First Passage Time Calculations

Transcription

1 First Passage Time Calculations Friday, April 24, :01 PM Homework 4 will be posted over the weekend; due Wednesday, May 13 at 5 PM. We'll now develop some framework for calculating properties of when and where the solution to a stochastic differential equation first leaves some ``domain.'' All this theory actually generalizes to Markov processes in general (those for which the noise is independent of the past, given the current state). The key random variable here is the first passage/escape/exit time The location where the first passage/escape/exit is made is There are two general classes of methods for computing properties of these two random variables. The first class of methods, which is often favored by physicists, is to set up an analogy between the probability density of the state variable and a mass density. Take the Fokker-Planck equation and write it in conservation form AppSDE15 Page 1

2 To determine when and where the state variable leaves the domain D, make the domain boundary an absorbing boundary. That is, when one writes down the Fokker-Planck equation, put a Dirichlet boundary condition on the domain. By solving this Fokker-Planck equation, one can recover several interesting statistical AppSDE15 Page 2

3 quantities: The probability that the state has not left the domain by time is: From that, you can recover the full first passage time distribution, i.e., And can be recovered from solving the FPE for and combining it with the drift and diffusion coefficients. If you care about where on the boundary the state is exiting, then one can use a related concept: This framework is useful, but takes some work to make fully rigorous. There is another AppSDE15 Page 3

4 framework that is more closely connected to the mathematics, and also has the virtue of giving simpler equations to solve for certain classes of first passage time problems. Note that the above approach requires the solution of the time-dependent Fokker-Planck equation, so even if the state is one-dimensional, one has to solve a 1+1 dimensional PDE, for which exact solutions are possible only for simple drift and diffusion coefficients. Here are the kinds of questions for which simpler equations can be developed: 1) 2) If you don't need the full probability density for the first passage time through, but rather just some low order statistics, such as and. If you don't care about the joint distribution of first passage time and first passage location, but only desire to know where the state leaves the domain, when it leaves the domain, i.e.,. Some applications where such questions are of interest: Literal escape problems, i.e. ecology or predator-prey. McKenzie, Lewis, and Merrill, "First Passage Time Analysis of Animal Movement and Insights into the Functional Response" Physical chemistry, where one is interested in computing the amount of time required for some molecules to reach some state where a reaction or binding event is possible. Also molecular motors. Neuroscience, time until a neuron reaches its threshold firing voltage. Surface diffusion Finance, for pricing options that are triggered by special events The analysis of first passage time problems relies on the fact that the first passage time is a Markov time (aka stopping time). What this means is that a Markov time is known to occur when it occurs. Counterexample: Last passage time is not a Markov time. Dynkin's Formula Start by writing out Ito's lemma for a general nice function solution to an SDE: and a AppSDE15 Page 4

5 Now let the upper time t be random, i.e.. AppSDE15 Page 5

6 Can't quite say this is zero, just because it is the average of a Ito stochastic integral, because the upper limit is random. To make progress, we use a standard trick for switching a random limit with a random factor in the integrand. This is a random indicator function. So then we obtain the Dynkin formula for Markov times : AppSDE15 Page 6

7 This operator is known as the infinitesmal generator, and is the adjoint of the Fokker-Planck operator Useful formulas for first passage questions arise by making suitable artistic simplifications to Dynkin's formula. These simplifications only work for the special case in which the drift and diffusion coefficients are time-independent (autonomous SDE): First, let's suppose that the function satisfied the following boundary value problem: This is an elliptic boundary value problem which would like a Poisson equation for the cases where the SDE is just ordinary Brownian motion. It is a deterministic problem. Apply Dynkin's formula to this function Markov time satisfying that BVP, and the AppSDE15 Page 7

8 Now suppose we can find a function on D such that the following boundary value problem is satisfied: This is a generalized Laplace equation. Then Dynkin's formula simplifies to: AppSDE15 Page 8

9 The summary of this derivation gives us the following equations for computing some basic first passage properties regarding 1) The mean first passage time, starting from initial state satisfies: where the deterministic function f satisfies the following deterministic (elliptic) boundary value problem: 2) The probability that the first passage through the boundary occurs over some subdomain starting from an initial state is given by the following formula: where is a deterministic function which satisfies the following deterministic boundary value problem: Here is the infinitesmal generator of the Markov process. For an SDE model: the infinitesmal generator is: AppSDE15 Page 9

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