V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V

I (..2) (0 < d < + r < u) X 0, X X = 0 S + ( + r)(x 0 0 S 0 ) () X 0 = 0, P (X 0) =, P (X > 0) > 0 0 H, T () X 0 = 0, X (H) = 0 us 0 ( + r) 0 S 0 = 0 S 0 (u r) X (T ) = 0 ds 0 ( + r) 0 S 0 = 0 S 0 (d r) u r > 0, d r < 0, 0 > 0 X (H) > 0, X (T ) < 0 0 = 0 X (H) = 0, X (T ) = 0 0 < 0 X (H) < 0, X (T ) > 0,...20, X 0 = 0, 0, Γ 0 (4 0 +.20Γ 0 ), X = 0 S + Γ 0 (S 5) +.25(4 0 +.20Γ 0 ) H, T, X (H) = 8 0 + 3Γ 0 (5 0 +.5Γ 0 ) = 3 0 +.5Γ 0 X (T ) = 2 0 + 0 (5 0 +.5Γ 0 ) = 3 0.5Γ 0,X (H) = X (T ) X (T ) 0, X (H) > 0 X (H) 0, X (T ) > 0

V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V n+ (T ) (d r) u d = ( + r)v n pv n+ (H) + pv n+ (T ) = pv n+ (H) + qv n+ (T ) pv n+ (H) + pv n+ (T ) = V n+ (T ) (H) = (.2.4) 5 X 2 (HH) = 3.2, X 2 (HT ) = 2.4. 2 (HT ) = (.2.4) X 3 (HT H) = 0, X 3 (HT T ) = 6. ( 4 ). V V (H) = 3 + 8 +.25 ( 4 ) = 3 + 3 V (T ) = 0 + 2 +.25 ( 4 ) = 3..5 = 0.5. 2

2.6875 V 3 (HHH) = 2.6875, V 3 (HHT ) = 5.325, V 3 (HT H) = 2.6875, V 3 (HT T ) = 3.325, V 3 (T HH) = 2.6875, V 3 (T HT ) = 0.6875, V 3 (T T H) = 0.6875, V 3 (T T T ) = 0.825 0. (.2.6),(.2.7) 2 (HH) = 3, 2(HT ) =, 2 (T H) = 3, 2(T T ) =, (H) = 5, (T ) = 3 30, 0 = 3 75. v n (s, y) = 2 5 (v n+(2s, y + 2s) + v n+ (0.5s, y + 0.5s)). v 3 (32, 60) =, v 3 (8, 36) = 5, v 3 (8, 24) = 2, v 3 (2, 8) = 0.5, v 3 (8, 8) = 0.5, v 3 (2, 2) = v 3 (2, 9) = v 3 (0.5, 7.5) = 0, v 2 (6, 28) = 6.4, v 2 (4, 6) =, v 2 (4, 0) = 0.2, v 2 (, 7) = 0, v (8, 2) = 2.96, v (2, 6) = 0.08, v 0 =.26.26 δ n (s, y) = v n+(2s, y + 2s) v n+ ( s 2, y + s 2 ) 2s 2 s V n (ω ω 2... ω n ) = + r n (ω ω 2... ω n ) p n(ω ω 2... ω n )V n+ (ω ω 2... ω n H) + q n (ω ω 2... ω n )V n+ (ω ω 2... ω n T ) p n (ω ω 2... ω n ) = + r n(ω ω 2... ω n ) d n (ω ω 2... ω n ) u n (ω ω 2... ω n ) d n (ω ω 2... ω n ) q n (ω ω 2... ω n ) = u n(ω ω 2... ω n ) r n (ω ω 2... ω n ) u n (ω ω 2... ω n ) d n (ω ω 2... ω n ),.2.2. 3

