Weak Taylor Approximations
In this chapter we shall use truncated stochastic Taylor expansions as we did in Chapter 10, but now to derive time discrete approximations appropriate for the weak convergence criterion. We shall call the approximations so obtained weak Taylor approximat
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Weak Taylor Approximations
In this chapter we shall use truncated stochastic Taylor expansions as we did in Chapter 10, but now to derive time discrete approximations appropriate for the weak convergence criterion. We shall call the approximations so obtained weak Taylor approximations and shall investigate the corresponding weak Taylor schemes. As with strong approximations, the desired order of weak convergence also determines the truncation that must be used. However, this will be different from the truncation required for strong convergence of the same order, in general involving fewer terms. In the final section we shall state and prove a convergence theorem for general weak Taylor approximations. Throughout the chapter we shall use the notation and abbreviations introduced in Chapter 10.
14.1
The Euler Scheme
We have already considered the Euler scheme in varying degrees of detail in Chapters 9 and 10. The Euler approximation is the simplest weak Taylor approximation and attains the order of weak convergence {J 1.0 under suitable smoothness and growth conditions on the drift and diffusion coefficients, which was first shown by Milstein and Talay. We recall from (10.2.5) that for the general multi-dimensional case d, m 1,2, ... the kth component of the Euler scheme has the form
=
=
m
(1.1)
y'A: n+1 -
y'A: n
j + ak a + '"' L...J bA:J aw '
j=l
with initial value Yo = X o, where
a = Tn+! -
Tn
and
aw; = wt"+1 - wt...
In what follows we shall use the Euler approximation yeS = { yeS(t), t E [0, T]} defined in (10.2.16), that is the solution of the Ito equation formed by freezing the drift and diffusion coefficients at their values at the beginning of each subinterval of the time discretization (T)eS. For notational clarity we shall often omit the superscript 6 from y6 in complicated expressions. The Euler scheme (1.1) corresponds to the truncated Ito-Taylor expansion (5.5.3) which contains only the ordinary time integral and the simple Ito integral. We shall see from a general convergence result for weak Taylor approximations, to be stated in Theorem 14.5.1 of Section 5, that the Euler approximation P. E. Kloeden et al., Numerical Solution of Stochastic Differential Equations © Springer-Verlag Berlin Heidelberg 1992
458
CHAPTER 14. WEAK TAYLOR APPROXIMATIONS
has order of weak convergence P == 1.0 if, amongst other assumptions, a and b are four times continuously differentiable. This means that the Euler scheme (1.1) is the order 1.0 weak Taylor approximation. Exercise 14.1.1
Show that the I-dimensional Ito process X satisfying dXt
3
1
1
1
2
== 2' X t cit + 10 X t dWt + 10 X t dWt
with the initial value Xo == 0.1, where Wl and W 2 are two independent standard Wiener processes, has expectation
E(XT ) = O.lexp
(~T).
For the Ito process X in Exercise 14.1.1 simulate M using the Euler scheme (1.1) with equidistant time steps of step size 6 = ~ == 2- 2 • Determine the 90%-confidence interval for the mean error
PC-Exercise 14.1.2
= 20 batches each of N = 100 trajectories of the Euler appr
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