STOCHASTIC CALCULUS 01
Why Ordinary Calculus Is Not Enough
Ordinary calculus describes smooth or sufficiently regular change. Stochastic calculus extends these ideas to systems containing continuous random fluctuations.
The main idea
In ordinary calculus, we often model a changing quantity \(X(t)\) using a differential equation such as
Once the initial condition is known, the equation determines how the system evolves.
This is a deterministic model.
1. Deterministic change
Consider
\[ \frac{dX}{dt}=2X. \]If
\[ X(0)=1, \]then the solution is
\[ \boxed{ X(t)=e^{2t} } \]The same initial condition and the same equation always give the same trajectory.
2. But many real systems contain randomness
In many systems, the direction of average change may be known, but the exact evolution is uncertain.
Examples include population growth, epidemics, financial prices, molecular motion and environmental systems.
The observed path may fluctuate around the behaviour predicted by a deterministic model.
3. A deterministic epidemic example
Suppose \(I(t)\) represents the number or proportion of infectious individuals.
A simple deterministic model may have the form
\[ \frac{dI}{dt} = \text{infection rate} - \text{recovery rate}. \]For an SIS model,
This equation describes the net average direction of change.
4. What the deterministic equation does not describe
Actual infection and recovery events do not occur as smooth fractions of events.
During a short period, there may be more infections than expected, fewer infections than expected, or more recoveries than expected.
Therefore two epidemics with the same parameters and initial conditions can follow different paths.
5. Average behaviour and random behaviour
Suppose the deterministic model predicts a change
\[ \Delta I_{\text{det}}. \]A stochastic model allows the actual change to differ from this average direction:
This simple idea is the starting point for stochastic differential equations.
6. From an ODE to an SDE
An ordinary differential equation may be written in differential form as
\[ dX_t = a(X_t,t)\,dt. \]The term
\[ a(X_t,t)\,dt \]describes the deterministic change over a small time interval.
To include continuous random fluctuations, we introduce another term:
This is a stochastic differential equation, or SDE.
7. The two parts of an SDE
The term
\[ a(X_t,t)\,dt \]is called the drift term.
It represents the systematic or average local direction of change.
The term
\[ b(X_t,t)\,dW_t \]is called the diffusion term.
It represents continuous random fluctuations.
8. Drift
The drift coefficient
\[ a(X_t,t) \]plays a role similar to the rate appearing in an ordinary differential equation.
If the stochastic term were removed,
\[ dX_t=a(X_t,t)\,dt, \]we would again have deterministic evolution.
9. Diffusion
The coefficient
\[ b(X_t,t) \]controls the strength of the random fluctuations.
If \(b\) is larger, random deviations are generally larger.
If
\[ b=0, \]the stochastic equation reduces to an ordinary deterministic equation.
10. What is \(W_t\)?
The symbol
\[ W_t \]denotes Brownian motion, also called a Wiener process.
Brownian motion provides the basic source of continuous randomness used in many SDE models.
Its change over a time interval is
\[ \Delta W = W(t+\Delta t)-W(t). \]11. Brownian increments
For a time interval of length \(\Delta t\),
This means the Brownian increment is normally distributed with mean
\[ 0 \]and variance
\[ \Delta t. \]Therefore its standard deviation is
\[ \boxed{ \sqrt{\Delta t} } \]12. Generating a Brownian increment
If
\[ Z\sim N(0,1), \]then we can generate a Brownian increment using
The random number \(Z\) determines the random direction and relative size of the fluctuation.
13. Why \(\sqrt{\Delta t}\) rather than \(\Delta t\)?
Brownian motion has the property
\[ \operatorname{Var}(\Delta W)=\Delta t. \]Since standard deviation is the square root of variance,
\[ \operatorname{SD}(\Delta W) = \sqrt{\Delta t}. \]Therefore Brownian fluctuations naturally scale like
\[ \boxed{ \sqrt{\Delta t} } \]rather than like \(\Delta t\).
14. A crucial difference in scale
Over a small time interval,
\[ \text{drift change} \sim \Delta t, \]while
\[ \text{Brownian change} \sim \sqrt{\Delta t}. \]For example, if
\[ \Delta t=0.0001, \]then
\[ \Delta t=0.0001 \]but
\[ \sqrt{\Delta t}=0.01. \]This difference becomes fundamental in stochastic calculus.
15. Ordinary smooth paths
For a smooth deterministic function \(x(t)\), a small change behaves approximately like
\[ \Delta x \approx x'(t)\Delta t. \]Dividing by \(\Delta t\),
\[ \frac{\Delta x}{\Delta t} \]approaches a finite derivative as
\[ \Delta t\to0. \]16. Brownian paths behave differently
A Brownian increment has typical size
\[ \Delta W\sim\sqrt{\Delta t}. \]Therefore,
\[ \frac{\Delta W}{\Delta t} \]has typical size
\[ \frac{\sqrt{\Delta t}}{\Delta t} = \frac{1}{\sqrt{\Delta t}}. \]As
\[ \Delta t\to0, \]this does not approach an ordinary finite derivative.
17. Brownian motion is not differentiable in the ordinary sense
Brownian paths are continuous, but with probability one they are nowhere differentiable.
but
\[ \boxed{ \frac{dW_t}{dt} \text{ does not exist as an ordinary derivative} } \]This is the main reason ordinary differential calculus cannot simply be applied to Brownian motion.
18. Continuous does not mean smooth
This distinction is important.
A function can be continuous without being differentiable.
Brownian motion is an extreme example: its path does not jump, but it remains irregular at every time scale.
19. Diagram: deterministic and stochastic paths
20. Why not just use the deterministic average?
A deterministic model can be very useful for describing typical or average behaviour.
But the average alone does not describe uncertainty.
Two stochastic systems can have similar expected behaviour but very different probabilities of extreme outcomes.
21. Example: epidemic decision making
Suppose a deterministic model predicts that the epidemic peak remains below hospital capacity.
That result alone does not tell us the probability that random fluctuations could push the actual peak above capacity.
A stochastic model allows questions such as
to be investigated.
22. One stochastic trajectory is not the final answer
A single simulation shows only one possible realisation.
We normally generate many trajectories to study the distribution of possible outcomes.
From this distribution we can estimate quantities such as means, variances and probabilities.
23. Deterministic versus stochastic questions
A deterministic model may answer:
\[ \boxed{ \text{What trajectory does the model predict?} } \]A stochastic model can additionally ask:
\[ \boxed{ \text{What outcomes are possible?} } \] \[ \boxed{ \text{How variable are they?} } \] \[ \boxed{ \text{What is the probability of an important event?} } \]24. Why new calculus is required
The problem is not simply that randomness has been added.
The deeper issue is that Brownian motion has mathematical behaviour different from ordinary differentiable functions.
For ordinary differentials, terms such as
\[ (dt)^2 \]become negligible compared with \(dt\).
For Brownian motion, however,
\[ (dW_t)^2 \]behaves at the scale of \(dt\).
Symbolically, stochastic calculus uses the fundamental scaling rule
This is not ordinary algebra. It expresses the quadratic-variation behaviour of Brownian motion.
25. Ordinary calculus and stochastic calculus
Ordinary calculus is not discarded. Stochastic calculus extends it to a new type of random, irregular motion.
26. The central idea
contains two fundamentally different forms of change:
Understanding the unusual behaviour of \(dW_t\) is the key step from ordinary calculus to stochastic calculus.