Finding the time at which an Ito process satisfies a constraint

Question

I want to run an Ito stochastic process. I have the following parameters

b1b = 0.9;
b3b = .8;
a1b = 0.1;
a3b = 0.2;
eps = 0.1;
G = (1/eps^2)*b1b ; a1 = (1/eps^2)*a1b; a3 = (1/eps^2)*a3b;
xc = Sqrt[a1/a3];
Uc = a1*xc^2/2 - a3*xc^4/4

I want to stop the process when U[x[t]]=U[xc] and y[t]=0 (or at least very close for each of these parameters: say in the vicinity of 10^-4*U[xc] for U[x[t]] and 10^-8 for y[t]). Please note U[x] is the integral of U'[x].

I am trying to write code that simulates dz[t] for many realisations (say 100) and calculates the average of times at which the aforementioned constraint is satisfied (i.e., average of 100 times). However, I have no idea how to incorporate the constraints and extract the time at which the constraints are satisfied.

I tried to use the ItoProcess[] function, but had no fruitful outcome. Any help would be much appreciated.

Answer 1

Not sure I got your process right and not a very elegant solution : setup the process, do the simulations first, then look at the outcome. First off define your constraints :

const[x_, y_] := And[10^-8 <= y <= 10^-3, 0.9*(Uc) <= a1*x^2 - a3*x^4/4 <= 1.1*(Uc)]

I am outputting

{t, x[t], y[t], Boole[const[x[t], y[t]]]}

which contains all the information we need.

Your process :

x0 = 0.35;   (* starting point for x[t] *)
y0 = 0.0005; (* starting point for y[t] *)

proc = ItoProcess[
 {\[DifferentialD]x[t] == y[t] \[DifferentialD]t, 
  \[DifferentialD]y[t] == (-G*y[t] - (a1*x[t] + a3*x[t]^3) - eps*b3b*y[t]^3) \[DifferentialD]t + Sqrt[2*eps*G] \[DifferentialD]w[t]}, 
  {t, x[t], y[t], Boole[const[x[t], y[t]]]}, 
  {{x, y}, {x0, y0}}, {t, 0}, w \[Distributed] WienerProcess[]]

A sample run of the simulation : Q1

SeedRandom[3]
sim = RandomFunction[proc, {0, 1, 0.001}];

ListLinePlot[{sim[[2, 1, 1, All, {1, 2}]], sim[[2, 1, 1, All, {1, 3}]]}] 

Find the first exit time :

First@Select[sim[[2, 1, 1]], #[[4]] == 0 &]
(* {0.001, 0.350001, 0.158739, 0} *)

So for this specific run your process exited at the first step past the initial condition. This is not very surprising since your constraints are quite tight; for instance : Q2

Reduce[const[x, y][[2]], x, Reals]/. Or -> List
(* { -1.37126 <= x <= -1.36069, 
    -0.385397 <= x <= -0.345918, 
     0.345918 <= x <= 0.385397, 
      1.36069 <= x <= 1.37126} *)