به انگلیسی

Brownian increments and continuous paths

Mathematics · Probability · ages 23-24

Name ______________________ Date ____________

For standard Brownian motion, positions B_1 and B_3 are independent because they occur at different times.
Circle one: True False
Let F_2 be the full history of standard Brownian motion through time 2. What is E[B_7-B_2 | F_2]?
Answer: ______________
Standard Brownian motion starts at 0. What is the variance of B_4?
Answer: ______________
Which distribution does every positive-length increment of standard Brownian motion have?
- A uniform law with mean zero
- A two-point law with mean zero
- A normal law with mean zero
A walk has independent fair +1 or -1 steps. Which time and space steps, with linear interpolation, give the Brownian limit as n grows?
- Time step 1/n; space step 1/sqrt(n)
- Time step 1/n; space step 1/n
- Time step 1/sqrt(n); space step 1/n
B is standard Brownian motion. Define W_t=B_(4t)/2 for all t>=0. What is the law of W as a process?
- Brownian motion with variance 4t
- Standard Brownian motion
- Brownian motion with positive drift
- A deterministic straight-line path
For standard Brownian motion, B_4-B_1 and B_6-B_4 are independent even though their intervals share endpoint 4.
Circle one: True False
For standard Brownian motion, let X=3+B_6-B_2. What is E[X^2]?
Answer: ______________
Kim obtains a position path by integrating a smooth finite velocity. Which property conflicts with using this as a standard Brownian path?
- Its increments must be uniform rather than normally distributed
- Its starting value must be random rather than fixed at zero
- Its paths are almost surely nowhere differentiable
A Brownian increment over length h>0 has standard deviation sqrt(h). As h tends to 0, what happens to the standard deviation of the difference quotient, increment/h?
- It shrinks as sqrt(h), approaching zero.
- It grows as 1/sqrt(h), without bound.
- It stays equal to 1 at every positive scale.

Answer key

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False · Their covariance is min(1,3)=1. Independence would imply covariance 0, so the positions are dependent.
0 · The future increment B_7-B_2 is independent of the history through time 2 and has mean 0. Its conditional mean is therefore 0.
4 · Variance grows linearly with time.
A normal law with mean zero · For s<t, the increment B_t-B_s has normal law N(0,t-s). Its distribution is normal, not uniform or two-point.
Time step 1/n; space step 1/sqrt(n) · Time step 1/n puts n steps in one time unit. Scaling positions by 1/sqrt(n) makes endpoint variance n/n=1. Donsker's theorem gives the Brownian path-law limit.
Standard Brownian motion · Brownian scaling with c=4 divides positions by sqrt(4)=2. The resulting process has the standard Brownian law.
True · These are successive Brownian increments. Sharing only an endpoint creates no positive-length overlap, and the independent-increments property applies.
13 · The increment has mean 0 and variance 4. Thus X has mean 3 and variance 4, giving E[X^2]=4+3^2=13.
Its paths are almost surely nowhere differentiable · Integrating a smooth finite velocity gives a differentiable position path. A Brownian path almost surely has no finite derivative anywhere.
It grows as 1/sqrt(h), without bound. · Division by h divides standard deviation by h. Thus the quotient's standard deviation is sqrt(h)/h=1/sqrt(h), which grows without bound.