Page 342 - Applied Probability

15. Diffusion Processes
of special functions
x
1

k −y /2
Φ k (x)
=
√
2π
−
through the integration-by-parts recurrence y e 2 dy 331
1 k−1 −x /2
2
Φ k (x) = −√ x e +(k − 1)Φ k−2 (x) (15.23)
2π
beginning with Φ 0 (x)= Φ(x). We can likewise evaluate the related integrals
x
j
Ψ jk (x) = y Φ k (y) dy
−
via the integration-by-parts reduction
1 j+1 1
Ψ jk (x) = x Φ k (x) − Φ j+k+1 (y). (15.24)
j +1 j +1
Based on the definition of Φ(x), the integral (15.18) becomes
a i,j+1 4

a i+1,k+1
= Φ z − µ ij0 − µ ij1 x 4 (f ij0 + f ij1 x) dx,
p ij→i+1,k
4
σ ij a i+1,k
a ij
and based on the recurrence (15.23), the integral (15.19) becomes
c ij→i+1,k p ij→i+1,k

a i,j+1 4
a i+1,k+1
= (µ ij0 + µ ij1 x)Φ z − µ ij0 − µ ij1 x 4 (f ij0 + f ij1 x) dx
4
σ ij a i+1,k
a ij
a i,j+1 4

σ ij 2 2 a i+1,k+1
ij
− √ e −(z−µ ij0 −µ ij1 x) /(2σ )4 4 (f ij0 + f ij1 x) dx.
2π a i+1,k
a ij
To evaluate the one-dimensional integrals in these expressions for p ij→i+1,k
and c ij→i+1,k p ij→i+1,k , we make appropriate linear changes of variables so
2
that e −x /2 and Φ(x) appear in the integrands and then apply formulas
(15.23) and(15.24) as needed. Although the details are messy, it is clear that
these maneuvers reduce everything to combinations of elementary functions
and the standard normal distribution function.
To summarize, the algorithm presented approximates the probability p ij
and center of probability c ij of each interval [a ij ,a i,j+1 ] of a subdivision
of the range I of X t . Equation (15.22) converts these parameters into a
piecewise-linear approximation to the density of the process in prepara-
tion for propagation to the next subdivision. The actual propagation of
probability from an interval of the current subdivision to another inter-
val of the next subdivision is accomplished by computing p ij→i+1,k and