Lecture 13: Martingales

Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of independent random variables and related models 1.5 An exponential martingale 1.6 Likelihood ratios 3. Square Integrable Martingales 2.1 Doob s martingale 2.2 Doob decomposition 2.3 Doob decomposition for square integrable martingales 2.4 Doob-Kolmogorov inequality 4. Convergence of Martingales 3.1 A.s. convergence of martingales 3.2 Law of large numbers for martingales 3.3 Central limit theorem for martingales 5. Stopping times 4.1 Definition and basic properties of stopping times 4.2 Stopped martingales 1

4.3 Optional stopping theorems 4.4 Wald equation 4.5 A fair game example 1. Definition of a Martingale 1.1 Filtrations < Ω, F, P > be a probability space; S = {S 0, S 1,...} is a finite or infinite sequence of random variables defined on this probability space. F = {F 0, F 1,...} is a finite or infinite sequence of σ-algebras such that F 0, F 1,... F. (1) If FN = {F 0, F 1,..., F N } is a finite sequence of σ-algebras, then the infinite sequence of σ-algebras F = {F 0, F 1,..., F N, F N,...} = {F n N, n = 0, 1,...} is called a natural continuation of the initial finite sequence of σ-algebras. (2) If SN = {S 0, S 1,..., S N } is a finite sequence of random variables, then the infinite sequence of random variable S = {S 0, S 1,..., S N, S N,...} = {S n N, n = 0, 1,...} is called a natural continuation of the initial finite sequence of random variables. The continuation construction described above us restrict consideration by infinite sequences of random variables and σ- algebras. Definition 13.1. A sequence of σ-algebras F = {F 0, F 1,...} is a filtration if it is a nondecreasing sequence, i.e., F 0 F 1 2

. Examples (1) F 0 = F 1 = F 2 =. Two extreme cases are, where F 0 = {, Ω} or F 0 = F. (2) S = {S 0, S 1,...} is a sequence of random variables, and F n = σ(s k, k = 0,..., n), n = 0, 1,.... Then, F = {F0, F 1,...} is a filtration. It is called the natural filtration generated by the sequence S. Note that in this case the random variable S n is F n -measurable for every n = 0, 1,.... (3) S = {S 0, S 1,...} and X = { X 0, X 1,...} are, respectively, a sequence of random variables and a sequence of random vectors, and F n = σ(s k, X k, k = 0,..., n), n = 0, 1,.... Then, F = {F 0, F 1,...} is a filtration. Note that in this case the random variable S n is also F n -measurable for every n = 0, 1,.... (4) If F 0 F 1,, F N is a finite nondecreasing sequence of σ-algebras, then its natural continuation F = {F n N, n = 0, 1,...} is also a nondecreasing sequence of σ-algebras, i.e., it is a filtration. (5) If F n = σ(s k, k = 0,..., n), n = 0, 1,..., N is the finite sequence of σ-algebras generated by a finite sequence of random variables S 0,..., S N }, then F = {F n N, n = 0, 1,...} is a natural filtration for the infinite sequence of random variables S = S n N, n = 0, 1,...}. 3

1.2 Definition of a martingale and its basic properties < Ω, F, P > be a probability, S = {S0, S 1,...} is a sequence of random variables defined on this probability space, and F = {F 0, F 1,...} is a filtration at this probability space. Definition 13.2. A sequence of random variables S = {S 0, S 1,...} is F-adapted to a filtration F = {F 0, F 1,...} if the random variable S n is F n -measurable (i.e., event {S n B} F n, B B 1 ) for every n = 0, 1,.... Definition 13.3. A F-adapted sequence of random variables S = {S 0, S 1,...} is a F-martingale (martingale with respect to a filtration F) if it satisfies the following conditions: (a) E S n <, n = 0, 1,...; (b) E(S n+1 /F n ) = S n, n = 0, 1,.... (1) Condition (b) can be written in the equivalent form E(S n+1 S n /F n ) = 0, n = 0, 1,.... (2) A F-adapted sequence of random variables S = {S 0, S 1,...} is F-submartingale or F-supermartingale if (a) E S n <, n = 0, 1,... and, respectively (b ) E(S n+1 /F n ) S n, n = 0, 1,... or (b ) E(S n+1 /F n ) S n, n = 0, 1,.... (3) If S = {S 0, S 1,...} is a martingale with respect to a filtration F = {F n = σ(x 0,..., X n ), n = 0, 1,...} generated by the sequence of random variables X == {X 0, X 1,...}, then the notation E(S n+1 /X 0,..., X n ) = E(S n+1 /F n ) and one can speak about S as the martingale with respect to the sequence of ran- 4

