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.