( 24 1 A Ω A A c 1 = P (Ω) = P (A A c ) = P (A) + P (A c ) P (A) = 1 P (A c ) 1 A P (A) = P (A ) = P (A) + P ( ) P ( ) = 0 2 (2.2) 2 (2.3) n 1 A 1,, A

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1 24 1 A Ω A A c 1 P Ω P A A c P A + P A c P A 1 P A c 1 A P A P A P A + P P n 1 A 1,, A n A i A j A k A i A j A k P A 1 A s A n 1 n 1 P A i i<j n 1 P A i A j n 2 P A 1 A 2 A n 1 i<j<k n 1 P A i A j A k 2.2 P A 1 A 2 A n P A 1 A 2 A n 1 A n P A 1 A 2 A n 1 + P A n P A 1 A 2 A n 1 A n n 1 P A i i<j n 1 P A i A j n 2 P A 1 A 2 A n 1 +P A n P A 1 A n A 2 A n A n 1 A n n 1 P A i + P A n i<j n n 2 P A 1 A 2 A n n 1 P A 1 A 2 A n 1 A n n P A i P A i A j + + i<j n i 1 <i 2 < <i n 1 n P A i A j i n 1 P A i A n i 1 <i 2 < <i n 2 n 1 1 n 2 P A i1 A i2 A in 2 A n 1 n 2 P A i1 A i2 A in n 1 P A 1 A 2 A n 2.3 n 3 A B P A B 1/36 P A 1/6 B 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1 P B 6/36 1/6 P A B P AP B 1/36 A B A 1

2 A x 7 x x B 1/6 B 7 B B 6 B 4 E[X] n x n x p x q n x x npp + q n 1 np n n 1 nn 1! x 1!n x! ppx 1 q n x n 1! np k!n 1 k! pk q n 1 k x1 k E[X 2 ] E[XX 1 + X] E[XX 1] + E[X] E[XX 1] n x n xx 1 p x q n x x n x2 nn 1n 2! x 2!n x! p2 p x 2 q n x n 2 nn 1p 2 n 2! k!n 2 k! pk q n 2 k nn 1p 2 p + q n 2 nn 1p 2 k V ar[x] nn 1p 2 + np np 2 np1 p npq E[X] npq n 1 d p dq qn d p q n p d dq dq n1 n n 1 1 q p 1 q 2 1 p E[X 2 ] E[XX 1 + X] E[XX 1] + E[X] E[XX 1] nn 1pq n 1 nn 1pq n 1 pq d2 dq n1 n q n pq d2 1 dq n 1 q 2 pq 1 q 3 2q p 2 V ar[x] 2q p p 1 p 2 q p 2 E[X] xλe λx dx [ xe λx] + e λx dx [ 1 λ e λx ] 1 λ 2

3 E[X 2 ] x 2 λe λx dx [ x 2 e λx] + 2 xe λx dx 2 λ 2 V ar[x] E[X 2 ] E[X] 2 1 λ 2 E[X] µ x 1 exp 2πσ 1 y exp 2πσ x µ2 2σ 2 dx y2 2σ 2 dy + µ 1 exp 2πσ x µ + µ 1 exp 2πσ x µ2 2σ 2 dx x µ2 2σ 2 dx E[X 2 ] 2 [ x 2 1 exp 2πσ x µ2 2σ 2 dx x µ 2 + 2x µ + µ 2 1 2πσ exp y 2 1 exp y2 2πσ 2σ 2 2σy 2π exp y2 2σ 2 ] dy + µ 2 x µ2 2σ 2 dx 2σ 2 + exp y2 2πσ 2σ 2 dy + µ 2 σ 2 + µ 2 V ar[x] E[X 2 ] E[X] 2 σ ,.46, ,.46.7, P P P

