S n+1 (1 + r) n+1 S ] (1 + r) n + n X T = X 0 + ( S) T,

Transcription

1

2 (S K) + (S K) + (S K) + (X 0 0 S 0, 0 S 0 ) X 0 0 (X 0 0 S 0 ) X 0 ( + r)(x 0 0 S 0 ) + 0 S (S K) +. S (H) us 0 S (T ) ds 0 X 0 0 u d X (S K) + ( X S 0, 0 S 0 ) 0 0 S 0 X 0 X 0 C 0 X 0 C 0 X 0 (C 0 X 0 ) C 0 X 0 C 0 ( X S 0, 0 S 0 ) X 0 (X 0 C 0 ) ( X S, 0 S ) ( 0 N ) n n P (ω ω ω N ) > 0 n ω ω ω N n X n+ ( + r) n+ X n ( + r) n + n X n X T X 0 + ( S) T, Xn (+r) S n n Sn (+r) n,,, N n S n+ ( + r) n+ S n ( + r) n H T X 0 0 X 0 S + ( + r)(x 0 0 S 0 ), X X 0

3 H T X X X (H) > 0 X (T ) < 0 X (T ) > 0 X (H) < 0. X (H) > ( + r)x 0 X (T ) < ( + r)x 0 X (T ) > ( + r)x 0 X (H) < ( + r)x 0. X 0 0 X ( ) S S 0 X 0 S 0 r + ( + r)x 0. S 0 X (H) ( + r)x 0 0 S 0 u ( + r) X (T ) ( + r)x 0 0 S 0 d ( + r). d < + r < u S 0 u ( + r) S 0 d ( + r) X (H) > ( + r)x 0 0 > 0 X (T ) < ( + r)x 0 X (T ) > ( + r)x 0 0 < 0 X (H) < ( + r)x 0. d < + r < u d < + r < u d < + r < u X 0 S + ( + r)(x 0 0 S 0 ) 0 S ( + r)(x 0 0 S 0 ) X 0 0 P (X X 0 ( + r)) P (X > X 0 ( + r)) > 0 S(T ) S0 d < + r < u S 0 < r < S(H) S0 S 0 X Γ 0 0 Γ Γ 0 X 0 S + Γ 0 (S 5) ( Γ 0 ). H T X X

4 X (u) Γ ( Γ 0 ) Γ 0 X (d) ( Γ 0 ) 3 0.5Γ 0 X (u) X (d) X X C 0 X 0 α β ( αs 0 βc 0, αs 0 + βc 0 ) X αs + βv ( + r)(αs 0 + βc 0 ) V (S K) + P (X > 0) > 0 P (X < 0) > 0 C 0 δ > 0 ( + r)(c 0 δs 0 ) + δs V. X X αs + βv ( + r)(αs 0 + βc 0 ) αs + β( + r)(c 0 δs 0 ) + δs ( + r)(αs 0 + βc 0 ) αs + β( + r)c 0 βδs 0 ( + r) + δβs ( + r)αs 0 ( + r)βc 0 (α + δβ)s S 0 ( + r)(α + βδ) S (α + βδ)s 0 ( + r). S 0 S /S 0 u d d < + r < u X (H) X (T ) V S K 0 V 0 V 0 +r d +r u d S (H) + u r u d S (T ) S ( S0 +r ) +r d u d u + u r u d d S 0 X n+ (ω ω ω n T ) V n+ (ω ω ω n T ).

5 X n+ (T ) n ds n + ( + r)(x n n S n ) n S n (d r) + ( + r)v n V n+(h) V n+ (T ) (d r) + ( + r) pv n+(h) + qv n+ (T ) u d + r pv n+ (T ) V n+ (H) + pv n+ (H) + qv n+ (T ) pv n+ (T ) + qv n+ (T ) V n+ (T ). V V (H).4 (H) V(HH) V(HT ) S (HH) S (HT ) V (HH) 3.0 V (HT ).40 (HT ) V 3(HT H) V 3 (HT T ) S 3 (HT H) S 3 (HT T ) V 3 (HT H) 0 V 3 (HT T ) 6 ω H X (H) ( + r)(x 0 0 S 0 ) + 0 S (H) ( + 0.5)( ) V (H). ω H (H) V (HH) V (HT ) S (HH) S (HT ) 6 4 ω ω HH X (HH) ( + r)x (H) (H)S (H) + (H)S (HH) ( + 0.5)( ) V (HH), ω ω HT X (HT ) ( + r)x (H) (H)S (H) + (H)S (HT ) ( + 0.5)( ) V (HT ). ω ω HT (HT ) V 3(HT H) V 3 (HT T ) S 3 (HT H) S 3 (HT T ) ω ω ω 3 HT H X 3 (HT H) ( + r)x (HT ) (HT )S (HT ) + (HT )S 3 (HT H) ( + 0.5).4 ( ) 4 + ( ) 8 0 V 3 (HT H),

6 ω ω ω 3 HT T X 3 (HT T ) ( + r)x (HT ) (HT )S (HT ) + (HT )S 3 (HT T ) ( + 0.5).4 ( ) 4 + ( ) 6 V 3 (HT T ). K 5 V 0.0 V ( + r)(x 0 0 S 0 ) + 0 S (S K) +. X X ( + r).0( + r) V 3 V ( ) ( + r) 3

7 S 0 4 u d r 4 p q n 0,,, 3 Y n n k0 S k n K 4 ( 4 Y 3 4 ) + v n (s, y) n S n s Y n y v 3 (s, y) ( 4 y 4) + v n v n v n+ v n (s, y) + r pv n+(us, y + us) + qv n+ (ds, y + ds) ( s v n+ (s, y + s) + v n+ 5, y + s ). v 0 (4, 4) S 0 4 Y 0 4 v 0 (4, 4).6 S (H) 8 Y (H) v (8, ).96 S (T ) Y (T ) 6 v (, 6) 0.08 S (HH) 6 Y (HH) 8 S (HT ) 4 Y (HT ) 6 S (T H) 4 Y (T H) 0 v (4, 0) 0. S 3 (T HT ) Y 3 (T HT ) v 3 (, ) 0 S (T T ) Y (T T ) 7 S 3 (HHH) 3 Y 3 (HHH) 60 v 3 (3, 60) v (6, 8) 6.4 S 3 (HHT ) 8 Y 3 (HHT ) 36 v 3 (8, 36) 5 S 3 (HT H) 8 Y 3 (HT H) 4 v 3 (8, 4) v (4, 6) S 3 (HT T ) Y 3 (HT T ) 8 v 3 (, 8) 0.5 S 3 (T HH) 8 Y 3 (T HH) 8 v 3 (8, 8) 0.5 S 3 (T T H) Y 3 (T T H) 9 v 3 (, 9) 0 v (, 7) 0 S 3 (T T T ) 0.5 Y 3 (T T T ) 7.5 v 3 (0.5, 7.5) 0