dom variable X. (4) If S = {S 0, S 1,...} is a martingale with respect to a natural filtration F = {F n = σ(s 0,..., S n ), n = 0, 1,...} then the notation E n S n+m = E(S n+m /F n ) may be used and one can refer to the sequence S as a martingale without a specification of the corresponding filtration. The martingales possess the following basic properties: 1. If S = {S 0, S 1,...} is F-martingale, then E(S n+m /F n ) = S n, 0 n < n + m <. 2. If S = {S 0, S 1,...} is F-martingale, then ES n = ES 0, n = 0, 1,.... (a) E(S n+m /F n ) = E(E(S n+m /F n+m 1 )/F n ) = E(S n+m 1 /F n ); (b) Iterate this relation to get the above formula. ES n = E(E(S n /F 0 )) = ES 0, n = 0, 1,.... 3. If S = {S 0, S 1,...} and S = {S 0, S 1,...} are two Fmartingales, and a, b R 1, then the sequence S = {S 0 = as 0 + bs 0, S 1 = as 1 + bs 1,...} is also a F-martingale. E(S n+1 /F n ) = E(aS n+1 + bs n+1/f n ) 5

4. If S = {S 0, S 1,...} is F-martingale and ES 2 n <, n = 0, 1,..., then this sequence is non-decreasing, i.e., ES 2 n ES 2 n+1, n = 0, 1,.... = ae(s n+1/f n ) + be(s n+1/f n ) = as n + bs n = S n, n = 0, 1,.... (a) ES n (S n+1 S n ) = E(E(S n (S n+1 S n )/F n )) = E(S n E(S n+1 S n )/F n )) = E(S n 0) = 0; 1.3 Sums of independent random variables and related models (1) Let X 1, X 2,... is a sequence of independent random variables and S = {S n, n = 0, 1,...}, where S n = S 0 + X 1 + + X n, n = 0, 1,..., S 0 = const. Let also F = {F n, n = 0, 1,...}, where F n = σ(x 1,..., X n ) = σ(s 0, S 1,..., S n ), n = 0, 1,..., F 0 = {, Ω}. In this case, the sequence S is a F martingale if and only if EX n = 0, n = 1, 2,.... (b) 0 E(S n+1 S n ) 2 = ES 2 n+1 2ES n+1 S n + ES 2 n = ES 2 n+1 ES 2 n 2ES n (S n+1 S n ) = ES 2 n+1 ES 2 n. (a) In this case, the sequence S is F-adapted; 6

(b) E(S n+1 S n /F n ) = E(X n+1 /F n ) = EX n+1, n = 0, 1,.... (2) Let X n, n = 1, 2,... is a sequence of independent random variables taking values +1 and 1 with probabilities p n and q n = 1 p n, respectively. In this case, EX n = p n q n and, therefore, the following compensated sequence is a F-martingale, where S n = S 0 + n X k A n, n = 0, 1,..., k=1 A n = n (p k q k ), n = 0, 1,.... k=1 1.4 Products of independent random variables and related models (1) Let X 1, X 2,... is a sequence of independent random variables and S = {S n, n = 0, 1,...}, where S n = S 0 n k=1 X k, n = 0, 1,..., S 0 = const. Let also F = {F n, n = 0, 1,...}, where F n = σ(x 1,..., X n ) = σ(s 0, S 1,..., S n ), n = 0, 1,..., F 0 = {, Ω}. In this case, the sequence S is a F martingale if EX n = 1, n = 1, 2,.... (a) In this case, the sequence S is F-adapted; 7

(b) S n+1 S n = S n (X n+1 1), n = 0, 1,...; (c) E(S n+1 S n /F n ) = E(S n (X n+1 1)/F n ) = S n E(X n+1 1/F n ) = S n (EX n+1 1), n = 0, 1,.... (d) If random variables X n are a.s. positive, for example if X n = e Y n, n = 1, 2,..., and S 0 > 0, then a.s. S n > 0, n = 0, 1,.... In this case condition EX n = 1, n = 1, 2,... is also necessary condition for the sequence S to be a F-martingale. (2) S n = S 0 exp{ n k=1 Y k }, n = 0, 1,..., where S 0 = const and Y k = N(µ k, σk), 2 k = 1, 2,... are independent normal random variables. In this case, E exp Y n = e µ n+ 1 2 σ2 n, n = 1, 2,... and therefore, the following condition is necessary and sufficient condition under which the sequence S = {S n, n = 0, 1,...} is a F-martingale, µ n + 1 2 σ2 n = 0, n = 1, 2,.... (3) S n = S 0 ( q p ) n k=1 Y k, n = 0, 1,..., where S 0 = const and Y 1, Y 2,... is a sequence of Bernoulli random variables taking values 1 and 1 with probabilities, respectively, p and q = 1 p, where 0 < p < 1. In this case, E( q p )Y n = ( q p )p + ( q p ) 1 q = 1, n = 1, 2,... and, therefore, the sequence S = {S n, n = 0, 1,...} is a F-martingale. 1.5 An exponential martingale Let X 1, X 2,... are i.i.d. random variables such that ψ(t) = Ee tx 1 < for some t > 0. Let also Y n = X 1 + + X n, n = 8

