V6
发帖数: 5 | 1 Denote \tau_a=inf{t>0, B_t+t=a}, where B_t is Brownian Motion.
What is P(\tau_a<\tau_b) for a<0
Thanks a lot! | H****h
发帖数: 1037 | 2 Find f such that f(B_t+t) is a local martingale.
【在 V6 的大作中提到】
: Denote \tau_a=inf{t>0, B_t+t=a}, where B_t is Brownian Motion.
: What is P(\tau_a<\tau_b) for a<0: Thanks a lot!
| V6
发帖数: 5 | 3 可以具体一点吗？是不是要用到generator之类的
【在 H****h 的大作中提到】
: Find f such that f(B_t+t) is a local martingale.
| Q***5
发帖数: 994 | 4 Here is my understanding of Health's suggestion:
Suppose you can find such a f (to avoid some tech issue, let's just assume
without proof that f(t+B_t) is a martingale, instead of just local
martingale).
Let T be the first time that t+B_t hit either a or b, then by optional
sampling Thm, you have the equation:
E(f(T+B_T)) = f(0+B_0) = f(0)
On the other hand, let p = prob(\tau_a<\tau_b), so
E(f(T+B_T)) = p f(a) + (1-p)f(b)
From these two equations, you can solve for p.
You may use Ito's lemma to
【在 V6 的大作中提到】
: 可以具体一点吗？是不是要用到generator之类的
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