8 δ n (s, y) n S n s Y n y δ n (s, y) v n+(us, y + us) v n+ (ds, y + ds). (u d)s n u n (ω ω ω n ) d n (ω ω ω n ) r n (ω ω ω n ) n n ω ω ω n u 0 d 0 r 0 u 0 S 0 ω H S (ω ) d 0 S 0 ω T n n + u n (ω ω ω n )S n (ω ω ω n ) S n+ (ω ω ω n ω n+ ) d n (ω ω ω n )S n (ω ω ω n ) ω n+ H ω n+ T + r 0 n n + r n (ω ω ω n ) n + n ω ω ω n 0 < d n (ω ω ω n ) < + r n (ω ω ω n ) < u n (ω ω ω n ) 0 < d 0 < + r 0 < u 0 N N V N N r u d r n u n d n p n +rn dn u n d n q n p n V n p nv n+ (H) + q n V n+ (T ) + r n. n 0 n N V N n V n+(h) V n+ (T ) S n+ (H) S n+ (T ) V n+(h) V n+ (T ) (u n d n )S n S 0 80 S (H) 90 S (T ) 70 S (HH) 00 u n S n+(h) S n S n+0 S n + 0 S n d n S n+(t ) S n n p n d n q n u n d n S n 0 S n 0 S n p n q n H

9 H 5! 3 ( ) !! ( ) ( ) S n+ S (+r) n+ n+ (+r 0 ) (+r n ) P P(w n+ H ω,, ω n ) : p n + r n d n u n d n P(w n+ T ω,, ω n ) : p n q n. P Ω w : (w,, w N )} ( p n, q n ) N n Sn+ Ẽ n u n p n + d n q n ( + r n ). S n P r n ω n (V n ) N n0 X n+ n S n+ + ( + r)(x n C n n S n ), (X n ) N n C n n V n n X n X n V n. X n n V n X n V n X n V n X n.

10 X N C N X N N X N N S N + ( + r)(x N C N N S N ). N (ω ω N ) C N(ω ω N H) C N (ω ω N T ) S N (ω ω N H) S N (ω ω N T ) X N C N + + r px N (ω ω N H) + qx N (ω ω N T ) XN C N + Ẽn + r N C k Ẽn. ( + r) k (N ) kn X n n n N Xn+ C k X n C n + Ẽn + r Ẽn ( + r) k n. C 0 C C N 0 (Ω, F, F t } t0,,t, P) S S t } t 0,,T } R d k P P (Ω, F, P) P P ϱ : dp /dp L (Ω, F T, P) S t L (Ω, F t, P ) t 0,, T } P kn E P S t F t a.s. S t t,, T }. k B E t k,, T } P (S t B F t ) a.s. P (S t B S t, S t,, S t k ). f g X t X 0 + t 0 f(s, X s )dw s + t 0 g(s, X s )ds f g f(t, x) f(x) g(t, x) g(x) X A A c P(A c ) P(A)

11 P(A c ) + P(A) ω A P(ω) + c ω A P(ω) ω Ω P(ω) A A A N N P( N na n ) P(A n ). n A A A N N A A P(A A ) ω A A P(ω) ω A P(ω) + ω A P(ω) P(A ) + P(A ) A A P(A A ) P((A A ) A ) P(A A ) + P(A ) P (A ) + P(A ) S 3 S 3 p q ω n+ ω ω n P(ω n+ H ω ω ω n ) : p + r d u d, P(ωn+ T ω ω ω n ) : q u r u d. ω n P P(S 3 3) p 3 8, P(S3 8) 3 p q 3 8, P(S3 ) 3 p q 3 8, P(S3 0.5) q 3 8. ẼS ẼS ẼS 3 P ẼS 8 P(S 8) + P(S ) 8 p + q 5 ẼS 6 p + 4 p q + q 6.5 ẼS P r 0 ẼS S , r ẼS 6.5 ẼS 5 0.5, r ẼS ẼS S n Ẽ ( + r) n S 0 Ẽ S n ( + r) n S 0, p 3 q 3 Ẽ S n ( + r). Ẽ S n P(S 3 3) ( 3 )3 8 7 P(S 3 8) 3 ( 3 ) P(S 3 ) 9 9 P(S 3 0.5) 7 ES 6 ES 9 ES P r r r

12 E n S n+ S n E n S n+ /S n S n (pu+qd) M 0 M M N φ φ(m 0 ) φ(m ) φ(m N ) X j j X j j M 0 M M M 0 0 M n n X j, n. j M 0 M M E n M n+ M n + E n X n+ M n + EX n+ M n σ n 0 ( ) n S n e σm n e σ + e σ. S 0 S S M n e σm n e σ +e σ σ 0 Sn+ E n E n e σx n+ S n e σ + e σ e σ + e σ E e σx n+. M 0 M M I 0 0 n I n M j (M j+ M j ), n,,. j0 I n M n n. n I n j0 n M j (M j+ M j ) j0 n Mn (M j+ M j ) Mn j0 n M j M j+ + Mn j0 n Xj+ Mn n. j0 n Mj+ j0 M j

13 T MT M0 M t dm t + M, M T. 0 I T T 0 M tdm t n f(i) i n f g(i) E n f(i n+ ) g(i n ). g(i n ) n I n M n I 0 I I 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+ ) f(i n + M n ) + f(i n M n ) g(i n ), g(x) f(x + x + n) + f(x x + n) I n + n M n M 0 M M N 0 N I 0 I I N I 0 0 n I n j (M j+ M j ), n,, N. j0 I 0 I I N E n I n+ I n E n n (M n+ M n ) n E n M n+ M n 0 X n n X n ω n H X n ω n T P(X ) P(X ) S X S n+ S n + b n (X,, X n )X n+ b n ( ), } n (S n ) n E n S n+ S n b n (X,, X n )E n X n+ 0. f E n f(s n+ ) f(s n + b n (X,, X n )) + f(s n b n (X,, X n )) E n f(s n+ S n b n (S n ) n

14 X n S n n b n (n + ) N M 0 M M N M 0 M M N P M N M N n 0 N M n M n M n E n M N E n M N M n n 0,,, N V N N N V N V N V 0 V V 0, + r,, V N ( + r) N, V N ( + r) N P X n V n X n X n+ n S n+ + ( + r)(x n n S n ) Vn (+r) n }0 n N P Xn (+r) n }0 n N V n V N Ẽn ( + r) N n, n 0,,, N. P V 0, V + r,, V N ( + r) N, V N ( + r) N V n V n n P(HH), P(HT ), P(T H), P(T T ), V V 0 Ẽ V. ( + r 0 )( + r )

15 S (HH) S (H) 8 r (H) 4 S 0 4 r 0 4 S (HT ) 8 S (T H) 8 S (T ) r (T ) S (T T ) u 0 S (H) S 0 u (H) S (HH) S (H) u (T ) S (T H) S (T ), d 0 S (H) S 0,.5, d (H) S (HT ) S (H), 4, d (T ) S (T T ) S (T ). p 0 + r 0 d 0 u 0 d 0, q 0 p 0 p (H) + r (H) d (H) u (H) d (H), q (H) p (H), p (T ) + r (T ) d (T ) u (T ) d (T ) 6, q (T ) p (T ) 5 6. P(HH) p 0 p (H) 4, P(HT ) p 0 q (H) 4, P(T H) q 0 p (T ), P(T T ) q 0 q (T ) 5. P P(ω n+ H ω,, ω n ) : p n P(ω n+ T ω,, ω n ) : q n V V 0 Ẽ V (+r 0 )(+r ) V (S 7) + V 0 V (H) V (T ) V (HH) 5 V (HT ) V (T H) V (T T ) 0 V (H) p (H)V (HH) + q (H)V (HT ).4 + r (H) V (T ) p (T )V (T H) + q (T )V (T T ) + r (T ) 9 V 0 p 0V (H) + q 0 V (T ) r 0