1, 2,... and S n = ety n ψ(t) n = n k=1 Let also F = {F n, n = 0, 1,...}, where e tx n ψ(t), n = 1, 2,..., S 0 = 1. F n = σ(x 1,..., X n ) = σ(s 0, S 1,..., S n ), n = 0, 1,..., F 0 = {, Ω}. In this case, S = {S0, S 1,...} is a F-martingale (known as an exponential martingale). (a) S = {S 0, S 1,...} is a F-adapted sequence; e (b) S n+1 = S txn n ψ(t), n = 0, 1,... e (c) E(S n+1 /F n ) = E(S txn n ψ(t) /F n) = S n E( etxn ψ(t) /F n)) = S n 1 = S n 1.6 Likelihood ratios Let X 1, X 2,... be i.i.d. random variables and let f 0 (x) and f 1 (x) are different probability density functions. For simplicity assume that f 0 (x) > 0, x R 1. Let us define the so-called likelihood ratios, S n = f 1(X 1 )f 1 (X 2 ) f 1 (X n ) f 0 (X 1 )f 0 (X 2 ) f 0 (X n ), n = 1, 2,..., S 0 = 1. Let also the filtration F = {F 0, F 1,...}, where F n = σ(x 1,..., X n ), n = 0, 1,..., F 0 = {, Ω}. (1) S = {S 0, S 1,...} is F-adapted sequence of random variables since, S n is a non-random Borel function of random variables X 1,..., X n. 9

(2) Due to independence of random variables X n, we get f E(S n+1 /F n ) = E(S 1 (X n+1 ) n f 0 (X n+1 ) /F n) = S n E f 1(X n+1 ) f 0 (X n+1 ), n = 0, 1,.... (3) Thus, the sequence S = {S 0, S 1,...} is a F-martingale under hypopiesis that the common probability density functions for random variable X n is f 0 (x). Indeed, E f 1(X n+1 ) f 0 (X n+1 ) = f 1 (x) f 0 (x) f 0(x)dx = f 1 (x)dx = 1, n = 0, 1,.... 2. Square Integrable Martingales 2.1 Doob s martingale Definition 13.4. Let S ba a random variable such that E S < and F = {F 0, F 1,...} is a filtration. In this case the following sequence is a F-martingale (Doob s martingale) S n = E(S/F n ), n = 0, 1,.... E(S n+1 /F n ) = E(E(S/F n+1 )/F n ) = E(S/F n ) = S n, n = 0, 1,.... 2.2 Doob decomposition Definition 13.5. A sequence of random variables A = {A n, n = 0, 1,...} is a predictable with respect to a filtration F = {F 0, F 1,...} ( F-predictable sequence) if A 0 = 0 and the random variable A n is F n 1 -measurable for every n = 1, 2,.... Theorem 13.1 (Doob decomposition). Let S = {S 0, S 1,...} be a sequence of random variables adapted to a filtration F = 10

{F 0, F 1,...} and such that E S n <, n = 0, 1,.... Then, S can be uniquely decomposed in the sum of two sequences M = {M 0, M 1,...}, which is a F-martingale, and A = {A 0, A 1,...}, which is a F-predictable sequence, S n = M n + A n, n = 0, 1,.... (a) Let define M 0 = S 0, A 0 = 0 and M n = M 0 + n (S k E(S k /F k 1 )), A n = S n M n, n = 1, 2,.... k=1 (b) M n is F n -measurable since S k, k = 0, 1,..., n and E(S k /F k 1 ), k = 1,..., n are F n -measurable. (c) E(M n+1 /F n ) = M 0 + n+1 k=1 E((S k E(S k /F k 1 ))/F n ) = M 0 + n k=1 E((S k E(S k /F k 1 ))/F n ) +E((S n+1 E(S n+1 /F n ))/F n ) = M n + 0 = M n. (d) A n = S n S 0 n k=1 (S k E(S k /F k 1 )) = n k=1 E(S k /F k 1 ) n 1 k=0 S k is a F n 1 -measurable random variable. (d) Note also that A n+1 A n = E(S n+1 /F n ) S n, n = 0, 1,.... (f) Suppose that S n = M n + A n, n = 0, 1,... is another decomposition in the sum of a F n -martingale and F n -predictable sequence. Then A n+1 A n = E(A n+1 A n/f n ) = E((S n+1 S n ) (M n+1 M n)/f n ) = E(S n+1 /F n ) S n (M n M n) = E(S n+1 /F n ) S n = A n+1 A n, n = 0, 1,.... (g) Since A 0 = A 0 = 0, we get using (e), A n = A 0 + n 1 k=0(a k+1 A k ) 11

