STOCHASTIC CALCULUS 02
Brownian Motion and Brownian Increments
Brownian motion is a continuous random process whose changes over time are normally distributed.
The main idea
Brownian motion is usually written as
or equivalently
\[ W_t. \]It starts at
\[ \boxed{ W(0)=0 } \]and then changes randomly through time.
1. Brownian motion is a random process
A single random variable gives one random value.
Brownian motion gives a random value at every time:
\[ W(0),\; W(t_1),\; W(t_2),\; \ldots \]So \(W(t)\) is a stochastic process.
2. A single path
If we generate Brownian motion once, we obtain one possible trajectory.
Another simulation gives a different trajectory.
Each individual path is called a realisation or sample path.
3. The Brownian increment
Suppose we look at two times:
\[ t \]and
\[ t+\Delta t. \]The change in Brownian motion is
This is called a Brownian increment.
4. Distribution of the increment
A Brownian increment over an interval of length \(\Delta t\) satisfies
This means:
\[ E[\Delta W]=0 \]and
\[ \operatorname{Var}(\Delta W)=\Delta t. \]5. Standard deviation
Since standard deviation is the square root of variance,
\[ \operatorname{SD}(\Delta W) = \sqrt{\Delta t}. \]Therefore,
6. Generating an increment
Suppose
\[ Z\sim N(0,1). \]Then
has the required distribution
\[ N(0,\Delta t). \]7. Why the square root appears
If
\[ Z\sim N(0,1), \]then
\[ \operatorname{Var}(Z)=1. \]For a constant \(a\),
\[ \operatorname{Var}(aZ) = a^2\operatorname{Var}(Z). \]Take
\[ a=\sqrt{\Delta t}. \]Then
\[ \operatorname{Var} \left( \sqrt{\Delta t}\,Z \right) = \Delta t. \]So the square-root scaling is required by the Brownian variance rule.
8. Example
Suppose
\[ \Delta t=0.04. \]Then
\[ \sqrt{\Delta t}=0.2. \]So
\[ \Delta W=0.2Z. \]If a generated standard normal value is
\[ Z=1.5, \]then
\[ \Delta W = 0.2(1.5) = 0.3. \]If instead
\[ Z=-0.8, \]then
\[ \Delta W=-0.16. \]The increment can be positive or negative.
9. Mean zero does not mean every increment is zero
The statement
\[ E[\Delta W]=0 \]means that the average increment over many independent realisations is zero.
An individual increment will usually not equal zero.
10. Independent increments
Brownian motion has independent increments over non-overlapping time intervals.
For example,
\[ W(t_2)-W(t_1) \]and
\[ W(t_4)-W(t_3) \]are independent when the intervals
\[ [t_1,t_2] \]and
\[ [t_3,t_4] \]do not overlap.
11. Why independence matters
Knowing the random change over one non-overlapping interval gives no information about the random change over another.
A positive Brownian increment does not make the next increment more likely to be positive.
12. Stationary increments
The distribution of a Brownian increment depends only on the length of the interval.
For example,
\[ W(5.1)-W(5) \]and
\[ W(100.1)-W(100) \]both correspond to an interval of length
\[ 0.1. \]Therefore both have distribution
\[ N(0,0.1). \]This property is called stationary increments.
13. Brownian motion at a fixed time
Since
\[ W(0)=0, \]we may regard
\[ W(t)=W(t)-W(0). \]Therefore,
for \(t>0\).
14. Example at time \(t=4\)
At
\[ t=4, \]we have
\[ W(4)\sim N(0,4). \]Therefore,
\[ E[W(4)]=0 \]and
\[ \operatorname{SD}(W(4))=2. \]The spread of possible Brownian positions becomes larger as time increases.
15. Variance grows linearly with time
A fundamental Brownian-motion property is
So uncertainty increases as time passes.
But standard deviation grows like
\[ \sqrt{t}. \]16. Building a Brownian path numerically
Choose time points
\[ t_0,\; t_1,\; t_2,\; \ldots \]separated by
\[ \Delta t. \]Begin with
\[ W_0=0. \]At each step, generate
\[ Z_n\sim N(0,1) \]and calculate
\[ \Delta W_n = \sqrt{\Delta t}\,Z_n. \]Then update
17. Why a computer plot looks discrete
Mathematical Brownian motion is defined in continuous time.
But a computer can store only finitely many values.
So a simulation uses a time grid:
\[ t_0,\; t_1,\; t_2,\; \ldots,\; t_n. \]The plotted line connects these simulated values.
The grid is only a numerical approximation to an underlying continuous-time process.
18. Brownian path diagram
19. Brownian motion is continuous
Brownian motion has continuous sample paths.
This means that it does not jump instantaneously from one value to another.
However, continuity does not mean smoothness.
20. Brownian motion is highly irregular
Although Brownian paths are continuous, with probability one they are nowhere differentiable.
There is no ordinary finite slope
\[ \frac{dW}{dt} \]at any time.
This irregularity is central to stochastic calculus.
21. Why the ordinary derivative fails
For Brownian motion,
\[ \Delta W \]typically has size
\[ \sqrt{\Delta t}. \]So the difference quotient has typical size
\[ \frac{\Delta W}{\Delta t} \sim \frac{1}{\sqrt{\Delta t}}. \]As
\[ \Delta t\to0, \]this does not settle to a finite value.
22. Brownian motion has quadratic variation
Suppose the interval \([0,T]\) is divided into many small subintervals.
Consider the sum of squared increments:
\[ \sum_i (\Delta W_i)^2. \]As the partition becomes finer, this sum approaches
in the appropriate stochastic sense.
This property is called quadratic variation.
23. Why \((dW)^2=dt\)
A Brownian increment has size
\[ dW\sim\sqrt{dt}. \]Squaring gives a quantity of order
\[ (dW)^2\sim dt. \]This motivates the symbolic stochastic-calculus rule
The rule represents quadratic variation. It is not ordinary algebraic squaring of differentials.
24. Compare with ordinary differentials
For ordinary calculus,
\[ dx\sim dt \]for a smooth time-dependent function.
Therefore,
\[ (dx)^2 \]is of order
\[ (dt)^2, \]which becomes negligible compared with \(dt\).
For Brownian motion,
\[ dW\sim\sqrt{dt}, \]so
\[ (dW)^2 \]remains of order \(dt\).
25. The important stochastic scaling rules
These are understood at the level of stochastic differential calculus, where terms smaller than order \(dt\) are discarded.
26. Brownian motion versus white noise
Brownian motion itself is continuous.
The formal expression
\[ \frac{dW_t}{dt} \]is sometimes called white noise.
But this is not an ordinary function or ordinary derivative.
For this reason, SDEs are normally written using
\[ dW_t \]rather than trying to treat white noise as an ordinary time-dependent function.
27. Why Brownian motion is useful
Brownian motion provides a mathematical model for many small random influences acting continuously through time.
Instead of modelling every microscopic random event separately, their aggregate effect can sometimes be represented by a Brownian diffusion term.
28. Brownian motion inside an SDE
A stochastic differential equation may be written as
\[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t. \]Over a small interval \(\Delta t\), this becomes approximately
with
\[ \boxed{ \Delta W = \sqrt{\Delta t}\,Z, \qquad Z\sim N(0,1). } \]29. Deterministic and random contributions
The first contribution,
\[ a\Delta t, \]is systematic.
The second,
\[ b\Delta W, \]is random.
30. The central idea
These properties distinguish Brownian motion from ordinary smooth functions and provide the random building block used in stochastic differential equations.