Wt stops when it hits a<0 or b>0
Is there a formula to determin P(Wt=a, t<T) and P(Wt=b, t<T)?
Any help would be appreciated
Would the probability in case 3 be 1? since by definition, T is a stopping time when the BM hits either a or b, hence W(T) is either a or b, given T, P(W(T) hits a or b) = 1.
Let me clarify the symbols a little bit.
The Process（t,W(t)） stops at
(Ta, a)
(Tb, b)
or (T,W(T))
for the first two occasions, W determines the stop, and for the last, t determines the stop
The question is:
1) Prob(0=<Ta<u<=T)
2) Prob(0=<Tb<u<=T)
3) Prob( a=<W(T)<=u, u∈[a,b])
Of course, differentiate them with the variable u we can generate probability density functions for random variable Ta, Tb & W(T).
I am particularly interested in problem 3, for the first two can be solved by this problem 3 alone.
I am not sure but you may find this book useful: Stochastic Optimization in Continuous Time by F.R. ChangWt stops when it hits a<0 or b>0
Is there a formula to determin P(Wt=a, t<T) and P(Wt=b, t<T)?
Any help would be appreciated
Okay, suppose you are observing a ball running along the time horizon before a right bound T.
However it must stays between the line y=a and y=b, once it touches either line, it is finished and no longer runs.
_________y=b__________
O O
0- - - - O- - - - - - - >- - - - - -- -T
O
_________y=a__________
Then we have three senarios:
1) it ends at (t,a) , in which stopping time t is undetermined
2) it ends at (t,b), in which stopping time t is undetermined
3) Luckily it makes to the end and at (T,w), in which the ending position w is undetermined
I am not sure but you may find this book useful: Stochastic Optimization in Continuous Time by F.R. Chang
chapter 6: 6 Boundaries and Absorbing Barriers
Okay, its clear now, but its better to use correct mathematical notation/definitions
I defined two stopping times above, the first time BM hits a and b:
Ta = min{t:W(t)=a} Tb = min{t:W(t)=b}
There are three possible mutually exclusive events:
E1 = {Ta < min(Tb, T)} (a is hit before b and before time T)
E2 = {Tb < min(Ta, T)} (b is hit before a and before time T)
E3 = {T <= min(Ta, Tb)} (neither a nor b is hit before time T)
P[E1] + P[E2] + P[E3] = 1
I assume you are asking about P[E1], P[E2] and P[E3]?
P[Ta < T] and P[Tb < T] follow from the reflection principle (a standard quant interview question). Thus P[E3] follows from my question about about the probability of hitting both a and b before time T:
P[E3] = 1 - P[Ta < T or Tb < T] = 1 - P[Ta < T] - P[Tb < T] + P[Ta < T and Tb < T].
I'll post this for the special case b = -a later.
Thank you.
Thanks for your notation, but I have seen the formula in infinite sum.