= A 0 + n 1 k=0(a k+1 A k) = A n, n = 0, 1,.... (h) M n = S n A n = S n A n = M n, n = 0, 1,.... (1) If S = {S 0, S 1,...} is a submartingale, then A = {A 0, A 1,...} is an a.s. nondecreasing sequence, i.e., P (0 = A 0 A 1 A 2 ) = 1. Indeed, according (d) A n+1 A n = E(S n+1 /F n ) S n 0, n = 0, 1,.... < Ω, F, P > be a probability space; F = {F 0, F 1,...} is a finite or infinite sequence of σ-algebras such that F 0, F 1,... F. S = {S 0, S 1,...} is a F-adapted sequence of random variables defined on this probability space. Lemma 13.1. Let a sequence S is a F-martingale such that E S n 2 <, n = 0, 1,.... Then the sequence S 2 = {S 2 0, S 2 1,...} is a submartingale. 2.3 Doob decomposition for square integrable martingales E(S 2 n+1/f n ) = E((S n + (S n+1 S n )) 2 /F n ) = E(S 2 n + 2S n (S n+1 S n ) + (S n+1 S n ) 2 /F n ) = E(S 2 n/f n ) + 2S n E(S n+1 S n /F n ) + E((S n+1 S n ) 2 /F n ) 12

= S 2 n + E((S n+1 S n ) 2 /F n ) S 2 n, n = 0, 1,.... (1) According Doob decomposition theorem the sequence S 2 can be uniquely decomposed in the sum of a F-martingale M = {M 0, M 1,...}, and a F-predictable sequence A = {A 0, A 1,...}, where S 2 n = M n + A n, n = 0, 1,..., M n = S0 2 + n (Sk 2 E(Sk/F 2 k 1 )), A n = Sn 2 M n. k=1 (2) A n = n k=1 E((S k S k 1 ) 2 /F k 1 ), n = 0, 1,.... A n = S n S 2 0 n k=1 (S 2 k E(S 2 k/f k 1 )) = = n k=1 (E(S 2 k/f k 1 ) S 2 k 1) = n k=1 (E(S 2 k 1 +2S k 1 (S k S k 1 )+(S k S k 1 ) 2 )/F k 1 ) S 2 k 1) = n k=1 E((S k S k 1 ) 2 /F k 1 ). (3) The F-predictable sequence A is a.s. nonnegative and nondecreasing in this case, i.e. P (0 = A 0 A 1 ) = 1. (4) ES 2 n = EM n + EA n = ES 2 0 + EA n, n = 0, 1,.... (5) The sequence {ES 2 0, ES 2 1,...} is nondecreasing, i.e., ES 2 n ES 2 n+1, n = 0, 1,.... 13

(a) EM n = EM 0 = ES 2 0, n = 0, 1,...; (b) 0 = EA 0 EA 1 EA 2 ; (c) ES 2 n = EM n + EA n = ES 2 0 + EA n ES 2 n+1, n = 0, 1,.... Definition 13.6. In this case, the notation S = { S 0, S 1,...} is used for F-predictable sequence A and it is called a quadratic characteristic of the martingale S, i.e., S n = n k=1 E((S k S k 1 ) 2 /F k 1 ), n = 0, 1,.... (6) Since S is a nondecreasing sequence, there exists with probability 1 a finite or infinite limit, Example S = lim n S n. If the martingale S is the sum of independent random variables X 1, X 2,..., i.e., S n = X 1 + + X n, n = 0, 1,... such that E X k 2 <, EX k = 0, k = 1,..., then the quadratic characteristic S n = EX 2 1 + EX 2 n, n = 0, 1,... = V arx 1 + V arx n. The quadratic characteristic is a non-random sequence. 2.4 Doob-Kolmogorov inequality Theorem 13.2 (Doob-Kolmogorov inequality). If S = {S 0, S 1,...} is a square integrable F-martingale, then P ( max 0 k n S n ε) 1 ε 2ES2 n, ε > 0, n = 0, 1,.... 14

(a) A i = { S j ε, j = 1,..., i 1, S i ε}, i = 0, 1,..., n; (b) n i=0a i = {max 0 k n S n ε}; (c) A = Ω \ ( n i=0a i ); (d) ES 2 n = n i=0 ES 2 ni Ai + ES 2 ni A n i=0 ES 2 ni Ai ; (e) E(S n S i )S i I Ai = EE(S n S i )S i I Ai /F i ) = ES i I Ai E(S n S i /F i ) = 0; (f) ES 2 ni Ai = E((S n S i + S i ) 2 I Ai = E(S n S i ) 2 I Ai + 2E(S n S i )S i I Ai + ES 2 i I Ai ES 2 i I Ai ε 2 EI Ai = ε 2 P (A i ); 3. Convergence of Martingales 3.1 A.s. convergence of martingales Theorem 13.3. Let a sequence S is a F-martingale such that ESn 2 < M, n = 0, 1,..., where M = const <. Then there exists a random variable S such that S n a.s. S as n. (g) E(S 2 n) n i=0 ES 2 ni Ai n i=0 ε 2 P (A i ) = ε 2 P (max 0 k n S n ε). (a) Since ES 2 n is bounded and non-decreases in n, we can choose M = lim n ES 2 n; (b) S (m) n = S m+n S m, n = 0, 1,..., for m = 0, 1,...; 15

