[?]  First passage time distribution with moving boundary

Questions
1

Hello everyone! I’d like to investigate a first passage problem with moving boundary. I’m not good in Wolfram Mathematica and I don’t see how I can modify this code (from Mathematica Help, but I’m interested not only in Wiener process)

sample = Map[FirstCase[#, _?(Last[#] >= 2 &)] &, 
   RandomFunction[WienerProcess[1, 1], {0, 15, 0.01}, 10^4]["Paths"]];

        data0 = DeleteCases[sample, {}][[All, 1]];

        data = Cases[data0, _?Positive];

        Show[Histogram[data, 30, "PDF"]]

I want to replace “2” in the first code line to some function f(t)f(t) and I understand I’ve got to calculate the boundary at times f(ti)f(ti) and then check the condition X(ti)?f(ti)X(ti)?f(ti) and if it’s true, titi is a hitting time for this trajectory, but what should I do in code?

Thanks for help and I’m sorry for bad English.

2

Dear Anonymous (nice name!),

Can you explain what you want more clearly? I am not sure I understand. If you want to change the threshold to some f[t]^2, you can do that with:

ff[t_] := 2 + .1*t
sample = Map[FirstCase[#, _?(Last[#] >= ff[First[#]]^2 &)] &, 
  RandomFunction[WienerProcess[1, 1], {0, 15, 0.01}, 5]["Paths"]]

(shown for 5 paths instead of 10^4). I do not understand why you have f(t)f(t) and titi-- do you mean f(t)^2 and ti^2 or are those typos? Above I assumed you meant f(t)^2. What is X(ti)? I don’t understand how you want to use this X(ti)>=f(ti)X(ti)>=f(ti)

3

Oh, sorry, I copied my question from another site (and there I used TeX). I mean I want to find a first passage time distribution with moving boundary (and boundary is some function f(t)) and I don’t know how to do it. In example from help f(t) is just a constant “2”, but I want, for example, a + \sqrt t . And I understand I’ve got to calculate boundary at each step $f({t_i})$ and if at some step stochastic process $X({t_i}) > f({t_i})$ then f({t_i}) is a hitting time.

4

I think this is what you want

ff[t_] := 2 + Sqrt[t]
sample = Map[FirstCase[#, _?(Last[#] >= ff[First[#]] &)] &, 
  RandomFunction[WienerProcess[1, 1], {0, 15, 0.01}, 500]["Paths"]]

data0 = DeleteCases[sample, Missing[_]][[All, 1]]
data = Cases[data0, _?Positive];

Histogram[data, 30, "PDF"]

Regards

5

Thank you, it works!