(.2.7).,. p n = Sn 0 S n S n+0 S n Sn 0 S n = 2 0.5. 9.375 4

I Ω = A A c, A A c = φ = P (Ω) = P (A) + P (A c ) P (A c ) = P (A). P ( m n=) m P (A n ) n=. P ( m+ n= ) = P ( m n=a n ) + P (A m+ ) P (( m n=a n ) A m+ ) m P ( m n=a n ) + P (A m+ ) P (A n ) + P (A m+ ) = m+ n= P (A n ). n= P (S 3 = 32) = 8, P (S 3 = 8) = 3 8, P (S 3 = 2) = 3 8, P (S 3 = 0.5) = 8 ẼS = 8 2 + 2 2 = 5 ẼS 2 = 6 4 + 4 2 4 + 4 = 25 4 ẼS 3 = 32 8 + 8 3 8 + 2 3 8 + 0.5 8 = 25 6 25. 5

P (S 3 = 32) = 8 27, P (S 3 = 8) = 2 27, P (S 3 = 2) = 6 27, P (S 3 = 0.5) = 27 ES = 8 2 3 + 2 3 = 6 ES 2 = 6 4 9 + 4 4 9 + 9 = 9 ES 3 = 32 8 27 + 8 2 27 + 2 6 27 + 0.5 27 = 27 2 50. E n ϕ(m n+ ) ϕ(e n M n+ ) = ϕ(m n ) Jensen, (M n ). n+ E n M n+ = E n X j = M n + E n X n+ = M n + EX n+ = M n j= (M n ). ( E n S n+ = E n e σm 2 ) n+ ( 2 ) n+ n+ = e σmn En e σx n+ e σ + e σ e σ + e σ ( 2 ) n+ ( = e σmn Ee σx n+ 2 ) n+( e σ + e σ ) = e σmn = S e σ + e σ e σ + e σ n 2 (S n ). 6

n 2 = n (M j+ M j ) 2 = 2 2 j=0 j=0 n n Mj+ 2 M j+ M j + n 2 j=0 = M n 2 n 2 + n n Mj 2 M j+ M j + j=0 j=0 j=0 j=0 M 2 j = M n 2 n 2 + M j (M j M j+ ) = M n 2 2 I n I n = 2 M 2 n n 2. j=0 M 2 j E n f(i n+ ) = E n f(i n + M n (M n+ M n )) = E n f(i n + M n X n+ ) h(x, y) = Ef(x + yx n+ ), 2.5.3 E n f(i n + M n X n+ ) = h(i n, M n ). h(i n, M n ) = g(i n ). E n I n+ = I n + E n n (M n+ M n ) = I n + n E n M n+ M n = I n + n EX n+ = I n (I n ). p = 0.5. 7

. E f(x 2 ) = g(x ) f, (f(2) + f(0)) = f() = g() 2. f(x) = x 2,. (). M n = ẼnM = ẼnM = M n Ẽ n V n+ ( + r) n+ = = { Vn }. (+r) n Ẽ n V n+ ( + r) n+ V n ( + r) ẼnV n+ n+ ( + r) pv V n n+(h) + qv n+ n+ (T ) = ( + r) n = Ẽn = Ẽn = = Ẽn+ V ( + r) ( + r) n Ẽn V n ( + r) n { }. (+r) n (iv) V ( + r) V ( + r) n V (+r) = V (+r) V n ( + r) = V n n ( + r), n V n n = V n, n. 8

p(ω ω 2 ) = + r(ω ω 2 ) d(ω ω 2 ) u(ω ω 2 ) d(ω ω 2 ), P H (HH) = 2, P H (HT ) = 2, P T (T H) = 6, P T (T T ) = 5 6 p H (HH) (H). P (HH) = 4, P (HT ) = 4, P (T H) = 2, P (T T ) = 5 2. V (H) =.25 2 5 + 2 = 2 5 V (T ) =.5 6 + 5 6 0 = 9. P (H) = P (T ) = 0.5 V 0 =.25 2 2 5 + 2 = 226 9 225. (iv) 0 = V (H) V (T ) S (H) S (T ) = 03 270 (H) = V 2(HH) V 2 (HT ) S 2 (HH) S 2 (HT ) = X n+ Ẽ n ( + r) n+ = Ẽn n Y n+ S n ( + r) + X n n S n = ns n n+ ( + r) n ( + r) ẼnY n+ n+ + X n n S n ( + r) n = ns n ( + r) n+ pu + qd + X n n S n ( + r) n = X n ( + r) n 9