(c) F n (m) = σ(s (m) k = S m+k S m, k = 0,..., n), n = 0, 1,...; (d) F (m) = {F (m) 0, F (m) 1,...}; (e) F (m) n F m+n, n = 0, 1,...; (f) E(S (m) n+1/f n (m) = E(S m+n /F (m) n (g) {S (m) n ) = E(E(S m+n+1 /F m+n )/F n (m) ) ) = E(S (m) /F (m) ) = S n (m), n = 0, 1,...;, n = 0, 1,...} is a F (m) -martingale; n n (i) E(S m+n S m ) 2 = ES 2 m+n 2ES m+n S m + ES 2 m = ES 2 m+n ES 2 m 2E(S m+n S m )S m = ES 2 m+n ES 2 m. (h) P (max m i m+n S i S m ε) 1 ε 2 (ES 2 m+n ES 2 m). (j) P (max m i< S i S m ε) 1 ε 2 (M ES 2 m); (k) P ( m=0 i=m { S i S m ε} = lim m P (max m i< S i S m ε) lim m 1 ε 2 (M ES 2 m) = 0; (l) P ( S i S m ε for infinitely many i, m) = 0, for any ε > 0; (m) P (ω : lim m max i m S i (ω) S m (ω) = 0) = 1; (n) A non-random sequence a n a as n if and only if max i m a i a m 0 as m. (o) P (ω : lim n S n (ω) = S(ω)) = 1. Theorem 13.4**. Let a sequence S is a F-martingale such that E S n < M, n = 0, 1,..., where M = const <. Then n 16

there exists a random variable S such that Example S n a.s. S as n. As is known the series 1 + 1 2 + 1 3 + diverges the alternating series 1 1 2 + 1 3 converges. Let X 1, X 2,... be i.i.d. random variables taking value +1 and 1 with equal probabilities. Let also, S n = n X i i=1 i, n = 1, 2,.... This sequence is a martingale (with respect to the natural filtration generating by this sequence). Now, ESn 2 = V ars n = n 1 i=1 i 1 2 i=1 <. Thus, a.s. there exists a random variable S such that S i2 n S as n, i.e., the random harmonic series X i i=1 i converges with probability 1. 3.2 Law of large numbers for martingales Theorem 13.5. Let a sequence S is a F-martingale such that ESn 2 a.s. <, n = 0, 1,... and S n. Then, for any nondecreasing function f(x) such that f(x) 1 and 0 f(x) 2 dx <, S n a.s. 0 as n. f( S n ) (a) Y n = n S i S i 1 i=1 f( S i ), n = 1, 2,..., Y 0 = 0; (b) E(Y n+1 Y n /F n ) = E( S n+1 S n f( S n+1 ) /F n) = 1 E( S n f( S n+1 ) /F n) = (c) EY n = n i=1 E S i S i 1 f( S i ) S n f( S n+1 ) S n f( S n+1 ) f( S n+1 ) E(S n+1/f n ) = 0, n = 0, 1,...; = n i=1 EE( S i S i 1 f( S i ) /F i 1) = 0, n 1; (d) Y n+1 Y n = E(Y n+1 Y n ) 2 /F n ) = E( (S n+1 S n ) 2 f( S n+1 ) 2 /F n ) 17

= 1 f( S n+1 ) 2 E(S n+1 S n ) 2 /F n ) = S n+1 S n f( S n+1 ) 2 ; (e) Y n = n 1 S k+1 S k k=0 f( S k+1 ) S k+1 2 S k f(x) 2 dx 0 f(x) 2 dx = M a.s., for n = 1, 2,.... (f) EY 2 n = EY 2 0 + E Y n M, n = 0, 1,.... (g) By Theorem 1, Y n = n i=1 S i S i 1 f( S i ) a.s. Y as n ; (h) Lemma (Kronecker). If a n as n and n x k k=1 a k c as n where c < then 1 nk=1 a n x k 0 as n. 1 (i) ni=1 f( S n ) (S i S i 1 ) = 1 f( S n ) (S n S 0 ) a.s. 0 as n. 1 (j) f( S n ) S n a.s. 0 as n. Example Let X 1, X 2,... be independent random variables such that σ 2 n = V arx n <, EX n = µ n, n = 1, 2,.... Denote b n = nk=1 σ 2 k, a n = n k=1 µ k. Assume that b n as n. Let also, S n = n (X k µ k ), n = 1, 2,..., S 0 = 0. k=1 This sequence is a martingale (with respect to the natural filtration generating by this sequence) and ES n = 0, S n = b n, n = 0, 1,.... Choose, f(x) = max(x, 1). This function is non-decreasing and 0 f(x) 2 dx = 1 + 1 x 2 dx <. 18

