A stochastic epidemic model may satisfy

\[ E[I_t] while still having

\[ P(I_t>C)>0. \]

If the decision concerns risk of overload, the probability may be more relevant than the mean alone.

15. Example of two models with the same mean

Imagine two stochastic models whose infectious populations have the same expected value:

\[ E[I_t]=1000. \]

One might have

\[ \operatorname{SD}(I_t)=20, \]

while another has

\[ \operatorname{SD}(I_t)=500. \]

Their average behaviour is identical, but their uncertainty and risk profiles are very different.

16. Deterministic models remain useful

The existence of stochastic models does not make deterministic models obsolete.

Deterministic models are often:

\[ \text{simpler to analyse}, \] \[ \text{faster to simulate}, \] \[ \text{easier to interpret}, \]

and useful for understanding the underlying mechanism.

17. Stochastic models require more information

A stochastic model needs assumptions about the random structure as well as the deterministic mechanism.

For an SDE, we must specify both

\[ a(X,t) \]

and

\[ b(X,t). \]

The diffusion term must be scientifically and mathematically justified.

18. Computational cost

A deterministic ODE normally requires one numerical solution for a fixed parameter set.

A stochastic simulation may require thousands of trajectories to estimate a probability accurately.

\[ \boxed{ \text{more uncertainty information} \quad\Longleftrightarrow\quad \text{greater computational cost} } \]

19. Population size matters

In large populations, individual random events may partly average out, making deterministic approximations useful.

In small populations or near extinction, stochastic effects can be much more important.

Therefore model choice can depend strongly on scale.

20. Early outbreak behaviour

Suppose only a few infectious individuals are present.

A deterministic epidemic model may predict growth whenever the average reproduction mechanism favours infection.

But in a stochastic model, the first few infectious individuals may recover before transmitting.

The outbreak can therefore disappear by chance.

21. Deterministic growth does not guarantee a major outbreak

A positive deterministic growth rate means the expected mechanism favours increase.

It does not imply

\[ P(\text{major outbreak})=1. \]

\[ \boxed{ \text{positive average growth} \neq \text{certain growth in every realisation} } \]

22. Long-run behaviour

Deterministic models often describe equilibria and their stability.

Stochastic models may instead have stationary distributions, extinction probabilities or random fluctuations around deterministic equilibria.

So the idea of long-run behaviour also changes when randomness is introduced.

23. Different forms of stochastic modelling

Stochastic modelling is broader than SDEs.

Other stochastic models include

\[ \text{discrete-time Markov chains}, \] \[ \text{continuous-time Markov chains}, \] \[ \text{jump processes}, \]

and

\[ \text{stochastic differential equations}. \]

The appropriate choice depends on the nature and scale of the random events.

24. Jump process versus diffusion model

A continuous-time Markov chain can represent individual discrete events such as one infection or one recovery.

An SDE represents continuous-valued random fluctuation.

\[ \boxed{ \text{CTMC} \rightarrow \text{discrete jumps} } \] \[ \boxed{ \text{SDE} \rightarrow \text{continuous random paths} } \]

25. Diffusion approximation

In suitable large-population settings, a jump process can sometimes be approximated by an SDE.

The drift reflects the average event rates, while the diffusion reflects variation generated by those random events.

\[ \boxed{ \text{random event model} \rightarrow \text{diffusion approximation} } \]

26. Deterministic limit

Under suitable scaling, stochastic fluctuations may become relatively small as system size increases.

The stochastic model may then approach a deterministic mean-field model.

This gives a mathematical connection between deterministic and stochastic descriptions rather than treating them as unrelated alternatives.

27. Model comparison should depend on the objective

If the goal is to understand the basic mechanism, a deterministic model may be enough.

If the goal is to quantify the chance of an extreme outcome, a stochastic model is often more appropriate.

\[ \boxed{ \text{choose the model according to the question} } \]

28. A useful modelling hierarchy

\[ \boxed{ \text{deterministic ODE} } \] \[ \downarrow \] \[ \boxed{ \text{stochastic jump process} } \] \[ \downarrow \] \[ \boxed{ \text{SDE diffusion approximation} } \] \[ \downarrow \] \[ \boxed{ \text{moments or probability density} } \]

These are different mathematical views of dynamical systems, each useful for different purposes.

29. Main comparison

\[ \boxed{ \begin{array}{c|c} \text{Deterministic} & \text{Stochastic}\\ \hline \text{one trajectory} & \text{distribution of trajectories}\\ \text{fixed outcome for fixed inputs} & \text{random outcome for fixed inputs}\\ \text{ODEs} & \text{SDEs or jump processes}\\ \text{trajectory and equilibrium} & \text{mean, variance, probability}\\ \text{usually cheaper computationally} & \text{often more computationally demanding} \end{array} } \]

30. The central idea

Deterministic and stochastic models are not competitors in every situation.

They answer different levels of the same modelling problem.

\[ \boxed{ \text{deterministic model} \rightarrow \text{systematic behaviour} } \] \[ \boxed{ \text{stochastic model} \rightarrow \text{systematic behaviour plus uncertainty} } \]

The important question is not which model is universally better, but which mathematical description is appropriate for the scientific question being asked.