. V. (.2.6) V n., n (.2.7)., (2.8.2), (.2.4). (2.8.2) (.2.4) X n+ (ω,..., ω n, H) = n us n + ( + r)(x n n S n ) X n+ (ω,..., ω n, T ) = n ds n + ( + r)(x n n S n )..2.2 X n (ω,..., ω n ) = V n (ω,..., ω n ), n, (ω,..., ω n ). (2.8.2) X n., V n 8 V V n = Ẽn ( + r) n. Ẽ n S n+ = ( + r) n+ Ẽn ( An+ )Y n+ S n = ( + r) n+ ( A n+ (H)) pu + ( A n+ (T )) qd + r =,. S n ( + r) n+ ( A n+(h)) pu+( A n+ (T )) qd S (S K) + (K S ) + K + K S = 0 if K > S = S K if K < S 0

. C C n = Ẽn ( + r) n F F 0 = Ẽ0 ( + r) = Ẽ0 = Ẽn F + P = F ( + r) n n + P n S K K = S ( + r) 0 ( + r) (iv) F 0,, ( + r) (F 0 S 0 ) + S = K + S.. (v) n = 0, F 0 = 0 C 0 = F 0 + P 0 = P 0. (vi) S K F n = Ẽn ( + r) n = Ẽn S ( + r) S 0 = S ( + r) n n ( + r)ns 0, n C n = P n. (v) K = ( + r) m S m m K > ( + r) m S m F F m = Ẽm ( + r) m = ( + r) m S K Ẽ m ( + r) ( + r) m K = S m ( + r) < 0 m

C m = F m + P m < P m C m < P m if K > ( + r) m S m C m = P m C m > P m if K = ( + r) m S m if K < ( + r) m S m m S m K 2 3 0 K m m K (+r) m K ( S m > ) K < ( + r) m S (+r) m m S m P m + S m K (+r) m K (+r) m = P m + F m = C m K K ( S m < ) K > ( + r) m S (+r) m m 0 K (v) m 0 Ẽ n f(s n+, Y n+ ) = Ẽnf(S n X n+, Y n + Sn + ),X n+ (H) = u, X n+ (T ) = d. 2.5.3 g(s, y) = Ef(sX n+, y + S n+ ) Ẽnf(S n+, Y n+ ) = g(s n, Y n ). (S n, Y n ). 2

v n (s, y) = + r pv n+(us, y + us) + qv n+ (ds, y + ds) ( y ) v (s, y) = f + n M +.n < M Ẽ n f(s n+, Y n+ ) = Ẽnf(S n+, 0) = g(s n, 0) = g(s n, Y n ).n = M Ẽ M f(s M+, Y M+ ) = ẼMf(S M+, S M+ ) = g(s M, S M ), (S n, Y n ). v n (s, y) = pv +r n+(us, y + us) + qv n+ (ds, y + ds) if n M + v M (s) = +r pv M+(us, us) + qv M+ (ds, ds) if n = M v n (s) = pv(us) + qv(ds) if n < M +r y v (s, y) = f( M ) 3

I (i ) Z(ω) = P (ω) P (ω) > 0, ω Ω. P ( Z > 0) = (ii ) (iii ) Ẽ = P (ω) P (ω) = P (ω) = Z P (ω) ω Ω ω Ω Ẽ Z Y = P (ω) P (ω) Y (ω) P (ω) = P (ω)y (ω) = EY ω Ω ω Ω P (Ω) = ω Ω Z(ω)P (ω) = EZ = ẼY = ω Ω Y (ω) P (ω) = ω Ω Y (ω)(ω)p (ω) = EZY P (A) = ω A P (ω) = 0, P (ω) 0, ω Ω P (ω) = 0, ω A. P (A) = ω A Z(ω)P (ω) = 0 (iv) P (A) = ω A Z(ω)P (ω) = 0, Z(ω)P (ω) 0, ω Ω 4