(1) By Theorem 2, S n = f(b n) b n b n S n f(b n ) a.s. 1 0 = 0 as n. (2) If a n bn c, then, also, 1 n X k b n k=1 a.s. c as n. (3) If σn 2 = σ 2, EX n = µ, n = 1, 2,..., then b n = nσ 2, a n = µn, n = 1, 2,..., c = µ σ and 2 1 nσ 2 n X k k=1 a.s. c as n. 3.3 Central limit theorem for martingales Theorem 13.6**. Let a sequence S is a F-martingale such that ES 2 n <, n = 0, 1,... and the following condition hold: (1) S n n (2) Then, P σ 2 > 0 as n ; n i=1 E((S i S i 1 ) 2 I( S i S i 1 ε n) n 0 as n, ε > 0. S n nσ d S as n, where S is a standard normal random variable with the mean 0 and the variance 1. 4. Stopping times 19

4.1 Definition and basic properties of stopping times < Ω, F, P > be a probability space; F = {F 0, F 1,...} is a finite or infinite sequence of σ-algebras such that F 0, F 1,... F; T = T (ω) be a random variable defined on a probability space < Ω, F, P > and taking values in the set {0, 1,..., + }; S = {S 0, S 1,...} is a F-adapted sequence of random variables defined on the probability space < Ω, F, P >. Definition 13.7. The random variable T is called a stopping time for filtration F if, {T = n} F n, n = 1, 2,.... (1) The equivalent condition defining a stopping time T is to require that {T n} F n, n = 1, 2,... or {T > n} F n, n = 1, 2,.... (a) Events {T = k} F k F n, k = 0, 1,..., n, thus, event {T n} = n k=0{t = k} F n and in sequel {T > n} F n ; (2) Let H 0, H 1,... be a sequence of Borel subsets of a real line. A hitting time T = min(n 0 : S n H n ) is an example of stopping time. (b) Events {T > n 1} F n 1 F n, thus, {T = n} = {T > n 1} \ {T > n} F n. 20

(3) If T and T are stopping times for a filtration F then T +T, max(t, T ), min(t, T ) are stopping times. {T = n} = {S 0 / H 0,..., S n 1 / H n 1, S n H n } F n, n = 0, 1,.... (a) {T = T + T = n} = n k=0({t = k, T = n k} F n ; (b) {T = max(t, T ) n} = {T n, T n} F n ; (b) {T = min(t, T ) > n} = {T > n, T > n} F n ; 21

4.2 Stopped martingales Theorem 13.7. If S = {S n, n = 0, 1,...} is a F-martingale and T is a stopping time for filtration F, then the sequence S = {S n = S T n, n = 0, 1,...} is also a F-martingale. (a) S T n = n 1 k=0 S k I(T = k) + S n I(T > n 1); (b) E(S n+1/f n ) = S T (n+1) /F n ) = n k=0 E(S k I(T = k)/f n )+ E(S n+1 I(T > n)/f n ) = n k=0 I(T = k)s k + I(T > n)e(s n+1 /F n ) = n 1 k=0 I(T = k)s k + I(T = n)s n + I(T > n)s n = n 1 k=0 I(T = k)s k + S n I(T > n 1) (1) ES n = ES 0, n = 0, 1,... Theorem 13.8. If S = {S n, n = 0, 1,...} is a F-martingale and T is a stopping time for filtration F such that P (T N) = 1 for some integer constant N 0, then = S T n = S n ES n = ES 0 = ES T 0 = ES 0. (a) S T = S T N = S N a.s. (b) ES T = ES N = ES 0. ES T = ES 0. 22

4.3 Optional stopping theorems Theorem 13.9. If S = {S n, n = 0, 1,...} is a F-martingale and T is a stopping time for filtration F such that (1) P (T < ) = 1; (2) E S T < ; (3) E(S n /T > n)p (T > n) = ES n I(T > n) 0 as n. Then, ES T = ES 0. (a) ES T = ES T I(T n) + ES T I(T > n); (b) E S T = k=0 E( S T /T = k)p (T = k) < ; (c) ES T I(T > n) k=n+1 E( S T /T = k)p (T = k) 0 as n ; (d) ES T I(T n) = ES T I(T n) + ES n I(T > n) ES n I(T > n) = ES T n ES n I(T > n); (e) ES n I(T > n) 0 as n ; (f) ES T = lim n ES T n = ES 0. Example Let consider so-called symmetric random walk that is S n = ni=1 X i, n = 0, 1,..., where X 1, X 2,... are i.i.d. Bernoulli random variables taking value 1 and 1 with equal probabilities. 23