16 V 0 0 V 0 V (H) V (T ) S (H) S (T ) (H) (S 7) + (H) V (HH) V (HT ) S (HH) S (HT ) 5 8 u, ω n+ H Y n+ (ω ω n ω n+ ) d, ω n+ T Y n+ (n + ) Y n+ S n n + A n+ (ω ω n ω n+ ) (0, ) n + A n+ Y n+ S n n + S n+ ( A n+ )Y n+ S n. X 0 n n n n X n+ n S n+ + ( + r)(x n n S n ) + n A n+ Y n+ S n n Y n+ S n + ( + r)(x n n S n ). p + r d u d, q u r u d. Ẽ n X n+ ( + r) n+ Ẽn n Y n+ S n ( + r) n+ + ( + r)(x n n S n ) ( + r) n+ n S n ( + r) n+ ẼnY n+ + X n n S n ( + r) n n S n ( + r) n+ (u p + d q) + X n n S n ( + r) n ns n + X n n S n ( + r) n X n ( + r) n. X n+ X n n(s n+ S n) + r(x n ns n) + na n+ Y n+ S n,

17 n us n + ( + r)(x n n S n ) X n+ (H) n ds n + ( + r)(x n n S n ) X n+ (T ). n X n+(h) X n+ (T ), X n us n ds Ẽn n Xn+. + r N X N V N X n Ẽn X N (+r) N n V Ẽ N n (X (+r) N n n ) 0 n N V N n V n X n Ẽn V N (+r) N n A n+ a (0, ) n ω ω n+ < S n ( a) n (+r) n S n+ Ẽ n ( + r) n+ ( + r) n+ Ẽn( A n+ )Y n+ S n S n ( + r) n+ p A n+(ω ω n H)u + q A n+ (ω ω n T )d} S n pu + qd ( + r) n+ S n ( + r) n. A n+ a S n+ S n Ẽ n ( + r) n+ ( + r) n+ ( a)( pu + qd) S n ( a). ( + r) n Ẽn S n+ (+r) n+ ( a) n+ S n (+r) n ( a) n S n ( a) n (+r) n N C N (S N K) + N C n C N C n Ẽn ( + r) N n, n 0,,, N. P N (K S N ) + N P N P n Ẽn ( + r) N n, n 0,,, N.

18 N K N F N S N K F N F n Ẽn ( + r) N n, n 0,,, N. N K S N < K C N F N + P N F N + P N S N K + (K S N ) + S N K + K S N K > S N (S N K) + C N. S N K K S N C n P n F n C n F n + P n n C N C n Ẽn (+r) Ẽn + N n (+r) Ẽn F N n (+r) N n n + P n F N P N F 0 S 0 K (+r) N F N F 0 Ẽ (+r) N (+r) ẼS N N K S 0 K (+r) N F 0 N F N F 0 F 0 S 0 N (F 0 S 0 )( + r) N + S N K + S N F N K ( + r) N S 0 F 0 S 0 (+r)n S 0 (+r) N 0 C 0 F 0 + P 0 P 0 N S 0 S 0 ( + r) N N N K K S 0 ( + r) N K ( + r) N S 0 C 0 P 0 C n P n n S N K C n P n F n 0 F n Ẽn S (+r) N n n (+r)n S 0 (+r) N n F n n 0 r 0 S n S 0 ( + r) n

19 m N K > 0 m m N K N K m K (+r) N m m (C m, P m ) (SN K) + (K SN ) + C m Ẽm ( + r) N m P m Ẽm ( + r) N m. C m m P m m N K (C m, P m ) m 0 Ẽ (C m,p m ) (+r) m K C m S m + P (+r) N m m (C m, P m ) P m + ( ) + ( K P m Ẽ ( + r) m + Ẽ S m (+r) N m (K ( + r) m Ẽ SN ) + K ( + r) N + Ẽ S m ( + r) m S m (+r) N m ) + + K (+r) N m N K m. K (+r) N m N ( ) N V N f S n, N + f Y n n k0 S k S n, Y n } n 0,,, N n0 P(ω n+ i ω,, ω n ) +r d u d u 4 u d i H i T ω ω n ω n P P P g E n g(s n+, Y n+ ) ( Sn+ E n g S n, Y n + S ) n+ S n S n S n pg(us n, Y n + us n ) + qg(ds n, Y n + ds n ), (S n, Y n ) (S n, Y n ) 0 n N P V n n v n S n Y n V n v n (S n, Y n ), n 0,,, N. v N (s, y) v n (s, y) v n+

20 y v N (s, y) f( N+ ) v N(S N, Y N ) f Vn+ ( N ) n0 S n N+ V N v n+ V n Ẽn + r vn+ (S n+, Y n+ ) Ẽn + r + r pv n+(us n, Y n + us n ) + qv n+ (ds n, Y n + ds n ). v n (s, y) v n+ v n (s, y) pv n+(us, y + us) + qv n+ (ds, y + ds). + r N M 0 N N ( ) N V n f S n, N M nm+ f 0, 0 n M Y n n km+ S k, M + n N (S n, Y n ) n 0,,, N P n M (S n, Y n ) (S n, 0) ω n P (S n, Y n ) 0 n M P h Ẽnh(S n+ ) ph(us n ) + qh(ds n ) n 0,,, M g n M + Ẽ M g(s M+, Y M+ ) ẼMg(S M+, S M+ ) pg(us M, us M ) + qg(ds M, ds M ). Ẽ n g(s n+, Y n+ ) Ẽn (S n, Y n ) 0 n N P ( Sn+ g S n, Y n + S ) n+ S n S n S n pg(us n, Y n + us n ) + qg(ds n, Y n + ds n ). V n n v n S n Y n V n v n (S n, Y n ), n 0,,, N. n M Y n n v n S n v n (S n ), 0 n M V n v n (S n, Y n ), M + n N v N (s, y) v n v n+ n < M n > M v M (s) v M+ (, )

21 y v N (s, y) f( N M ) v N (S N, Y N ) f v n+ n > M ( N ) KM+ S k N M V N Ẽ n v n+ (S n+, Y n+ ) pv n+ (us n, Y n + us n ) + qv n+ (ds n, Y n + ds n ). v n (s, y) pv n+ (us, y + us) + qv n+ (ds, y + ds) n M Ẽ M v M+ (S M+, Y M+ ) pv M+ (us M, us M ) + ṽ n+ (ds M, ds M ). v M (s) pv M+ (us, us) + qv M+ (ds, ds) n < M Ẽ n v n+ (S n+ ) pv n+ (us n ) + qv n+ (ds n ). v n (s) pv n+ (us) + qv n+ (ds) P p + r d u d, q u r u d ( + r) n } N n0 P ζ n } N n0 P ( Z > 0) Ẽ Z Y EY Ẽ Z Y. Z Ẽ E Z E Ẽ Z(ω) : P(ω) P(ω) Z(ω) P P Z P P Z P Ω P(ω) 0 ω Ω Z Ω P(Z 0) EZ ω Ω P(ω) Z(ω)P(ω) A Ω P(A) ω A P(ω) P P(Ω)

