金融随机分析 II Stochastic process

n
i
bn 0
P 1 (ii) bn E X | F ni i1 0, and 2 n
(iii ) b
EX
n i 1
2 ni
E E X ni | Fi 1
2
0
12
Ref: Hall . P, Heyde C., Martingale limit theory and its applicatiom.1980, Academic Press. Inc
M k M a.s and
M
k
M dP 0 as k
10
Corollary 2.2.9 Let X L P ,N k k 1 be an
1

increasing family of σ-algebras, N k F and define N to be the σ-algebra generated by
h X k 1 k h X k 1 X k (b) for every bounded Borel-measurable function h :
(c) (d)
euX k1 k euX k1 X k , u



(Agreement of Laplace transforms)
2.3 Markov Processes
Definition 2.3.1 Let (, , ) be a filtration under . Let {Xk,k=0,1,…}be a stochastic process on (, , ) . This process is said to be Markov if:
if n is nonnegative then {Xn,n∈N} is a submartingale.
Example 2.2.3 Let n be an independent random process with mean 1, X then {Xn,n∈N} is a martingale.
N k k 1 .

Then
E X | N k E X | N as k , a.e.P and in L1 P .
11
* Weak Law of Large Number
n Let Sn X i , Fn , n 1 be a martingale and {bn} a sequence i 1
eiuXk1 k eiuXk1 X k , u



(Agreement of characteristic functions) 14
• More generally, Conditions (a) – (d) can be stated with the process at time k and multiple future times. (A)
n n
K2
for all m N ,where (of course ) the right hand side has a total of k+m factors. Then there exists a probability space (,F, P) n X : R , and a stochastic process {Xt} on , t s.t.
M t M a.s and
M
t
M dP 0 as t
9
Theorem 2.2.8 Let Mk be a discrete time martingal such that
p sup E M k for some p 1 k
1 Then there exists M L P such that
X t ( );
t X t ( ); t T
The (finite-dimensional) distributions of the
t ,
1
X={Xt}t T are the measures t ,...,t 1 k defined on Rnk k=1,2,..., by process
K1
vt (1) ,...,t ( k ) ( F1 ... F ) t1 , ( F 1 (1)
F （ ) 1 k）
for all permutations on {1,2,...k}
vt1 ,tk (F1 Fk ) vt1 ,,tk ,tk1 ,,tkm (F1 Fk R R )
of positive constants with bn ↑∞ as n→∞. Then, writing P 1 X ni X i I X b ,1 i n , we have that bn Sn 0 as n→∞ if
i n
(i )
P X
i 1 n i 1
X k 1 Bk 1 ,
X k 1 Bk 1 ,
, X k j Bk j k
subsets of Rn for n 1.
Note that for each t fixed we have a random variable
On the other hand, fixing we can consider the function Which is called a path of Xt.
n n i 1 i
Example 2.2.4 Let be a random variable on (, F ,P) , Fn be a filtration on (,F ), then{Xn=E(| Fn), n∈N} is a martingale.
7
PROPERTIES OF MARTINGALE
X • k is adapted to k k 0
be a probability space. k k 0

• The distribution of Xk+1 conditioned on X0, X1, …, Xk is the same as the distribution of Xk+1 conditioned on Xk . That is
THEOREM 2.2.5 Let Mt be a submartingale (resp. martingale). Then E(Mt), as a function of t, is nondereasing. (resp. a constant) THEOREM 2.2.6 Let Xt,Yt be Ft -submartingales (resp. martingales). Then i)for all a≥0,b≥0, aXt +bYt is Ft -submartingale (resp. martingale). ii){ Xt ∨Yt} is Ft -submartingale. iii)Let : R→R a nondereasing convex function ( resp. convex function) such that E[ (Xt)] exists for all t≥0. Then ( Xt) is a submartingale. In particular, when Xt is a martingale and E[ |Xt |p ]<∞ for some p≥1. Then {|Xt |p} is a submartingale.
defined on a probability space (,F, P) and assuming values in Rn , The parameter space T is usually the halfline ,but it may also be an interval ,the non-negative and even
X k 1 B k X k 1 B X k , B ()
13
Different ways to write the Markov property • X k , k is Markov, i.e., (a)
X k 1 B k X k 1 B X k , B ()
Stochastic calculus for finance
2. Stochastic process
1
2.1 Stochastic process Def 2.1.1
A stochastic process X t tT is a
parametrized collection of random variables
,tk
( F1 F2
Fk ) P[ X t1 F1,
, X tk Fk ]; ti T
Here F1, ...,Fk denote Borel sets in Rn. THEOREM (Kolmogor, , t k T , k N let vt ,..., t k be probability 1 measuers on Rnk s.t.
vt1 ,,tk ( F1 Fk ) P[ X t1 F1 ,, X t Fk ],
k
for all ti T and all Borel sets Fi.
2.2 martingales DEFINTION 2.2.1 An n-dimensional stochastic process {Mt}t≥0 on (, F ,P) is called a martingale (resp. submartingale, supermartingale) with respect to a filtration {Ft}t≥0 (and with respect to P0) if (Ⅰ) {Mt} is Ft -adapted (Ⅱ) E[| Mt |]<∞ for all t, and (III) E[Mt | Fs]= Ms(resp. ≥,≤), a.s. , for all s≤t . (Note: If t∈T={0,1,2,….},then {Mt} is a martingale (resp. submartingale, supermartingale) if and only if E[Mk+1 | Fk]= Mk(resp. ≥,≤), a.s. It is clear that any martingale must be both a sub- and a 6 uper- martingale.