smfd12(13a).tex
Day 12. 21.11.2013.
General case: M A(q ). As above,
q
Xt = t +
j tj = (B )t ,
where () = 1 +
j =1
q
j =1
j j .
So formally, if we invert this we obtain
t = (B )1 Xt ,
and as () = 1 + 1 + , 1/() = 1 + c. + . So
Xt = 1 Xt1 + + i Xti + + t ,
smfd1(13a).tex
Day 1. 11.11.2013
I. ESTIMATION OF PARAMETERS
1. PARAMETERS; LIKELIHOOD
To do Statistics to handle the mathematics and data analysis of situations involving randomness we need to model the situation. Here we
conne ourselves to models that c
smfd3(13a).tex
Day 3. 18.11.2013
6. Complements
1. CAPM.
All of this is highly relevant to Mathematical Finance. Finance was an
art rather than a science before the 1952 PhD thesis of Harry MARKOWITZ
(1927-; Nobel Prize 1990). Markowitz gave us two insigh
smfd4.tex
Day 4. 24.10.2013.
3. Likelihood Ratio Tests
We turn now to the general case: composite H0 v. composite H1 . We may
not be able to nd UMP (best) tests. Instead, we seek a general procedure
for nding good tests.
Let be a parameter, H0 be a null h
smfd5.tex
Day 5. 25.10.2013.
5. Multiplication Theorem for Determinants.
If A, B are n n (so AB , and BA, are dened),
|AB | = |A|.|B |.
Proof. We can display a matrix A as a row of its columns, A = [a1 , . . . , an ]
(or as a column of its rows). The k th
smfd6.tex
Day 6. 31.10.2013.
As always, n may be large the larger the better, as large samples are
more informative than small ones. The size of p varies with the problem.
But typically p might be of the order of 10 or 12, say. A 12-dimensional
variable s
smfd7(13a).tex
Day 7. 1.11.2013.
IV. REGRESSION
1. Least Squares
The idea of regression is to take some sample of size n from some unknown
population (typically n is large the larger the better), and seek how best to
represent it in terms of a smaller num
smfd8(13a).tex
Day 8. 7.11.2013.
We turn now to a technical result, which is important in reducing ndimensional problems to one-dimensional ones.
Theorem (Cramr-Wold device). The distribution of a random n-vector
e
X is completely determined by the set of
smfd9(13a).tex
Day 9. 8.11.2013.
The MLE for is x, as this reduces the last term (the only one involving
) to its minimum value, 0. For a square matrix A = (aij ), its determinant
is
aij Aij
|A| =
j
for each i, or
|A| =
aij Aij
i
for each j , expanding by
smfd10(13a).tex
Day 10. 14.11.2013.
V. TIME SERIES (TS).
1. Stationary processes and autocorrelation
A TS - a sequence of observations indexed by time - may well exhibit, on
visual inspection after plotting, a trend - a tendency to increase or decrease
wi
smfd11(13a).tex
Day 11 15.11.2013.
4. General autoregressive processes, AR(p).
Again working with the zero-mean case for simplicity, the extension of
the above to p parameters is the model
Xt = 1 Xt1 + 2 Xt2 + + p Xtp + t ,
()
with (t ) WN as before. Sinc
smfd13(13a).tex
Day 13. 22.11.2013.
10. ARCH and GARCH; Econometrics ([BF, 9.4.1, 220-222)
There are a number of stylised facts in mathematical nance. E.g.:
(i). Financial data show skewness. This is a result of the asymmetry between
prot and loss (large
smfd2(13a).tex
Day 2. 17.11.2013
3. Large-sample properties of maximum-likelihood estimators
We assume the following regularity conditions:
(i) dierentiability under the integral sign twice (as before);
(ii) nite positive Fisher information per reading I