22 P(Ω) P(ω) ω Ω ω Ω Z(ω)P(ω) EZ Y ẼY EZY ẼY ω Ω Y (ω) P(ω) ω Ω Y (ω)z(ω)p(ω) EY Z A P(A) 0 P(A) 0 P(A) ω A Z(ω)P(ω) P(A) 0 P(ω) 0 ω A P(A) 0 P(Z > 0) A P(A) 0 P(A) 0 P(A) ω A Z(ω)P(ω) 0 P(Z > 0) P(ω) 0 ω A P(A) ω A P(ω) 0 P(A) 0 P(A) 0 P P P(Z > 0) P P P(A) P(A) P(A) P(A c ) 0 P(A c ) 0 P(A) P(Z 0) P P ω 0 > P(ω 0 ) > 0 Z(ω) P(Z 0) EZ P(ω 0) P(ω 0) P(Ω \ ω 0 }) 0 ω ω0 P(ω 0 ) ω ω 0. ω ω 0 Z(ω)P(ω) 0. P(Ω \ ω 0 }) P(ω 0 ) > 0 P(ω 0 ) < P P p 3 q 3 S 3 M n E n S 3, n 0,,, 3. M n M n n 0,,, 3

23 M 3 (HHH) 3 M (HH) 4 M (H) 8 M 3 (HHT ) M 3 (HT H) M 3 (T HH) 8 M M (HT ) M (T H) 6 M (T ) 4.5 M 3 (HT T ) M 3 (T HT ) M 3 (T T H) M (T T ).5 M 3 (T T T ).50 M 3 S 3 M (HH) E S 3 (HH) ps 3 (HHH) + qs 3 (HHT ) M (HT ) E S 3 (HT ) ps 3 (HT H) + qs 3 (HT T ) M (T H) E S 3 (T H) ps 3 (T HH) + qs 3 (T HT ) M (T T ) E S 3 (T T ) ps 3 (T T H) + qs 3 (T T T ) M (H) E S 3 (H) p S 3 (HHH) + pqs 3 (HHT ) + qps 3 (HT H) + q S 3 (HT T ) ( ) ( ) (8 + 8) M (T ) E S 3 (T ) p S 3 (T HH) + pqs 3 (T HT ) + qps 3 (T T H) + q S 3 (T T T ) ( ) ( ) ( + ) ( ) 3 ( ) M 0 E 0 S ( ) + ( ) ( ) 3 3 ( + + )

24 (M n ) 3 n0 M E S 3 E M 3 M 3 S 3 E M (H) pm (HH) + qm (HT ) M (H) E M (T ) pm (T H) + qm (T T ) M (T ) E 0 M (T ) pm (H) + qm (T ) M 0. (M n ) 3 n0 Z n ζ 3 (HHH), ζ 3 (HHT ) ζ 3 (HT H) ζ 3 (T HH), ζ 3 (HT T ) ζ 3 (T HT ) ζ 3 (T T H), ζ 3 (T T T ) ζ 3 (HHH) 7 64 ( + 0.5) ζ 3 (HHT ) ζ 3 (HT H) ζ 3 (T HH) 7 3 ( + 0.5) ζ 3 (HT T ) ζ 3 (T HT ) ζ 3 (T T H) 7 6 ( + 0.5) ζ 3 (T T T ) 7 8 ( + 0.5) v 0 (4, 4) p 3 q 3 ( ) + V 0 E ζ 3 4 Y 3 4 ( ) + 4 Y 3(ω) 4 ζ(ω)p(ω) ω Ω ( ) + ( ) 3 ( ) + ( ) ( ( ) + ( ) ( ) ( ) + ( ) ( ) + ( ) ( ) ) + ( ) + 3

25 S 0 4 Y 0 4 S (H) 8 Y (H) S (T ) Y (T ) 6 S (HH) 6 Y (HH) 8 S (HT ) 4 Y (HT ) 6 S (T H) 4 Y (T H) 0 S (T T ) Y (T T ) 7 S 3 (HHH) 3 Y 3 (HHH) 60 S 3 (HHT ) 8 Y 3 (HHT ) 36 S 3 (HT H) 8 Y 3 (HT H) 4 S 3 (HT T ) Y 3 (HT T ) 8 S 3 (T HH) 8 Y 3 (T HH) 8 S 3 (T HT ) Y 3 (T HT ) S 3 (T T H) Y 3 (T T H) 9 S 3 (T T T ) 0.5 Y 3 (T T T ) 7.5 ζ (HT ) ζ (T H) ζ Z Z (+r) E (+r) ( + r) 3 ζ (HT ) ( + r) pz(ht H) + qz(ht T ) ζ (T H) ( + r) pz(t HH) + qz(t HT ) pz(ht H) + qz(ht T ) 0.7. ( + r) V (HT ) ζ (HT ) E ζ 3 V 3 (HT ), V (T H) ζ (T H) E ζ 3 V 3 (T H) V (HT ) V (T H) V (HT ) v (4, 6) V (T H) v (4, 0) v (s, y) V (HT ) V (T H) V (HT ) v (4, 6) V (T H) v (4, 0) 0.

26 V (HT ) V (T H), ζ (HT ) E ζ 3 V 3 (HT ) ζ (HT ) pζ 3(HT H)V 3 (HT H) + qζ 3 (HT T )V 3 (HT T ) ( ) ( ) ζ (T H) E ζ 3 V 3 (T H) ζ (T H) pζ 3(T HH)V 3 (T HH) + qζ 3 (T HT )V 3 (T HT ) ( ) ( ) P(HH) 4 9, P(HT ) 9, P(T H) 9, P(T T ) 9. Z(HH) Z(HT ) Z(T H) Z(T T ) P P P(HH) 4, P(HT ) 4, P(T H), P(T T ) 5. Z(HH) P(HH) P(HH) 9 6 Z(HT ) P(HT ) P(HT ) 9 8 Z(T H) P(T H) P(T H) 3 8 Z(T T ) P(T T ) P(T T ) 5 4. Z 0 Z Z Z Z Z (H) Z (T ) Z 0 Z 0 EZ P(HH) P(HT ) P(T H) P(T T ) ω ω P p : P(ω H) P(ω H) 3 q : P(ω T ) P(ω T ) 3 Z (H) E Z (H) Z (HH)p + Z (HT )q , Z (T ) E Z (T ) Z (T H)p + Z (T T )q , Z 0 EZ pz (H) + qz (T )