In this case, sequence S = {S n, n = 0, 1,...} is a martingale with respect to the natural filtration F generated by random variables X 1, X 2,.... Let a, b be strictly positive integers and T = min(n 1 : S n = a or b). It is a hitting time, and, therefore, a stopping time for the filtration F. (1) P (S T = a) = b a+b. (2) ET = ab. (a) S T can take only two values a or b with probabilities P a = P (S T = a) and 1 P a respectively; (b) Thus, E S T <, i.e., condition (2) of Theorem 13.9 holds. (c) A = {X k = 1, k = 1,... a + b}, B c = { a < c + S k < b, k = 1,... a + b}, a < c < b; (d) P (A) = ( 1 2 )a+b = p > 0; (e) B c A, a < c < b; (f) max a<c<b P (B c ) P (A) 1 p < 1; (g) P (T > a + b) = P (B 0 ) 1 p; (h) P (T > 2(a + b)) = a<c<b P (T > a + b, S a+b = c)p (B c ) (1 p) a<c<b P (T > a + b, S a+b = c) = (1 p)p (T > a + b) (1 p) 2 ; (i) P (T > k(a + b)) (1 p) k, k = 1, 2,...; 24

(j) P (T > n) 0 as n, i.e P (T < ) = 1; (k) Thus, condition (1) of Theorem 13.9 holds. (l) ES n I(T > n) (a + b)ei(t > n) 0 as n ; (m) Thus condition (3) of Theorem 2 also holds; (n) ES T = ap a + b(1 P a ) = ES 0 = 0; (o) P a = b a+b. (p) Consider a random sequence V n = S 2 n n, n = 0, 1,.... The non-random sequence n is a quadratic characteristic for the submartingale S 2 n and V n is a martingale with respect to the natural filtration F generated by random variables X 1, X 2,...; (q) ET < that follows from (i); (r) V T S T 2 + T max(a, b) 2 + T ; (s) E V T max(a, b) 2 + ET < ; (t) EV T I(T > n) (max(a, b) 2 + n)p (T > n) 0 as n ; (u) By Theorem 2, EV T = a 2 P a + b 2 (1 P a ) ET = EV 0 = 0; (w) ET = ab. Theorem 13.10. If S = {S n, n = 0, 1,...} is a F-martingale and T is a stopping time for filtration F such that (1) ET < ; (2) E S n+1 S n /F n ) K <, n = 0, 1,...; Then, ES T = ES 0. (a) Z 0 = S 0, Z n = S n S n 1, n = 1, 2,...; 25

(b) W = Z 0 + + Z T ; (c) S T W ; (d) EW = n=0 nk=0 EZ k I(T = n) = k=0 n=k EZ k I(T = n) = k=0 EZ k I(T k) = E S 0 + k=1 EZ k I(T k); (e) I(T k) is F k 1 -measurable, for k = 1, 2,...; (f) EZ k I(T k) = EE(Z k I(T k)/f k 1 ) = EI(T k)e(z k /F k 1 ) KP (T k); (g) EW E S 0 + k=1 KP (T k) = E S 0 + KET < ; (h) ES T n ES T E S T n S T I(T > n) 2EW I(T > n); (i) k=0 EW I(T = k) = EW < ; (j) EW I(T > n) = k>n EW I(T = k) 0 as n ; (k) ES T n = ES 0 ; (l) It follows from (h) - (k) that ES T = ES 0. 4.4. Wald equation Theorem 13.11 (Wald Equation). Let X 1, X 2,... be i.i.d. random variables such that E X 1 <, EX 1 = µ. Let also Y n = n k=1 X k, n = 1, 2,.... Then, the centered sequence S n = Y n µn, n = 1, 2,..., S 0 = 0 is a martingale with respects to F generated by random variables X 1, X 2,.... Lets also T is a stopping time with respect to filtration F such that ET <. Then, EY T = µet. 26

(a) E( S n+1 S n /F n ) = E X 1 µ 2µ, n = 1, 2,...; (b) S T = Y T µt ; (c) Theorem 4 implies that ES T = ES 0 = 0; (d) EY T µet = ES T = 0. 4.5. A fair game example A gambler flips a fair coin and wins his bet if it comes up heads and losses his bet if it comes up tails. The so-called martingale betting strategy can be viewed as a way to win in a fair gain is a to keep doubling the bet until the gambler eventually wins. (1) Let X n, n = 1, 2,... be independent random variables taking value 2 n 1 and 2 n 1 with equal probabilities. These random variables represent the outcomes of sequential flips of the coin. Thus, S n is the gain of the gambler after n flips. Since EX n = 0, n = 1, 2,..., the sequence S n = X 1 + + X n, n = 1, 2,..., S 0 = 0 is a martingale with respect to the natural filtration generating by the random variables X 1, X 2,.... (2) Let T be the number of flip in which the gambler first time win. By the definition, P (T = n) = ( 1 2 )n, n = 1, 2,.... Then, according the definition of T and the description of martingale betting strategy, S T = 2 T (1 + 2 + + 2 T 1 ) = 1. (3) This, seems, contradicts to optional stopping theorem since T is a stopping time with ET <, and, the equality ES T = 27

