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EdisonCruise
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2D martingale problem

December 27th, 2014, 9:30 am

This is an interview question. Circle A with radius a is inside Circle B with radius b. a 2D independent Brownian motion start from point M with distance r to center of circle A. point M locates between circle A and cirlce B. What's the probability of reaching circle A before reaching circle B?
Last edited by EdisonCruise on December 27th, 2014, 11:00 pm, edited 1 time in total.
 
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EdisonCruise
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2D martingale problem

December 29th, 2014, 1:32 am

Is there anything unclear on this question? The interviewer suggested a martingale be constructed in 2D to solve this problem, but I really cannot figure it out. Any further suggestions from you guys?
 
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Alan
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2D martingale problem

December 29th, 2014, 5:09 am

It is a standard fact that [$]f(x) = E_x[ \phi(B_{\tau})][$] is harmonic in an arbitrary region [$]\Omega[$], where [$]\tau[$] is the first time that [$]B_t[$] hits the boundary of [$]\Omega[$]. Apply that to [$]\Omega[$] the annulus with [$]\phi(x) = 1_{\{|x|=a\}}[$]. Since the problem is rotationally symmetric in 2D, solns to [$]\Delta f=0[$] are [$]f(r) = A + B \log r[$] with bc [$]f(a)=1[$], [$]f(b) = 0[$] determining [$](A,B)[$].The martingale is [$]f(B_t)[$] (The fact that it is a martingale follows from Ito's formula and the fact that f is harmonic).p.s. the 2D BM [$]B_t[$] should not be confused with the undetermined constant [$]B[$]Also notation: [$]x[$] is a 2-vector, while [$]|x|[$] is its length.BTW, for any students interested in some related textbook discussion, see Durrett's `Stochastic Calculus', Sec 4.4
Last edited by Alan on December 28th, 2014, 11:00 pm, edited 1 time in total.
 
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EdisonCruise
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2D martingale problem

December 31st, 2014, 8:10 am

Thank you Alan. I have sth to pick up.
 
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studenttt
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2D martingale problem

January 6th, 2015, 11:21 pm

Alan, do you mean this book? Section 4.4 doesn't look like to be relevant in this book.
Last edited by studenttt on January 6th, 2015, 11:00 pm, edited 1 time in total.
 
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Alan
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2D martingale problem

January 7th, 2015, 12:42 am

yes, that book -- it looks spot on to me for the martingale connection.