Z(ω)P (ω) = 0, ω A. Z(ω) > 0 P (ω) = 0 ω A. P (A) = 0. (v) P P P (A) = P (A c ) = 0 P (A c ) = 0 P (A) =. (vi) Z P (Z(ω ) = 3 2 ) = 3, P (Z(ω 2) = 3 2 ) = 3, P (Z(ω 3) = 0) = 3 EZ =, P (Z 0) =. P (ω 3 ) = 0 P (ω 3 ) = 3 0 P P.! (M n ). E n M n+ = E n E n+ S 3 = E n S 3 = M n (M n ). 5

( 3Z3 4 ζ 3 = 5) ζ 3 (HHH) = 27 25, ζ 3(HHT ) = ζ 3 (HT H) = ζ 3 (T HH) = 54 25 ζ 3 (HT T ) = ζ 3 (T HT ) = ζ 3 (T T H) = 08 25, ζ 3(T T T ) = 26 25 V 0 = ω Ω ζ(ω)p (ω)v (ω), V 0 = 27 8 54 4 08 2 26 + (5 + 2 + 0.5) + (0.5 + 0 + 0) + 25 27 25 27 25 27 25 27 0 =.26. (iv) ( 4 ) 2Z2 ζ 2 (HT ) = ζ 2 (T H) = (HT ) = 8 5 25 V 2 (HT ) = V 2 (T H) = ζ 2 (HT ) E 2ζ 3 V 3 (HT ) = 25 2 8( 3 ζ 2 (T H) E 2ζ 3 V 3 (T H) = 25 54 25 2 + 08 ) 3 25 0.5 = 2 54 8( 3 25 2 + 08 ) 3 25 0 = 0.2 Z(HH) = 9 6, Z(HT ) = 9 8, Z(T H) = 3 5, Z(T T ) = 8 4 Z (H) = E Z(H) = 2 9 3 6 + 9 3 8 = 3 4 Z (T ) = E Z(T ) = 2 3 3 8 + 5 3 4 = 3 2 Z 0 = E 0 Z = 2 3 3 4 + 3 3 2 = 6

V (H) = 6 2 9 5( 3 6 5 + 9 ) 3 8 = 2.4 V (T ) = 8 ( 2 3 9 3 8 + 5 ) 3 4 0 = 8 V 0 = 25( 6 4 9 9 6 5 + 2 9 ) 9 8 + 5( 8 2 3 9 8 + 9 3.3.4 X max Elog(X ), s.t.ẽ = X ( + r) 0 5 ) 4 0 = 226 225.. L = ( log(x )(ω) + λ X 0 X (ω) ) ( + r) P (ω)z(ω) ω Ω ω Ω. P (ω) X (ω) λζ (ω)p (ω) = 0 X = λζ.,λ = X 0. X 0 = X ζ. Z X 0 = E n X ( + r). 3.2.6 X Ẽ n = X Z E ( + r) n Z n ( + r), Z X E n X = Z ( + r) n Ẽ n ( + r). X 0 = X n ζ n, n. = Z n X n ( + r) n = X nζ n. L = ( log(x )(ω) + λ X 0 X (ω) ) ( + r) P (ω)z(ω) ω Ω ω Ω 7

X p (ω)p (ω) λp (ω)ζ(ω) = 0. λ = Xp ζ. λ p = X 0 Eζ p p.. X = λ X p ζ p 0 ζ p = Eζ = X0( + r) Z p p EZ p p p U(x) V (x) = U(x) yx. U(x) ( I ), I(y) V (I(y)) = 0. V (x) V (I(y)) + V (I(y))(x I(y)) = V (I(y)) U(x) yx I(y). x = X, y = λz (+r) (3.6.3) U(X ) λzx ( + r) U(X ) λz ( ( + r) I λz ) ( + r). λzx X ( ) = EU(X ) E = EU(X ( + r) ) λẽ = EU(X ( + r) ) λx 0 ( ) = EU(X) λz ( E ( + r) I λz ) = EU(X ( + r) ) λx 0 ( ) (3.3.26). EU(X ) EU(X ) 8

