Quant, Math & Computer Science Puzzles for Interview Preparation & Brain Teasing
A collection of ~225 Puzzles with Solutions (classified by difficulty and topic)

Sep 10, 2012

Brownian Motion in Circles Puzzle


Source: http://www.stanford.edu/~gowthamr/puzzles.html

Problem:
Suppose the starting point of a particle undergoing Brownian motion in 2 dimensions is chosen uniformly at random on an imaginary circle C_1. Suppose there is a solid circle C_2 completely inside C_1, not necessarily concentric. Show that the particle hits the boundary of C_2 with uniform distribution.

Book: I strongly recommend the book by Steven Shreve for understanding Brownian Motion and its applications in Financial Modelling (Its expensive! Flipkart Link: Stochastic Calculus for Finance II)

Update (4th Feb 2013):
Solution posted by me in comments!

4 comments:

  1. Does the particle 'interact' with any of the circles? How do you count 'hitting' the boundary - from outside, from inside or both?

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  2. @Asad.
    a) The question does not say it interacts, Why would you want to assume any interaction?
    b) Since the point is starting from C_1, and C_2 is inside C_1, its impossible to hit C_2 from inside first! So, that doubt also does not require any clarification.

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  3. This comment has been removed by a blog administrator.

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  4. Solution posted on Goutham's website: http://www.stanford.edu/~gowthamr/Answer2.html

    Assume that the particle starts (at a point randomly chosen with uniform distribution) on the boundary of a circle C_0 of very large radius that is concentric with C_1 . By symmetry, the particle hits C_1 with uniform distribution, hence one may rather assume the particle starts at random from infinity. It is clear then that the particle should hit C_2 with uniform distribution.

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