MAST30001 Chap.8 Brownian Motion
Brownian Motion
Brownian motion is a continuous process with independent increments and normally distributed changes whose variance equals elapsed time. The chapter builds finite-dimensional distributions from shared increments, yielding covariance min(s,t), and uses scaling to standardise probabilities. First-hitting times and path maxima require reflection arguments because a path can cross a barrier and later finish below it.
Random-walk limits connect the model back to discrete stochastic processes. The essential distinction is between values and increments: values at different times share history, whereas changes over disjoint intervals are independent. A labelled time line exposes that dependence and makes conditional normal calculations much safer.
What this chapter covers
- 01
Continuous paths and independent normal increments
- 02
Time scaling and marginal normal distributions
- 03
Joint Gaussian distributions and covariance min(s,t)
- 04
Conditional distributions from fresh increments
- 05
First-hitting times and the reflection principle
- 06
Path maxima and random-walk limits
Worked example · free
Condition on an observed Brownian value
- 1The fresh increment from time 2 to 5 is N(0,3).
- 2Therefore B(5) conditional on B(2)=1.20 is N(1.20,3).
- 1The z-score is (2.20-1.20)/sqrt(3)=0.577.
- 1The requested probability is the upper standard normal tail beyond 0.577.
Key terms
- Independent Increment
- A change over one time interval that is independent of changes over disjoint intervals.
- Brownian Covariance
- The covariance between B(s) and B(t), equal to the earlier time min(s,t) in the standard model.
- First-Hitting Time
- The earliest time at which a continuous path reaches a specified level.
- Reflection Principle
- A symmetry argument that links barrier-crossing probabilities to normal endpoint tails.
- Path Maximum
- The largest value attained by the process over a stated time interval.
- Scaling Property
- The relationship between elapsed time and the spatial scale of Brownian fluctuations.
- Quadratic Variation
- The limiting accumulated sum of squared increments, equal to elapsed time for standard Brownian motion.
- Drifted Brownian Motion
- A Brownian model with a linear mean trend and a chosen diffusion scale.
Brownian Motion FAQ
Are Brownian values at different times independent?
No. Values share the earlier path history, giving covariance equal to the earlier time. What is independent are increments taken over disjoint intervals. Decomposing the later value into the earlier value plus a fresh increment makes this distinction clear.
Which variance should I use for an increment?
For standard Brownian motion, the increment from s to t has variance t-s, not variance t and not standard deviation t-s. Standardise by dividing by the square root of the elapsed time.
Why is a maximum probability larger than an endpoint tail?
A path may cross the barrier at an earlier time and finish below it. The reflection principle accounts for those returning paths and, for a positive level in standard Brownian motion, doubles the corresponding upper endpoint-tail probability.
How do I find covariance between two Brownian values?
Assume the earlier time is s. Write the later value as B(t)=B(s)+[B(t)-B(s)]. The bracketed increment is independent of B(s), so the covariance is Var(B(s))=s. Symmetry then gives covariance min(s,t) for either time ordering.
What changes for Brownian motion with drift and scale?
For X(t)=mu t+sigma B(t), an increment over length h has mean mu h and variance sigma squared times h. Drift affects the centre while scale affects standard deviation. Subtract the drift contribution and divide by sigma times the square root of h when standardising.
When should I use the reflection principle?
Use it for a path event involving whether the process reaches or exceeds a positive level before a fixed time. It is not needed for a simple endpoint event. Translate the maximum or first-hitting event first, then match the reflected endpoint tail under the stated standard Brownian assumptions.
Exam move
Draw a time axis for every Brownian problem and mark the intervals used by each value or increment. Decompose later values into an earlier observation plus a fresh independent increment. Write variance as elapsed time and take its square root only when standardising. For joint questions, build covariance entries from min(s,t).
Keep endpoint events distinct from first-hitting and maximum events; use the reflection principle only after identifying the barrier-crossing event. Check probability direction by noting that a maximum event includes paths that crossed and returned. Drill conditional problems by rewriting all requested values in terms of independent increments over disjoint intervals.
For each probability, record its mean, variance and standard deviation on separate lines before using a normal table. Sketch at least one path that crosses a barrier and returns below it so that maximum and endpoint events cannot be confused. Finally, test Brownian scaling by changing the time unit and confirming that typical spatial size changes with the square root of time rather than linearly.
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