09Stoch7&8

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

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

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