09Stoch7&8

# 09Stoch7&8 - 2.3 SOME BASIC PROPERTIES 65 2.3 Some...

This preview shows pages 1–4. Sign up to view the full content.

2.3. SOME BASIC PROPERTIES 65 2.3 Some basic properties Let f : [0 ,t ] R be a real-valued function, we say that f is of bounded variation if V ( f ) = sup P n X i =1 | f ( t i ) - f ( t i - 1 ) | < . where the supremum is taken over all partition P = { 0 = t 1 < t 2 < ... < t n = t } of [0,t]. It is known that if f is of bounded variation, then f is diﬀerentiable a.e. A function f is said to have quadratic variation if the limit lim ||P||→ 0 n X i =1 | f ( t i ) - f ( t i - 1 ) | 2 exists , where ||P|| = max i {| t i - t i - 1 |} . The following shows that bounded variation and bounded quadratic variation are two non-compatible conditions. Proposition 2.3.1. If f : [0 ,t ] R is continuous and is of bounded variation, then f has zero quadratic variation. Proof . Observe that for any P = { 0 = t 1 < ... < t n = t } , n X i =1 | f ( t i ) - f ( t i - 1 ) | 2 max i | f ( t i ) - f ( t i - 1 ) | · V ( f ) By the uniform continuity of f , the above expression tends to 0 as ||P|| tends to 0. / Theorem 2.3.2. Let [ B ]( t ) denote the quadratic variation of B t , then [ B ]( t ) = t a.e. Proof Let δ n = ||P n || and satisﬁes n =1 δ n < . For a partition P n with ||P n || ≤ δ n , let T n = n k X i =1 | B ( t i ) - B ( t i - 1 ) | 2 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
66 CHAPTER 2. BROWNIAN MOTION Then the expectation E ( T n ) = E ( n k X i =1 | B ( t i ) - B ( t i - 1 ) | 2 ) = n k X i =1 ( t i - t i - 1 ) = t. (2.3.1) We claim that X n =1 E ( T n - E ( T n )) 2 = X n =1 Var( T n ) < a.e. It follows that E (∑ n =1 ( T n - E ( T n )) 2 ) < . Hence n =1 ( T n - E ( T n )) 2 < a.e., and lim n →∞ ( T n - E ( T n )) = 0 . This together with (2.3.1) implies that [ B ]( t ) = lim n →∞ E ( T n ) = t a.e. The claim follows from Var( T n ) = Var( n k X i =1 | B ( t i ) - B ( t i - 1 ) | 2 ) = n k X i =1 Var( | B ( t i ) - B ( t i - 1 ) | 2 ) = n k X i =1 E (( B ( t i ) - B ( t i - 1 )) 4 ) = n k X i =1 3 · ( t i - t i - 1 ) 2 3 ||P n || · n k X i =1 ( t i - t i - 1 ) 3 n and n =1 Var( T n ) 3 t n =1 δ n < . Recall that { X t } t 0 is a martingale if E ( | X t | ) < and for any s > 0 E ( X t + s | F t ) = X ( t ) a.e. Here {F t } t is ﬁltration (right continuous sub- σ -ﬁeld) generated by { X r : 0 r t } .
2.3. SOME BASIC PROPERTIES 67 Theorem 2.3.3. The following processes are martingales: (i) { B t } t 0 ; (ii) { B 2 t - t } t 0 ; (iii) { e ξB t - ξ 2 2 t } t 0 . Proof . The proof depends on the independence of B t + s - B t and B r , 0 r t , and also E ( g ( B t + s - B t ) | F t ) = E ( g ( B t + s - B t )) where g is a Borel measurable function. (i) Since B t N (0 ,t ), E ( | B t | ) < . By independence, E ( B t + s |F t ) = E ( B t + ( B t + s - B t ) |F t ) = E ( B t |F t ) + E ( B t + s - B t | F t ) = B t + B ( B t + s - B t ) = B t . (ii) Note that

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 05/23/2010 for the course MATH 987 taught by Professor Cheung,cecilia during the Spring '08 term at CUHK.

### Page1 / 17

09Stoch7&8 - 2.3 SOME BASIC PROPERTIES 65 2.3 Some...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online