Page 334 - Applied Probability
P. 334

15. Diffusion Processes
                                                                                            323
                              and setting
                                                              v(x) − v(c)
                                                     u(x)=
                                                                                         (15.10)
                                                                        .
                                                              v(d) − v(c)
                              Here the lower limit of integration l can be any point in the interval [c, d].
                              This particular solution also satisfies the boundary conditions.
                              Example 15.4.1 Fixation Probabilities in the Neutral Model
                              In the diffusion approximation to the neutral Wright-Fisher model with
                              constant population size N, we calculate
                                                        x
                                                           "  y
                                             v(x)=       e −  l  0 dz dy = x − l.
                                                       l
                              Thus, starting at a frequency of x for allele A 1 , allele A 2 goes extinct before
                              allele A 1 with probability
                                                           x − l − (c − l)
                                           u(x) =     lim                = x.
                                                    c→0,d→1 d − l − (c − l)
                              This example is typical in the sense that u(x)= (x − c)/(d − c) for any
                              diffusion process with µ(x)= 0.
                                Another important problem is to calculate the expectation

                                                   w(x)  =  E[g(T) | X 0 = x]

                                                                                      n
                              of a function of the exit time T from [c, d]. For instance, g(t)= t gives the
                              nth moment of T, and g(t)= e −θt  gives the Laplace transform of T.We
                              again derive an ordinary differential equation determining w(x), but now
                              the pertinent boundary conditions are w(c)= w(d)= g(0). To emphasize
                              the dependence of T on the initial position x, let us write T x in place of T.
                                We commence our derivation with the expansion

                                 w(x)  =  E[g(T x ) | X 0 = x]
                                                  + s) | X 0 = x]+ o(s)
                                       =E[g(T X s

                                       =E[g(T X s  )+ g (T X s )s | X 0 = x]+ o(s)

                                       =E{E[g(T X s  ) | X s ] | X 0 = x} +E[g (T X s ) | X 0 = x]s + o(s)
                                       =E[w(X s ) | X 0 = x]+ E[g (T x ) | X 0 = x]s + o(s).

                              Employing the same reasoning used in deriving the differential equation
                              (15.9) for u(x), we deduce that

                                                                          1
                                                                            2

                                 E[w(X s ) | X 0 = x]  = w(x)+ µ(x)w (x)s + σ (x)w (x)s + o(s).

                                                                          2
   329   330   331   332   333   334   335   336   337   338   339