STOCHASTIC CALCULUS 03
Stochastic Differential Equations
A stochastic differential equation combines systematic change with continuous random fluctuation.
The main idea
An ordinary differential equation may be written as
\[ dX_t=a(X_t,t)\,dt. \]A stochastic differential equation adds a Brownian-motion term:
The first term describes deterministic change and the second describes random fluctuation.
1. The state variable
The quantity
\[ X_t \]is the state of the system at time \(t\).
Unlike an ordinary deterministic solution, \(X_t\) is a random variable for each fixed time \(t\).
The collection
\[ \{X_t:t\ge0\} \]forms a stochastic process.
2. Drift term
The term
\[ a(X_t,t)\,dt \]is called the drift.
The function
\[ a(X_t,t) \]gives the systematic local rate of change.
If the diffusion term were removed, we would have the deterministic equation
\[ dX_t=a(X_t,t)\,dt. \]3. Diffusion term
The term
\[ b(X_t,t)\,dW_t \]is called the diffusion term.
The coefficient
\[ b(X_t,t) \]controls the strength of the random fluctuation.
4. Brownian motion supplies the randomness
The Brownian increment over a short interval satisfies
\[ \Delta W \sim N(0,\Delta t). \]Equivalently,
5. Small-time interpretation
Over a small time interval \(\Delta t\), the SDE
\[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t \]is approximated by
where
\[ \Delta W = \sqrt{\Delta t}\,Z. \]6. The two different scales
The deterministic contribution has size
\[ a\Delta t. \]The random contribution has typical size
\[ b\sqrt{\Delta t}. \]Therefore,
This difference in scaling is fundamental.
7. A simple SDE
Consider
\[ \boxed{ dX_t = \mu\,dt + \sigma\,dW_t } \]where \(\mu\) and \(\sigma\) are constants.
Here,
\[ \mu \]is the drift rate and
\[ \sigma \]is the diffusion strength.
8. Interpretation over a small interval
For a small time step \(\Delta t\),
\[ \Delta X \approx \mu\Delta t + \sigma\sqrt{\Delta t}\,Z. \]The first part shifts the process systematically.
The second part randomly moves it above or below that systematic change.
9. Example
Suppose
\[ \mu=2, \qquad \sigma=0.5, \qquad \Delta t=0.04. \]Then
\[ \mu\Delta t = 2(0.04) = 0.08. \]Also,
\[ \sigma\sqrt{\Delta t} = 0.5(0.2) = 0.1. \]So
\[ \Delta X \approx 0.08+0.1Z. \]If
\[ Z=1.3, \]then
\[ \Delta X \approx 0.08+0.13 = 0.21. \]If instead
\[ Z=-1, \]then
\[ \Delta X \approx 0.08-0.1 = -0.02. \]So the random term can overcome the positive drift during a particular short interval.
10. Same model, different paths
For a deterministic ODE, the same initial condition and parameters give the same trajectory.
For an SDE, different Brownian increments produce different trajectories.
11. But the model is still precise
The fact that trajectories differ does not mean the model is vague.
The SDE specifies the probability law governing those trajectories.
It tells us how the systematic and random components behave mathematically.
12. Additive noise
In
\[ dX_t = a(X_t,t)\,dt + \sigma\,dW_t, \]the diffusion coefficient is constant.
The random fluctuation has the same strength regardless of the current state.
This is often called additive noise.
13. State-dependent noise
Consider
\[ dX_t = a(X_t,t)\,dt + \sigma X_t\,dW_t. \]Now the random fluctuation depends on the current value \(X_t\).
When \(X_t\) is large, the diffusion term becomes larger.
This is an example of multiplicative noise.
14. Geometric Brownian motion
An important SDE is
Both the drift and diffusion are proportional to the current state.
This model is known as geometric Brownian motion.
15. Deterministic part of geometric Brownian motion
If
\[ \sigma=0, \]then
\[ dX_t = \mu X_t\,dt. \]This is simply exponential growth or decay.
The stochastic term introduces random variation around that growth mechanism.
16. Integral form of an SDE
The differential notation is compact, but an SDE is defined rigorously through an integral equation.
For
\[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t, \]the integral form is
The first integral is an ordinary time integral.
The second is a stochastic integral.
17. Why the stochastic integral needs a new definition
In ordinary integration,
\[ \int f(t)\,dt \]integrates against smooth time increments.
But
\[ \int b(X_t,t)\,dW_t \]integrates against Brownian motion, whose paths are nowhere differentiable.
So ordinary Riemann integration is not enough.
This leads to the Itô integral.
18. SDEs do not mean \(dW_t/dt\) exists
It may be tempting to write
\[ \frac{dX_t}{dt} = a(X_t,t) + b(X_t,t)\frac{dW_t}{dt}. \]But
\[ \frac{dW_t}{dt} \]does not exist as an ordinary derivative.
So the differential form of an SDE must be understood through stochastic integration, not ordinary differentiation.
19. Numerical simulation
A common numerical method for SDEs is Euler–Maruyama.
For
\[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t, \]the update is
with
\[ \boxed{ \Delta W_n = \sqrt{\Delta t}\,Z_n, \qquad Z_n\sim N(0,1) } \]20. Compare with Euler's method
For an ODE,
\[ dX_t=a(X_t,t)\,dt, \]Euler's method gives
\[ X_{n+1} = X_n+a(X_n,t_n)\Delta t. \]Euler–Maruyama adds the stochastic contribution:
\[ b(X_n,t_n)\Delta W_n. \]21. Diagram of possible trajectories
22. One path versus a distribution
A single simulated trajectory answers:
\[ \boxed{ \text{What could happen in one realisation?} } \]Many trajectories allow us to study:
\[ E[X_t], \] \[ \operatorname{Var}(X_t), \]and probabilities such as
\[ P(X_t>L). \]These quantities describe uncertainty across possible realisations.
23. Expected value
The expected value
\[ E[X_t] \]describes the average state across many hypothetical repetitions of the stochastic system.
It is not the same thing as one particular sample path.
24. Variance
The variance
\[ \operatorname{Var}(X_t) \]describes how widely possible values of \(X_t\) are spread around their mean.
A deterministic model contains only one trajectory for fixed inputs, while an SDE can describe both central behaviour and variability.
25. Epidemic example
Suppose the deterministic infectious population satisfies approximately
\[ dI_t = \left( \beta\frac{S_tI_t}{N} - \gamma I_t \right)dt. \]A stochastic version may include a diffusion term:
The drift retains the epidemic mechanism, while the diffusion represents random fluctuations.
26. The diffusion term should have meaning
In a well-constructed stochastic model, the function
\[ \sigma(S,I) \]should not be chosen arbitrarily.
Its form should reflect the source and scale of the uncertainty being represented.
For epidemic models derived from random infection and recovery events, the diffusion structure can be obtained from those event rates.
27. Why SDEs are useful
An SDE allows us to study more than a single average trajectory.
It can represent:
28. What an SDE does not automatically provide
Writing down an SDE does not immediately give its probability distribution.
We may need simulation, moment equations, the Fokker–Planck equation, or other analytical methods to study the distribution of \(X_t\).
29. The structure to remember
30. The central idea
An SDE is not simply an ODE with an arbitrary random number added.
It is a continuous-time stochastic model with a precise Brownian-motion structure.