ES 0 = 0 may be expected, while according (2) ES T = 1. (4) The explanation of this phenomenon is that conditions of optional stopping Theorems 13.8 13.10 does not hold. (a) Theorem 13.8 can not be implied, since T is not bounded random variable; (b) Theorem 13.9 can not be implied, since E(S n I(T > n)) = (1 + 2 + + 2 n 1 ) 1 2 = 2n 1 n 1 2 0 as n ; n (c) Theorem 13.10 can not be implied, since E( S n+1 S n /F n ) = E X n+1 = 2 n as n. LN Problems 1. Please, characterize a sequence of random variables S = {S 0, S 1,...} which is a martingale with respect to the filtration F = {F 0, F 1,...} for the case where F 0 = F 1 = F 2 =, in particular, if F 0 = {, Ω}. 2. Let X 1, X 2,... be i.i.d random variables, EX 1 = 0, V arx 1 = σ 2 <. Let also S 0 = 0 and S n = n k=1 X k nσ 2. Prove that the sequence S = {S 0, S 1,...} is a martingale with respect to filtration F = {F 0, F 1,...}, where F n = σ(x 1,..., X n ), n = 0, 1,..., F 0 = {, Ω}. 3. Suppose that a sequence S = {S 0, S 1,...} is a martingale and also a predictable sequence with respect to a filtration 28

F = {F 0, F 1,...}. Show that, in this case P (S n = S 0 ) = 1, n = 0, 1,.... 4. Suppose that a sequence S = {S 0, S 1,...} is a martingale with respect to a filtration F = {F 0, F 1,...}. Show that the sequence S = {S 0 = S 2 0, S 1 = S 2 1,...} is a submartingale with respect to a filtration F. 5. Suppose that sequences S = {S 0, S 1,...} and S = {S 0, S 1,...} are submartingales with respect to a filtration F = {F 0, F 1,...}. Show that the sequence S = {S 0 = S 0 S 0, S 1 = S 1 S 1,...} is also a submartingale with respect to a filtration F. 6. Let F = {F 0, F 1,...} is a filtration and let F be a minimal σ-algebra which contains all σ-algebra F n, n = 0, 1,.... Let also S a random variable such that P (S 0) = 1 and ES <. Define, S n = E(S/F n ), n = 0, 1,.... Prove that S n a.s. S as n. 7. Let F = {F 0, F 1,...} is a filtration and let F be a minimal σ-algebra which contains all σ-algebra F n, n = 0, 1,.... Let also S a random variable such that E S <. Prove that E(S/F n ) a.s. E(S/F) as n. 8. Let X 1, X 2,... be a sequence of independent random variables such that EXn 2 = b n <, EX n = a n 0, n = 1, 2,.... Let define random variables S n = n X k k=1 a k, n = 1, 2,..., S 0 = 1. Prove that the sequence S = {S 0, S 1,...} is a martingale with respect to filtration generated by random variables X 1, X 2,... and prove that condition b k k=1 m < implies that there exist 2 k 29

a random variable S such that S n a.s. S as n. 9. Let X 1, X 2,... be a sequence of independent random variables such that EX 2 n = b k <, EX n = 0, n = 1, 2,.... Define S n = nk=1 X k, n =, 1, 2,..., S 0 = 0. Prove that the sequence S = {S 0, S 1,...} is a martingale with respect to filtration generated by random variables X 1, X 2,... and that condition k=1 b k < implies that there exist a random variable S such that S n a.s. S as n. 10. Please, re-formulate Theorem 4 for the case where the random variables S n = X 1 + + X n, n = 1, 2,... are sums of independent random variables and F is the natural filtration generated by the random variables X 1, X 2,.... 11. Let X 1, X 2,... be non-negative i.i.d. random variables such that EX 1 = µ > 0. Let also Y n = n k=1 X k, n = 1, 2,... and T u = min(n 1 : Y n u), u > 0. Prove, that T u is a stopping time such that ET u <, u > 0. 12. Let X 1, X 2,... be non-negative i.i.d. random variables such that EX 1 = µ > 0. Let also Y n = n k=1 X k, n = 1, 2,... and T u = min(n 1 : Y n u), u > 0. Prove, EY Tu = µet u, u > 0. 30