Brownian motion properties - Ch 9 - Practice questions 9.4 (iv)

[Sandor Kelemen]

Dear all,

I think that the most interesting part of the question - namely to find the probability of being B_t for some 0 <= t <= 2 negative - is not answered properly in the CMP. Could anybody please give me a simple (exam style) but rigorous answer?

Thx in advance.

Sandor

 

[Calm]

Dear all,

I think that the most interesting part of the question - namely to find the probability of being B_t for some 0 <= t <= 2 negative - is not answered properly in the CMP. Could anybody please give me a simple (exam style) but rigorous answer?

Thx in advance.

Sandor

That's because (as I may have been thinking) that all we are looking for is B_1 and/or B_2 to be negative values, much like the previous part question. But don't forget that the Brownian motion is continuous state space, continuous time stochastic process. The thing is that while B_1 and B_2 can both be positive, no matter how positive the interval is at any one instant, there is non-zero probability that at the next instant it will drop below zero due to that distribution B_t-B_s~N(0,t-s).

 

[Sandor Kelemen]

That's because (as I may have been thinking) that all we are looking for is B_1 and/or B_2 to be negative values, much like the previous part question. But don't forget that the Brownian motion is continuous state space, continuous time stochastic process. The thing is that while B_1 and B_2 can both be positive, no matter how positive the interval is at any one instant, there is non-zero probability that at the next instant it will drop below zero due to that distribution B_t-B_s~N(0,t-s).

Hi Calm,

No. The question is to calculate the Prob("There is a t from [0,2] such that B_t <0 "). And the answer in the CMP is that this equals to 1 with really unclear and high-level argumentation.