27 V (H) + r 0 Z (H) E Z ( + r 0 )( + r ) V (H) Z (H)( + r (H)) E Z V (H), V (T ) + r 0 Z (H) E Z ( + r 0 )( + r ) V (T ) Z (H)( + r (H)) E Z V (T ), Z V 0 E ( + r 0 )( + r ) V. V (H) V (T ) V 0 V (S 7) + V (H) Z (HH)V (HH)p + Z (HT )V (HT )q Z (H)( + r (H)) V 0 V (T ) Z (T H)V (T H)p + Z (T T )V (T T )q Z (T )( + r (T )) 9 6 ( 7) ( (8 7)+ ) 3.4, 3 8 (8 7) ( 7)+ 3 3 ( ) + 9, Z (HH)V (HH) ( + r 0 )( + r (H)) P(HH) + Z (HT )V (HT ) ( + r 0 )( + r (H)) P(HT ) + Z (T H)V (T H) P(T H) + 0 ( + r 0 )( + r (T )) 9 6 ( 7)+ 4 9 ( + 4 )( + 4 ) (8 7)+ 3 9 ( + 4 )( + 4 ) + 8 (8 7)+ 9 ( + 4 )( + ) N U(x) x X n X 0 ζ n n 0,,, N ζ n U (x) x I(x) x E Z X 0 λ X 0 (+r) N (+r) N λz X N (+r)n λz X 0 Z ( + r)n X N X0 ( + r) n X n Ẽn ( + r) N n Ẽn X 0 ( + r) n Ẽ n Z Z X 0 ( + r) n E n Z X 0, Z n Z ζ n N U(x) p xp p < p 0 N X N X 0( + r) N Z p, E Z p p Z P P

28 U (x) x p I(x) x p E X N λz ( + r) N λ E X 0 Z p p (+r) Np p p λ p Z p ( + r) N p p ( Xp Z p p X 0( + r) Np p E Z p p ( ) Z λz (+r) N (+r) N 0 ( + r) Np ) p. E Z p ( + r) N p p ( + r)n X 0 Z p. E Z p p X 0 λ (p ζ,, p m ζ m ) X N XN X N ( XN I λ ( + r) N Z ), λ X N EU(X N ) EU(X N). y > 0 x U(x) yx x I(y) U(x) yx U(I(y)) yi(y) x d dx (U(x) yx) U (x) y x I(y) U(x) yx d dx (U(x) yx) U (x) 0 U x I(y) U(x) yx U(I(y)) yi(y) x x X N λz y (+r) N ( λz EU(X N ) E X N ( + r) N E U I ( )) λz ( + r) N E (3.3.9) X N EU(X N ) λx 0 EU(X N ) λẽ ( + r) ( ) N EU(XN Z λz ) λe ( + r) N I ( + r) N y I(x) λz ( + r) N I EU(X N ) E (3.3.6) EU(X N) λx 0. ( ) λz ( + r) N, X N λz ( + r) N

29 EU(X N ) EU(X N ) X 0 N N N X N γ P(X N γ), X N X 0 X n X n 0, n,,, N. P(X N γ) X N Ẽ ( + r) N X 0, X n 0, n,,, N. X N 0 X n 0 n X n Ẽn X N X (+r) N n N 0 X n 0 n y > 0 U(x) 0, 0 x < γ, x γ U(x) yx U(I(y)) yi(y) x 0, I(y) γ, 0 < y γ 0, y > γ 0 x < γ 0 < y γ U(x) yx yx 0 U(I(y)) yi(y) U(γ) yγ yγ 0 U(x) yx U(I(y)) yi(y) 0 x < γ y > γ U(x) yx yx 0 U(I(y)) yi(y) U(0) y 0 0 U(x) yx U(I(y)) yi(y) x γ 0 < y γ U(x) yx yx U(I(y)) yi(y) U(γ) yγ yγ yx U(x) yx U(I(y)) yi(y) x γ y > γ U(x) yx yx < 0 U(I(y)) yi(y) U(0) y 0 0 U(x) yx U(I(y)) yi(y) λ ( ) Z λz E ( + r) N I ( + r) N X 0. X N ( ) λz XN I ( + r) N.

30 x X N y λz X (+r) N N Ẽ XN X (+r) N 0 λz EU(X N ) E ( + r) N X N EU(XN λz ) E ( + r) N X N. EU(X N ) λx 0 EU(X N ) λx 0 EU(X N ) EU(X N ) M N ω ω M ζ m ζ(ω m ) p m P(ω m ) ζ m ζ ζ ζ M. λ K ζ K < ζ K+ K ζ m p m X 0 γ. p m ξ m X 0 N m m p m ξ m I(λξ m ) N m p m ξ m γ λξm γ }. X 0 γ N m p mξ m λξm γ } λ X 0 γ > 0 m : λξ m γ } K m : λξ m γ } λξ K γ < λξ K+ ξ K < ξ K+ X 0 γ K m p mξ m K N ξ K+ λ > 0 K ξ K < ξ K+ K m ξ mp m X 0 γ λ > 0 ξ K < λγ < ξ K+ λ Z E ( + r) N I( λz ( + r) N ) N m p m ξ m λξm γ } γ K p m ξ m γ X 0. m X N X N (ω m ) γ, m K 0, m K +. X N(ω m ) I(λξ m ) γ λξm γ } γ, m K 0, m K +. n τ τ S n

31 τ C n C N n N C n τn} G n C N τn} G N V n (τ) N kn G k G τ Ẽ n τk} ( + r) k n Ẽn τ N} ( + r) τ n. τ V n τ Sn Ẽ n τ N} (+r) G τ n τ (V n ) 0 n N n V n G n ( n ) 0 n N (C n ) 0 n N C n 0 V n+ n S n+ + ( + r)(v n C n n S n ) G n+ P V n+ Ẽ n ( + r) n+ V n ( + r) n C n ( + r) n 0, Sn (+r) n }0 n N Vn (+r) }0 n N P n (V n ) 0 n N Vn (+r) }0 n N P n V n G n (V n ) 0 n N V n G n V n (+r) n }0 n N P τ n} F n : σ(ω,, ω n ), n 0,,,, N. F n ω : ω A,, ω n A n } τ n} ω ω ω n (V n ) N n0 V n G n V P n (+r) }n n (V n ) n C n 0 Vn (+r) }n P n (V n ) n V n (+r) n }n P r 4 p q V P 0 g P (s) (4 s) +

32 V3 P (HHH) (4 3) + 0 V3 P (HHT ) V3 P (HT H) V3 P (T HH) (4 8) + 0 V3 P (HT T ) V3 P (T HT ) V3 P (T T H) (4 ) + V3 P (T T T ) (4 0.5) } V V P ( ) (4 S ) + P, 3 Ẽ (4 S ) +, pv 3 P ( H) + qv P } 3 ( T ). + r + r V P (HH) V P (HT ) V P (T H) V P (T T ) 4 S (HH) +, 5 V P 3 (HHH) + V P 4 S (HT ) +, 5 V P 3 (HT H) + V P 4 S (T H) +, 5 V P 3 (T HH) + V P } 3 (HHT ) } 4 S (T T ) +, 5 V P 3 (T T H) + V P 3 (T T T ) (4 6) +, 5 } (0 + 0) (4 4) +, 5 } (0 + ) 0 3 (HT T ) 0.8 } 3 (T HT ) (4 4) +, 5 } (0 + ) 0.8 } (4 ) +, 5 } ( + 3.5) 3 } V V P ( ) (4 S ) + P, Ẽ (4 S ) +, pv P ( H) + qv P } ( T ) + r + r (H) V P V P (T ) 4 S (H) +, 5 V P (HH) + V P } (HT ) } 4 S (T ) +, 5 V P (T H) + V P (T T ) V0 P (4 S 0 ) +, } 5 V P (H) + V P (T ) (4 8) +, 5 } ( ) (4 ) +, 5 } ( ) 0.3. (4 4) +, 5 } (0.3 + ) V C 0 g C (s) (s 4) + V3 C (HHH) (3 4) + 8, V3 C (HHT ) V3 C (HT H) V3 C (T HH) (8 4) + 4, V3 C (HT T ) V3 C (T HT ) V3 C (T T H) ( 4) + 0, V3 C (T T T ) (0.5 4) + 0.