. X X n = Ẽn 0 ( + r) n (0 < y ) γ y(γ x) 0 if 0 x < γ (RHS) (LHS) = U(x) y(γ x) = y(γ x) 0 if x γ (y > ) γ yx 0 (RHS) (LHS) = (U(x) yx) = + yx 0 if 0 x < γ if x γ x = X, y = λz (+r) U(X ) λzx ( + r) U(X ) λz ( ( + r) I λz ) ( + r). U(x) λzx X ( ) = P (X γ) E = P (X ( + r) γ) λẽ = P (X ( + r) γ) λx 0 ( ) = P (X λz ( γ) E ( + r) I λz ) = P (X ( + r) γ) λx 0 P (X γ) P (X γ). (iv) (3.6.4) Z ( X 0 = E ( + r) I λz ) = ( + r) M ζ m I(λζ m )p m = m= K ζ m γp m m= 9

. K ζ K γλ ζ K+. λ K m= ζ m p m = X 0 γ K. (v) X = I(λζ), K m K λζ m γ X (ω m ) = γ. m > K λζ m > γ X (ω m ) = 0 20

I! ". V P 0 = 0.928.! " V C 0 = 2.56. 2

!" # $ % & #!" # V S 0 = 3.296. (iv) V S 0 = 3.296 < V C 0 + V P 0 = 3.488..,,.,., (T ). 0 4, X X (H) = 3 0 + 0.4, X (T ) = 3 0 + 3..36.25 =.7 0 = 3. 30.7.25 = 7 8 6 (H) + 5 4 (.7 8 (H)) = 7 8 4 (H) + 5 4 (.7 8 (H)) = 7 8 4 (T ) + 5 4 (.7 4 (T )) = 7 8 2 (T ) + 5 4 (.7 4 (T )) = 7 8 22

#! (H), (T ). (H) = (T ) = 0.!" 0.4.. T.. (HH), (HT ),,.36. (T H),, 0.43. (T T ),, (T ) =.,.. ()τ(hh) =, τ(ht ) =, τ(t H) =, τ(t T ) = (2)τ(HH) = 2, τ(ht ) =, τ(t H) =, τ(t T ) = (3)τ(HH) =, τ(ht ) = 2, τ(t H) =, τ(t T ) = (4)τ(HH) =, τ(ht ) =, τ(t H) = 2, τ(t T ) = 23

(5)τ(HH) =, τ(ht ) =, τ(t H) =, τ(t T ) = 2 (6)τ(HH) =, τ(ht ) =, τ(t H) = 2, τ(t T ) = 2 (7)τ(HH) =, τ(ht ) = 2, τ(t H) =, τ(t T ) = 2 (8)τ(HH) =, τ(ht ) = 2, τ(t H) = 2, τ(t T ) = (9)τ(HH) = 2, τ(ht ) =, τ(t H) =, τ(t T ) = 2 (0)τ(HH) = 2, τ(ht ) =, τ(t H) = 2, τ(t T ) = ()τ(hh) = 2, τ(ht ) = 2, τ(t H) =, τ(t T ) = (2)τ(HH) =, τ(ht ) = 2, τ(t H) = 2, τ(t T ) = 2 (3)τ(HH) = 2, τ(ht ) =, τ(t H) = 2, τ(t T ) = 2 (4)τ(HH) = 2, τ(ht ) = 2, τ(t H) =, τ(t T ) = 2 (5)τ(HH) = 2, τ(ht ) = 2, τ(t H) = 2, τ(t T ) = (6)τ(HH) = 2, τ(ht ) = 2, τ(t H) = 2, τ(t T ) = 2 (7)τ(HH) =, τ(ht ) =, τ(t H) =, τ(t T ) = (8)τ(HH) =, τ(ht ) =, τ(t H) =, τ(t T ) = 2 (9)τ(HH) =, τ(ht ) =, τ(t H) = 2, τ(t T ) = (20)τ(HH) =, τ(ht ) =, τ(t H) = 2, τ(t T ) = 2 (2)τ(HH) =, τ(ht ) =, τ(t H) =, τ(t T ) = (22)τ(HH) =, τ(ht ) = 2, τ(t H) =, τ(t T ) = (23)τ(HH) = 2, τ(ht ) =, τ(t H) =, τ(t T ) = (24)τ(HH) = 2, τ(ht ) = 2, τ(t H) =, τ(t T ) = (25)τ(HH) =, τ(ht ) =, τ(t H) =, τ(t T ) = (26)τ(HH) = 0, τ(ht ) = 0, τ(t H) = 0, τ(t T ) = 0,,3,4,5,6,7,8,2,2,22,26. ( 4 ) τgτ Ẽ {τ 2} 5 24

