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Math 227C/CS 285 Problem Set 3
Consider the problem of determining the probablity that the solution of
dx = f(x)dt + g(x)dw x(0) ∈ S1
leaves an open connected set S ⊃ S1 before time t. Such problems arise in the analysis of the life expectancy of a machine, the time to financial ruin, etc, the evolution time to the fixation of mutations in a population, etc. One way to formulate this is to consider a modified process which satisifies the given equation as long as x ∈ S and satisfies dx = 0 once x reaches the boundary of S. The corresponding Kolmogorov forward equation in S is
∂ρ(x, t) ∂t = − Xn i=1 ∂ ∂xi fi(x)ρ(x, t) +Xn i=1 Xn j=1 1 2 ∂ 2 ∂xi∂xj gi(x)gj (x)ρ(x, t)
1. What would be an approriate boundary condition for ρ(x, t) on the modified process?
2. Let T = inf{t : x(t) ∈/ S}, which is often called exit time. Write down the probability distribution of exit times in term of ρ(x, t).
3. Now consider an example of one-dimensional process dx = −xdt + dw x(0) = 0.
We want to know the probability that x(t) has not left the interval [−π, π] over the period [0, t]. Write down the Kolmogorov forward equation with boundary conditions.
4. Solve the above Kolmogorov equation and find out the probablity distribution of exit times.