553.633/433 Homework #6

Description

Three problems:
A. For a special case of SDE,
𝑑𝑑𝑑𝑑(t) = λ𝑋𝑋(𝑡𝑡)𝑑𝑑𝑑𝑑 + µ𝑋𝑋(𝑡𝑡)𝑑𝑑𝑊𝑊(𝑡𝑡), 𝑋𝑋(0) = 1
we have the closed form solution:
𝑋𝑋(𝑡𝑡) = e
�λ−µ2
2 �𝑡𝑡+µ𝑊𝑊(𝑡𝑡)
.
Find the following expectations:
(a) 𝐸𝐸[𝑋𝑋(𝑡𝑡)]
(b) 𝐸𝐸 �λ ∫ 𝑋𝑋(𝑠𝑠)𝑑𝑑𝑑𝑑
𝑡𝑡
0 �
(c) 𝐸𝐸 �µ ∫ 𝑋𝑋(𝑠𝑠)𝑑𝑑𝑑𝑑(𝑠𝑠) 𝑡𝑡
0 �
(Hint: 𝐸𝐸 �λ ∫ 𝑋𝑋(𝑠𝑠)𝑑𝑑𝑑𝑑
𝑡𝑡
0 � = λ ∫ 𝐸𝐸[𝑋𝑋(𝑠𝑠)]𝑑𝑑𝑑𝑑
𝑡𝑡
0 .)
B. Consider the Ornstein–Uhlenbeck process:
dX(t) = θ(µ−X(t))dt + σdW(t),
where θ = 1, µ = 20, and σ = 10, and W(t) denotes the (standard) Wiener process.
Simulate the stochastic process X(t) using the Euler−Maruyama method described in the
slide “Numerical Solution of Itô SDE” from the class notes (see also Sect. 4 of Higham,
2001). (That is, use the full E-M form in the class notes as if no solution to the SDE were
available, rather than the solution of the O-U process discussed on the slide “Example 2
of Itô SDE: Ornstein–Uhlenbeck Process” in the class notes.) Run the E-M process 50
separate (statistically independent) times over the time interval [0, 5] with a ∆t = 0.01 and
X(0) = 0 (i.e., carry out the E-M process 50 independent times, each beginning at the
same initial condition X(0) = 0). From these 50 runs, do the following:
(a) Show the first 5 (of 50) solution paths on one plot (one solution path is the sequence
of Xj from the E-M process over all j, representing time from 0 to 5).
(b) Produce a separate line that represents the sample mean of the 50 paths and comment
on how the sample mean differs from its limiting value as a function of t (this line may be
on the plot in part (a) or on a separate plot). It is not necessary to run any formal
statistical (or other) tests for analyzing the difference; a brief “words-only” discussion is
sufficient.
(c) Perform a statistical t-test on whether the (unknown) true mean of the value of Xj
(from the E-M process) that represents X(2) is µ. That is, report a P-value and provide
some brief interpretation. Do the same for X(5). (Note: This part of the problem uses the
basic statistical material from slides 24 and 25 in Chap1_633_handout.pdf; you should be
familiar with that material from course prerequisites. The material will also be briefly
reviewed in class on Monday, 10/9/17.)
C. Consider the setting of Figure 4 in Higham (2001). This problem will be a statistical
test of strong convergence (5.2). Using the linear SDE in (4.5) of Higham and the same
coefficient settings, perform a statistical test of the accuracy of ∆t = 2−9 versus ∆t =
16×2−9 in the E-M method. In particular, compute the P-value from a two-sample
matched-pairs statistical t-test using 100 independent runs of the process at each of the
above two values of ∆t, with each run starting from the same X0 . Each of the 100 values
in the statistical test for each ∆t will be the absolute value of the difference between the
terminal (endpoint) E-M solution and endpoint “truth” as approximated by substituting
the values for the discretized Brownian path (using δt = 16×2−9
) into the exact solution
(4.6), as described in Higham. The P-value in the statistical test will be based on a null
hypothesis of no difference in accuracy. Because this is a matched-pairs test, the exact
solution used in the absolute value of the difference at each pair of ∆t will be the same.
(Summary: In run 1, you compute “truth” via (4.6) and use that truth to compute the two
absolute differences with E-M from the two values of ∆t; then in run 2, you generate a
new independent “truth” and repeat. This is done 100 times to get the data for the
statistical test, comprising two columns of numbers, each column representing the
absolute differences.) You may use the Matlab code of Higham if you wish. Two-sample
statistical tests are described in the file, Spall_ISSO_excerpt_Appendix B.pdf, at the
Materials link of the course page.