() ()0,(3)0.6,(4)0.6,(5)0.64,(6)0.8,(7)0.8,(8)0.32,(2)0.96,(2).2,(22).36,(26) (4.4.6), (22).. K S, + r Ẽ (K S ) = K + r S. K S K + r S (K S ) = rk + r < 0. 2..,.,. (4.8.4). V0 EC K S 0 + ( + r) = V EP 0. τ({k S > 0}) =, τ({k S 0}) =,V EP 0 V AP 0. (4.8.5).. S K. + r Ẽ S K = S K + r 25

. S K + r (S K) = rk + r > 0. 2. + r Ẽ 2 S K = S 2 K + r ( + r) 2. S 2 K (( ( + r) (S 2 2 K) = K ) 2 ) > 0 + r. l + S (l ) + r Ẽ l K ( + r) l l S (l ) S l K ( + r) l K (S ( + r) l l K) = K ( ( ) l ) > 0 + r. l.l = S 0 K ( + r).. 26

I Eα τ 2 = Eα τ 2 τ +τ = Eα τ 2 τ Eα τ = Eα τ 2 Eα τm = Eα P m k=0 (τ k+ τ k ) = Π m k=0 Eατ k+ τ k = Eα τ m {τ k+ τ k } k = 0 m.,. f (σ) = pe σ qe σ = 0, f (σ) > 0 e σ = (q/p) 0.5 f(σ), σ > 0, f(σ) > f(0) =. ( ) nen ( E n S n+ = e σmn e σx n+ ) ( pe σ + qe σ ) = S n = S n f(σ) f(σ) f(σ) = S 0 = ES n τ = E. lim S n τ = n ( e σmn τ ) n τ f(σ) 0 if τ = ( τ e σ f(σ)) if τ < lim E ( ) = E {τ < }e σ τ f(σ) 27

e σ = E ( ) τ {τ < } f(σ).σ 0 lim E = P (τ < ). (iv) α = f(σ) αpeσ + αqe σ = 0 e σ = 4α 2 pq 2αq. 2αq = E 4α 2 pq ατ.. Eα τ = 4α 2 pq 2αq (v) α Eτ α τ = ( ( 4α2 pq) 2 ) 2α 2 q( 4α 2 pq) 2 α Eτ = ( 4pq)0.5 2q( 4pq) 0.5. pe σ + qe σ = (e σ )(pe σ q) = 0 28

σ 0 = log(q/p). ( σ > σ 0 ) σ σ 0 e σ 0 = P (τ < ). P (τ < ) = p/q. α = f(σ) Eα τ = E {τ < }α τ + E {τ = }α τ = E {τ < }α τ = 4α 2 pq 2αq. (iv). (iv) (v) Eα τ E {τ < }τ, E {τ < }τ = ( 4pq)0.5 2q( 4pq) 0.5. Eα τ 2 = k= P (τ 2 = 2k)α 2k = k=. P (τ 2 = 2k) = (2k)! 4 k (k + )!k! ( α 2 ) 2k (2k)! (k + )!k! P (τ 2 2k) = P (M 2k = 2) + 2P (M 2k 4) = P (M 2k = 2) P (M 2k = 0) 29