33 V C (HH) C (8 + 4).8, V (HT ) V C (T H) C (4 + 0).6, V (T T ) (0 + 0) V C (H) C (.8 +.6) 5.76, V (T ) (.6 + 0) V0 C ( ) V S 0 g S (s) g P (s) + g C (s) g S (s) 4 s V3 S (HHH) 4 3 8, V3 S (HHT ) V3 S (HT H) V3 S (T HH) 4 8 4, V3 S (HT T ) V3 S (T HT ) V3 S (T T H) 4, V3 S (T T T ) V S (HH) 4 6, 5 } (8 + 4).8, V S (HT ) V S (T H) 4 4, 5 } (4 + ) V S (T T ) 4, 5 } ( + 0.5) 3. V S (H) 4 8, 5 } (.8 +.4) 6.08, V S (T ) 4, 5 } (.4 + 3).6. V0 S 4 4, 5 } ( ) , V S 0 < V P 0 + V C 0 V S < V P 0 + V C (a, b ) + (a, b ) (a + a, b + b ),

34 > b > a b < a b < a b > a N VN S g S(S N ) g P (S N ) + g C (S N ) VN P + V N C Vn S g S (S n ), pv n+ S + qv n+ S } + r g P (S n ) + g C (S n ), pv n+ P + qv n+ P + pv n+ C + qv C } n+ + r + r n 0,,, N g P (S n ), pv n+ P + qv n+ P + r V P n + V C n, } + g C (S n ), pv C n+ + qv C n+ + r < g C (S n ) < pv C n+ + qv C n+ +r n 0,,, N < g P (S n ) > pv n+ P + qv n+ P. + r V0 S < V0 P + V0 C 5.36 (H) V (HH) V (HT ) S (HH) S (HT ), (T ) V (T H) V (T T ) S (T H) S (T T ), 0 V (H) V (T ) S (H) S (T ) τ n : V n G n } τ(hh), τ(ht ), τ(t H) τ(t T ). X (T ) ( + r)( S 0 ) S (T ) ( + 4 )( ) X (H) ( + r)( S 0 ) S (H) 0.4 X (HH) ( + r)(x (H) S (H)) + S (HH) 0 X (HT ) ( + r)(x (H) S (H)) + S (HT ) }

35 r 4 p q n ( n 0,,, 3 4 n + n+ j0 j) S n G 0 0 G (T ) G (T H) 3 G (T T ) 5 3 G 3(T HT ) G 3 (T T H).75 G 3 (T T T ).5 G V V (T ) V (T H) 3 V (T T ) 5 3 V 3(T HT ) V 3 (T T H).75 V 3 (T T T ).5 V ω H τ(ω) ω T ρ H T H T T Y (HH), Y (HT ), Y (T H) 3, Y (T T ) 4. ( ) ρ 4 Ẽ Y X n+ n S n+ + ( + r)(x n C n n S n ), n 0,,, N, ( n ) n n F n S 0 S 0 τ Ẽ τ } ( 4 5) τ Gτ

36 S 0 τ 0 τ τ(ht ) τ(hh) τ(t H) τ(t T ), } τ(ht ) τ(hh), } τ(t H) τ(t T ) τ(ht ), τ(hh), τ(t H), τ(t T ), } τ 0 τ(ht ), } τ(hh) τ(t H), τ(t T ), } τ(ht ), } τ(hh) τ(t H) τ(t T ) Ẽ ( 4 τ τ } 5) Gτ G0 Ẽ ( 4 τ ( τ } 5) Gτ Ẽ 4 τ τ } 5) Gτ τ (HT ) τ (HH) τ (T H) τ (T T ) ( ) τ (4 4 Ẽ τ } G τ ) ( ) 4 ( + 4) Ẽ τ } ( 4 5) τ Gτ τ τ(ht ) τ(hh) τ(t H) τ(t T ) n n 0,,, N G n n n N G n N V 0 Ẽ τ S 0,τ N ( + r) τ G τ. 0,,, N G n K S n K N N N K S 0 N K S N } } VN K V N K S N, ẼN K S N, + r + r S N K S N. V n K S n (0 n N) K S 0 τ τ N G n, 0} G n K N K N V0 EC V0 AP V0 AP K S 0 + V0 EC V AP 0

37 N K S N < 0 K N V0 AP K S 0 + V0 EC V AP 0 K ( + r) N S 0 + V EC 0 V AP 0. V0 EP K N V0 AP V0 EP V0 EC Ẽ SN K ( + r) N V0 EC K S 0 + ( + r) N. G n S n K K N N V N S N K V N S N K, ẼN V n S n S 0 K (+r) N } VN S N K, S N + r K (+r) N n N K } S N K + r + r. (0 n N) V n > G n 0 n N τ m m τ τ τ τ τ τ,,, τ τ + τ + τ Eα τ (Eα τ ) α (0, ) Eα τ Eα (τ τ)+τ Eα (τ τ) Eα τ Eα τ τ τ τ m Eα τ m (Eα τ ) m α (0, ) M n (m) M n+τm M τm (m,, ) (M (m) ) m M τ m+ τ m n : M n (m) } τ Eα τ m Eα (τ m τ m )+(τ m τ m )+ +τ Eα τ m.

38 (M (m) ) m E : f : N Z} f g f n g n n M (m) n : Ω E (τ m+ τ m ) m0 p q p < p < 0 < q < τ 0 τ f(σ) pe σ + qe σ f(σ) > σ > 0 σ > 0 f (σ) pe σ qe σ > p q > 0 f(σ) > f(0) σ > 0 σ > 0 E Sn+ n S n E n e σx n+ f(σ) ( ) n S n e σmn f(σ) pe σ f(σ) + qe σ f(σ) σ > 0 P(τ < ) ( ) τ e σ E τ < }. f(σ) ES n τ ES 0 ( ) n τ S n τ e σm n τ e σ f(σ) ( n σ > 0 0 < τ }S n τ e σ f(σ)) 0 n E τ < }S τ E S n τ n ES n τ n. ( ) E τ < }e σ τ ( ) f(σ) e σ τ E τ < } f(σ) σ 0 ( ) τ E τ< } P (τ < ) f(0) Eα τ α (0, ) α f(σ) pe σ +qe σ σ 0 α (0, ) σ α 4pqα < 4( p+q ) e σ ± 4pqα. pα