P (τ 2 = 2k) = P (τ 2 2k) P (τ 2 2k 2) = P (M 2k 2 = 2) + P (M 2k 2 = 0) P (M 2k = 2) P (M 2k = 0) ( ) 2k 2 { (2k 2)! = 2 k!(k 2)! + (2k 2)! } ( 2k { (2k)! (k )!(k )! 2) (k + )!(k )! + (2k)! } k!k! (2k)! = 4 k (k + )!k!, m, M n = b M n = m + (m b) = 2m b.,(b m),. P (M n m, M n = b) = P (M n m, M n = 2m b) = P (M n = 2m b) = n! ( ) ( ) n b n+b + m! m!( 2 2 2 ) n, 2 n.. n! ( ) ( ) p n b 2 +m q n+b 2 m n b n+b + m! m! 2 2 (, ). 2 0.4 = 0.8. 30

0.928. 0.96896. (j 0) ( s v(2s) = 4 2s, v = 4 2) s 2 c(s) = 4 s 4 (4 5 ) 5 4 s = 0.8 (j = ) v(2s) = 4 ( s ) 2s =, v = 4 s 2 2 = 3 c(s) = 4 2 4 ( 5 2 + ) 2 3 = 0.4 3

(j 2) v(2s) = 4 2s = 2 ( s ) s, v = 4 2 s/2 = 8 s c(s) = 4 s 4 ( 5 s + 4 = 0 s) (j 0) δ(s) = 3 2 s 3 2 s = (j ) δ(s) = 2 3 (j 2) δ(s) = 6 s 3 2 s = 4 s 2 s x(s ) x(s ) = δ(s)s + ( + r)(v(s) c(s) δ(s)s) v(s ). (j 0) x(s ) = s + 5 4 (4 s 0.8 + s) = 4 s = v(s ). (j = ) x(s ) = 2 3 s + 5 4 (2 0.4 + 2 3 2) = 2 3 s + 3 = s = 4 3 s = v(s ). (j 2) x(s ) = 4 s 2 s + 5 ( 4 4 s + 4 ) s s = 4s 2 s + 0 2 s = 2/s s = 2s 8/s s = s/2 32

v(s ). v(s+ ) S + Ẽ n = ( + r) + Ẽn = ( + r) n+ v(sn) (+r) n S n ( + r) = v(s n) n ( + r) n.. n Sn K K Ẽ = S ( + r) n 0 ( + r) n. n S 0.. { max s K, } + r pus + qds = max { } s K, s = s = v(s) v(s) = s. (5.4.8). (iv) v(s n ) = S n,. v(s) = s p s p = 2 5 (2s)p + 2 ( s ) p s p (2 p+2p+ 5 ) = 0 5 2 s p > 0 2 p+2p+ 5 = 0 2(2 2p ) 5(2 p ) + 2 = (2 p 2)(2(2 p ) ) = 0 p =,. 33

v(s) = As + B s lim s = 0,A lim s s = 0 A = 0. f B (s) = B s 2 + = 0, f B(s) = 2B s 3 > 0 s = B f B (s). f B (s) = 2 B 4 > 0 B > 4 B > 4 f B (s). B 4 f B (s) f B (s) = 0. (iv) 5.2.3 ( 4 ) τ(4 ( 4 τ v B (S 0 ) = Ẽ sb ) = (4 s B )Ẽ = (4 s B ) 5 5) 2 = (4 2j )2 j 2 j 4. (4 2 j+ )2 j+ (4 2 j )2 j > 2 j < 4 3 v B (s) j =, s B = 2. B = 4. (v) (iv) s B, B v B (s B) = 4 2 2. = v B (s) s = s B 34

E n c X + c 2 Y (ω... ω n ) = I (c X(ω... ω ) + c 2 Y (ω... ω ))P (ω n+...ω ω...ω n ) ω n+...ω = c X(ω... ω )P (ω n+... ω )+ ω n+...ω c 2 Y (ω... ω )P (ω n+... ω ) ω n+...ω = c E n X(ω n+... ω ) + c 2 E n Y (ω n+... ω ) E n XY (ω... ω n ) = X(ω... ω n ) ω n+...ω Y (ω... ω )P (ω n+... ω ω... ω n ) = X(ω... ω n )E n Y (ω... ω n ) E n E m X = E n Z = Z(ω... ω m )P (ω n+... ω ω... ω n ) ω n+...ω = Z(ω... ω m )P (ω... ω m ) P (ω... ω n ) ω n+...ω m = X(ω... ω ) P (ω... ω ) P (ω... ω n ) P (ω ω m+...ω... ω m ) P (ω... ω m ) = E n X ω n+...ω m (v) 2.3.2. n, m.n k Sn B n,m ( Ẽ n S k S ) n B k,m D k = B ẼnS k D k Ẽn S n D k n,m D n Ẽ n D m = S n D n S nd n Ẽ n D m = 0 Ẽ n D m 35 Ẽ k D m D k