39 ( ) + 4pqα σ > 0 σ pα Eα τ pα + 4pqα 4pqα. qα Eτ α Eατ E α ατ Eτ α τ ( ) 4pqα q q qα ( 4pqα )α ( 4pqα ) ( 4pqα)α + ( 4pqα )( )α. Eτ α α Eατ q ( 4pq) ( 8pq) ( 4pq) p p q. 0 < p < < q < σ 0 f(σ) pe σ + qe σ f(σ 0 ) f(σ) > σ > σ 0 pe σ + qe σ + 4pq p p p q p. σ 0 q p f(σ 0 ) σ > σ 0 f(σ) > f(σ 0 ) σ > σ 0 f (σ) pe σ qe σ qe σ q p (σ e ) > 0 Pτ < } ( n S n e σmn f(σ)) ES 0 ES n τ E ( ) τ n e σm n τ. f(σ) σ > σ 0 f(σ) > ( ) n τ ( ) τ E n eσmn τ E τ < }e σ. f(σ) f(σ) σ σ 0 P(τ < ) e σ 0 p q <

40 Eα τ α (0, ) σ > σ 0 f(σ) > E E α τ τ< } e σ(α) τ< } ( f(σ) σ(α) e σ(α) ± 4pqα pα σ(α) > σ 0 σ(α) E α τ 4pqα τ < }. qα ) τ e σ α f(σ) ( ) + 4pqα pα E τ< }τ P(τ ) > 0 Eτ E ( τ τ < } E τ α τ ) E ( τ α τ ) α α α α Eατ 4pq ( 4pq) 4pq q 4pq q q + q p q q p. τ τ ( Eα τ ) α α (0, ) α ( ) α α α α α ( α ) j (j )! + j!(j )! j j ( α ) j (j )! j!(j )! ( α ) k (k)! (k + )!k!. P τ k} k,, Eα τ k P(τ k)α k ( α ) k k P(τ k)4 k P(τ k) (k)! 4 k (k+)!k! Pτ k} k,, k

41 P(τ ) 4 k P(τ k) P(τ k) P(τ k ) P(τ k) P(M k ) + P(M k 4) + P(τ k, M k 0) P(M k ) + P(M k 4) P(M k ) + P(M k 4) + P(M k 4) P(M k ) P(M k 0). P(τ k ) P(M k ) P(M k 0) P(τ k) P(M k ) + P(M k 0) P(M k ) P(M k 0) ( ) k ( ) k (k )! k!(k )! + (k )! (k)! (k )!(k )! (k + )!(k )! + (k)! k!k! (k)! 4 4 k (k + )!k! k(k ) (k + )k(k ) + 4 k(k ) (k + )k k (k + ) (k)! (k ) 4 k (k + )!k! k + (k + k) k 4k k (k)! 4 k (k + )!k!. M n M n k n M k. n m b m m n m b n PMn m, M n b} PM n m b} ( ) n n! ( n b + m )! ( n+b m ).! m n b n m b ( ) n P(Mn n! m, M n b) P(M n m b) ( ) ( m + n b! n+b m )!. p q p 0 < p < PM n m, M n b} p q x p + x q n x p x q m b. x p m + n b x q n+b m.

42 P(M n m, M n b) P(M n m b) ( m + n b n! ) (! n+b m )! pm+ n b q n+b m. n n S 0 4 j S 0 4 M n 0,,, n, } M 0,,, } V V Ẽ τ M τ< } (K S τ ) + ( + r) τ K n V (n) V (n) (K S τ ) Ẽ + τ< } τ M n ( + r) τ. (V (n) ) n 0 V (n) V n n V (n) n V (n) V, τ τ M τ (n) τ n, τ < τ (n) τ (n) M n n τ (n) τ (K S τ ) Ẽ + (K S τ< } ( + r) τ Ẽ τ (n)) + τ n (n) < } ( + r) V (n). τ (n) n V n V (n) V n V (n) s v(s) c(s) v(s) 4 5. v(s) + ( s ) v δ(s) v(s) v ( ) s s s C n n s S n C n c(s n ) n δ(s n ) c(s) s j j 0 j j v ( s )

43 v( j ) 4 j j 4 j j j 0 j j c( j ) v( j ) 4 5 v(j+ ) + v(j ) (4 j ) 5 (4 j+ + 4 j ) 4 5. c() v() 4 5 c( j ) v( j ) 5 v(4) + v() 5 ( + 4 ) 5. v( j+ ) + v( j ) 4 j ( 4 5 j+ + 4 ) j 0. δ(s) s j j 0 j j j 0 δ( j ) v(j+ ) v( j ) j+ j (4 j+ ) (4 j ) j+ j. j j δ() v(4) v() 4 (4 ) 4 3. δ( j ) v(j+ ) v( j 4 ) 4 j+ j j+ j j+ j 4 j. s j j 0 j j u d r 0 < d < + r < u p + r d u d, q u r u d. g(s) s K K > 0 ( v(s) ) s v(s n ) g(s n n ( ) n +r v(sn ) +r v(sn )

44 v(s n ) S n S n K g(s n ) (+r) v(s n n ) Sn (+r) n v(s) s n S 0 K (+r) n n K S 0 n ( + r) n S 0, S 0 n Ẽ Sn K (+r) S n 0 K (+r) n n S 0 K (+r) S n 0 n S 0 K (+r) n S 0 v(s) s g(s), pv(us)+ qv(ds) +r } s K, pu+ qv +r s} s K, s} s v(s) v(s) s τ Sτ K Ẽ ( + r) τ K τ< } S 0 n ( + r) n S 0, P(τ < ) 0 Ẽ K (+r) τ τ< } > 0 Ẽ Sτ K (+r) τ τ< } < Ẽ Sτ (+r) τ τ< } ( Sn (+r) )n 0 n Ẽ S τ ( + r) τ τ< } Ẽ n S τ n ( + r) τ n τ< } Ẽ n S τ n ( + r) τ n S 0 Ẽ Sτ K ( + r) τ τ< } < Ẽ S τ ( + r) τ τ< } S 0. ẼS 0 S 0. n Ẽ Sτ K (+r) τ τ< } < S 0 τ S 0 v(s) s s 4 5 v(s) + ( ) s v 5 v(s) + ( s ) 5 v.