,. B n,m B n,m+ = ẼnD m D n ẼnD m+ D n = D n Ẽ n D m+ (+R m ) D m+ = ẼnD m+ R m D n V = ) + Ẽ D 3 (R 2 3 D V (H) = 2 3 2 3 7 3 + 6 7 0 3 = 4 2, V (T ) = 0 (6.2.6), 2 ( 2 ) X = B,2 + 2 B 0,2 = ( 2 + + R 2 ) 4. H, T X = V = 4 3. 3 ( 2 ) ( 2 X = B,3 + 2 B 0,3 = B,3 + 2 4 ) 7. H, T 2 2 V. (T ) 0, (H)., 3 (H) ( 4 ) X 2 = (H)B 2,3 + 2 (H)B,3(H) ( + R (H)) = ( 4 + + R 2 2 4 ) 7 7 6. HH, HT X 2 = V 2 (H) = 2 3. 2 ( 4 ) X 2 = (H)B 2,2 + 2 (H)B,2(H) X 2 = V 2 (H). = (H) + 2 9 7 6 6 7 (H) = 2 9 36

Ẽ m+ k F n,m = = = D k B k,m+ Ẽ k D m+ F n,m Ẽ n D m Ẽ k D m+ Ẽ k D m+ Ẽ n D m+ Ẽ n D m Ẽ k Ẽn D m+ Ẽ k D m+ Ẽ n D m+ = ẼkD m Ẽ k D m+ = F k,m F n,m (6.3.3) F 0,2 = 0.05, F,2 (H) = 0.055, F,2 (T ) = 0.045. Ẽ 3 F,2 = P 3 (H)F,2 (H) + P 3 (T )F,2 (T ) = 0.05. n Sn B n,m, m S m Sn B n,m. n + V n+ ( D n+ V n+ = D Ẽn+ m S m S ) n = D n+ S n+ S n Ẽ n+ D m B n,m B n,m. = D n+ S n+ S nd n+ B n+,m B n,m V n+ = S n+ S nb n+,m B n,m r B n,m = ( + r) (n m), ( + r) m n (S n+ S n ( + r)) 37

. ( Fut n+ m Fut n,m = Ẽn+S m ẼnS m = ( + r) m S n+ ( + r) S ) n n+ ( + r) n = ( + r) m n (S n+ Sn(+r)).. ψ n+ (0) = ẼD n+v n+ (0) = 2 n+ ( + r 0 )... ( + r n (0)) ψ n (0) = ẼD nv n (0) = 2 n ( + r 0 )... ( + r n (0)) ψ n+ (0) =. ψ n (0) 2( + r n (0)) ψ n+ (k) = 2 n+ ( + r 0 )... ( + r n ) n + k. ψ n (k ) = 2 n ( + r 0 )... ( + r n ) n k. ψ n (k) = 2 n ( + r 0 )... ( + r n ) n k. n + k, ()n k n.(2)n k n., ψ n (k ), ψ n (k). n + r n (k ), + r n (k), 2( + r n (k )), 2( + r n (k)) 38

ψ n+ (k), ψ n+ (k) =.,. ψ n (k ) 2( + r n (k )) + ψ n (k) 2( + r n (k)) ψ n+ (n + ) = ẼD n+v n+ (n + ) = 2 n+ ( + r 0 )... ( + r n (n)) ψ n (n) = ẼD nv n (n) = 2 n ( + r 0 )... ( + r n (n )) ψ n+ (n + ) = ψ n (n) 2( + r n (n)) 39