45 v(s) 5 v(s) + 5 v ( s ). s p p s p p p p p p v (s) s v (s) v(s) s p s p 5 p s p + 5 p+ p p s p 5 + p 5 v (s) v (s) p p v(s) As + B s. s A B A s v(s) s (As + B s ) 0 A 0 s v(s) B s B s 4 s B B s > 0 f B (s) B s (4 s) B > 4 0 s > 0 B 4 f B (s) 0 f B (s) 0 B + s 4s 0 ( 4) 4B 4(4 B) B 4 B > 4 B 4 s B f B (s) 0 s B s B j j s B v B (S 0 ) v B (s) 4 s, s s B v B (s) B s, s s B B s B B 4 s 4s + B 0 ± 4 B 4 s B s v B(s) 4 s B s 4(4 B) 0 4 s B B s B. B 4 s B s < s B v B (s) v B (s) s > s B v B (s) B s B v B (s) s S B v B (s) s s B B B s s B 4 s B B s B B s B B 4

46 (Ω, F, F t } t0,,t, P) S S t } t 0,,T } R d d N P (Ω, F, P) P P ϱ : dp /dp L (Ω, F T, P) S t L (Ω, F t, P ) t 0,, T } P E P S t F t a.s. S t t 0,, T } M n ( + R 0 ) ( + R n ), n,,, N; M 0. Dm D n B n,m Ẽn D m B m,m B n,m Ẽn. n m n B n,m m m B n,m F n,m (m + ) n (m + ) F n,m ( + F n,m ). B n,m B n,m+ D n B n,m B n,m+ B n,m B n,m+ B n,m+. n m S n m Ẽ n D m (K S m ) KD n B n,m D n S n D n V n, S n B n,m

47 V n n V n 0 m n R m (m + ) m (m + ) m ( + R m ) (m + ) m n m (m + ) n (m + ) n Dm+ R m V n Ẽn D n V n B n,m B n,m+. Ẽn Dm D m+ B n,m B n,m+. Swap m K D n m +B 0,m n K m ( + R n ) X n+ ( + R n )(X n n B n,m ) + n B n+,m. V n+ B n+,m (H) B n+,m (T ) m (B n,m ) m n0 m B m,m V n B n,m Ẽ m n V m D n Ẽ n D m V m. (Z n,m ) m n0 Z m,m : d P m d P, Z n,m : ẼnZ m,m, n 0 Z m,m Dm B 0,m Z n,m ẼnD m B 0,m D nb n,m B 0,m, n 0,, m. t T > t S > T T S

48 m P m (B n,m ) m n0 m P m k n,, m k X k S k ẼkD m (S m K)D k S n B n,m B k,m. S n B n,m Ẽ k D k X k Ẽk D k S k ẼkD m (S m K) S n B k,m D k B n,m D k S k Ẽk D m (S m K) S n Ẽ k Ẽk D m B n,m D k S k Ẽk D m (S m K)D k S } n B k,m B n,m D k X k. P 0 n m N R m m + n D n Ẽ n D m+ R m D n Ẽ n D m+ R m B n,m B n,m+. Ẽ n D m+ R m Ẽ n D m ( + R m ) R m Ẽ n D m D m+ B n,m B n,m+. D n D n D n ( R 3) + V 3 (HH) 3, V 3(HT ) V 3 (T H) V 3 (T T ) 0. V (HH) + R (HH) V 3(HH) 3, V (HT ) V (T H) V (T T ) 0.

49 3 ω H ω H 3 3 V (H) V (T ) ω H ω T D V ẼD V D Ẽ V V D D Ẽ V +R Ẽ V V (H) V (HH)P (ω H ω H) + V (HT )P (ω T ω H) 6 + R (H) 7 V (T ) V (T H)P (ω H ω T ) + V (T T )P (ω T ω T ) 0. + R (T ) ( ) 4, X V X 0 0 X ( + R 0 )(X 0 0 B 0, ) + 0 B, V V (H) ( + R 0 )(X 0 0 B 0, ) + 0 B, (H) V (T ) ( + R 0 )(X 0 0 B 0, ) + 0 B, (T ). 0 V 4 (H) V (T ) B, (H) B, (T ) B 0, B,3 (H) B,3 (T )( 4 7 ) V V (H) V (T ) X X V V V ( + R )(X B,3 ) + B,3 (H) V (HH) V (HT ) B,3 (HH) B,3 (HT ) 3, (T ) V (T H) V (T T ) B,3 (T H) B,3 (T T ) 0. T H 3 V m 0 m N F n,m B n,m B n,m+ B n,m+, n 0,,, m. F n,m n 0,,, m (m + ) P m+

50 F n,m B n,m B n,m+ B n,m+ (B n,m+ ) n P m+ (B n,m+ ) n n m Ẽ m+ n F n,m Ẽn B ( Bn,m Ẽn n,m+ (B n,m B n,m+ )Z n,m+ Z ) Bn,m+ D n B n,m+ B n,m+ D n D n B n,m+ D n Ẽ n B n,m B n,m+ D n n,m+ Ẽ n Dn B n,m+ D ẼnD m Dn ẼnD m+ n Ẽn D m D m+ B n,m+ D n B n,m B n,m+ B n,m+ F n,m. F 0, F, (H) F, (T ) Ẽ 3 F, F 0,. F 0, B 0, B 0, F, (H) B,(H) B,3 (H) F, (T ) B,(T ) B,3 (T ) Ẽ 3 F, P 3 (H)F, (H) + P 3 (T )F, (T ) F 0,. S m m n 0,,, m n,m n,m ẼnS m S n B n,m n n + S n+ S nb n+,m B n,m n + n + D n+ Ẽ n+ D m (S m n,m ) S n+ n,m B n+,m S n+ S nb n+,m B n,m. n n + S n+ SnBn+,m B n,m S n

51 r n ( + r) m n n + n+,m n,m n + ( ( + r) m n S n+ S ) nb n+,m ( + r) m n (S n+ B n,m S n (+r) m n (+r) m n ( + r) m S n+ ( + r) n+ ( + S r)m n ( + r) n ( + r) m S m Ẽ n+ ( + r) m ( + r) m S m Ẽ n ( + r) m n+,m n,m. ) n n n r n (k) R n (ω,, ω n ) r n ( H(ω,, ω n )). p q n k n V n (k) H(ω,,ω n )k} ψ 0 (0) n,, ψ n (k) Ẽ D nv n (k), k 0,,, n, ψ n (k) ψ n (0) ψ n+ (0) ( + r n (0)) ψ n (k ) ψ n+ (k) ( + r n (k )) + ψ n (k), k,, n, ( + r n (k)) ψ n (n) ψ n+ (n + ) ( + r n (n)). ψ n+ (0) Ẽ D n+v n+ (0) Ẽ D n + r n (0) H(ω ω n+ )0} Ẽ D n + r n (0) H(ω ω n)0} ωn+t } D n H(ω ω ( + r n (0))Ẽ n)0} ψ n (0) ( + r n (0)).

52 k,,, n ψ n+ (k) Ẽ D n + r n ( H(ω ω n )) H(ω ω n+ )k} Ẽ D n + r n (k) H(ω ω n )k} ωn+ T } + Ẽ D n + r n (k ) H(ω ω n )k } ωn+ H} ẼD n V n (k) + ẼD n V n (k ) + r n (k) + r n (k ) ψ n (k) ( + r n (k)) + ψ n (k ) ( + r n (k )). ψ n+ (n + ) ẼD n+v n+ (n + ) Ẽ D n + r n (n) ψ n (n) H(ω ω n )n} ωn+ H} ( + r n (n)). ω n+ (ω,, ω n ) P P(ω n+ H ω,, ω n ) p P(ω n+ T ω,, ω n ) q u n d n P(ω n+ H ω,, ω n ) p n P(ω n+ T ω,, ω n ) q n p n +r n d n u n d n q n u n r n u n d n X F n σ(ω,, ω n ) ẼXf(ω n+) ẼXẼf(ω n+) F n ẼX( p nf(h) + q n f(t ))