4 rt R 4.9 2πR/u k 3 F C dt k3 R R 3 2πR u 2πk3 ur 2 P 1 exp 2πk3 ur 2 2 rt R + ut k 3 [ F C R + ut 3 dt k 3 ] 2uR + ut 2 k3 2uR 2 P 1 exp k3 2uR 2 A 点 R センサー C R A 点センサー 1: 2: 2 x rt x 2 + ut 2 F C 2k n ux n 1 k n k n dt 2 x 2 + ut 2 n/2 x n 1 + ut/x 2 1 z n 3/2 1 z 1/2 dz 2kn n 1 ux n 1 B, 2 dt 2 n/2 1 2 k n x n 1 + y 2 n/2 x u dy y ut/x z 1/1 + y P Lx r r R 2 W 2 dx + 2 r r + 2R2 ut θ R 2 ut 2 /4 R 2 x2 ut dx 2r + 4 θ 4R 2 sin2 θ dθ 2r + 2R2 [θ 1 ] θ ut ut 2 sin 2θ

5 x R cos θ θ cos 1 r/r sin θ 1 r/r 2 ut /2R W R 2 ut R2 sin 1 ut ut 2R 4 θ D/u cos θ L + D tan θ/v fθ L + D tan θ/v D/u cos θ θ f θ D v cos 2 θ D sin θ u cos 2 θ D L θ sin 1 u/v 85 1 fx 1 e x/α 1 e β/x α, β > x fx 1 e x/α + e β/x + e x/α+β/x e x/α + e β/x 2 e x/α e β/x e x/α e β/x x/α β/x x/α + β/x x α + β x 2 x β α x 2 β α x/α β/x fx x/α β/x x αβ θ 1 ξ 1 cos α ξ cosθ α cosθ α cos α/ξ 1 cosx α cos α/ξ x α < π θ θ [x, 2π + 2α x] 5

6 6.8 2π+2α x cos α ξ cosθ α dθ [θ cos α ξ sinθ α] 2π+2α x x x 2π + 2α 2x cos α ξ sinα x sinx α cos α 2 π cos 1 cos α + 2ξ sinx α ξ 2 cos 1 cos α cos α + 2 ξ ξ 2 cos 2 α Cos α/ x-α π 2π-x+α 2π -α -1 2 ξ < 1 cos α/ξ 1 cos α ξ cosθ α cosθ α cos α/ξ θ 6.8 [θ cos α ξ sinθ α] 2π 2π cos α 6.9 cos 1 ξ α cos 1 ξ cos 1 ξ α cos 1 ξ ξ cos α ξ 1 cos α/ξ 1 1 ξ 1 θ 6.1 α < cos 1 ξ α > cos 1 ξ cos α < ξ cos α ξ cosθ α cos α + ξ < θ [t, t + t] At Qt Qt t/at qt qt + t qt + t qt 1 Qt t At t dqt dt qt + t qt lim qt Qt t t At qt 1 t Qt qt exp t At dt 6

7 [t, t E ] te Qt pt, t E 1 exp t At dt At πu t 2, Qt Q te t Qt At dt te t Q πu 2 t2 dt Q 1 πu 2 1 t t E Z Q/π logt E /t [ 1 2πσ 2 exp z2 2σ 2 zdzdθ exp 1 exp Q 2πσ 2 log t Q/2πσ 2 E t 1 t t E 2π Z z2 2σ 2 ] Z 1 exp Z2 2σ Z [t, t E ] 1 exp Q logt E/t πz 2 Z 1 exp Z2 2σ 2 6.A 6.B 6.A 6.B 6.18 Z 5.22 Q πz 2 log t E t Z2 2σ 2 Z 2Qσ 2 π log t 1/4 E t 6.A 6.B W S 1 1 ξ λ 1ξ 1 + ξ 6.C 7

8 5.4 S W 1 W/S P W < S W/S < 1 P W/S 6.21 W 2r p 5.6 k P S W/S W < p S W/S < p 5.7 k P S p p W S S/2 r < S p W/S < 2p W/S 6.C i j x ij min x ij s.t. m n c ij x ij j1 n x ij a i, i 1,, m j1 m x ij b j, j 1,, n x ij, i 1,, m, j 1,, n 2 j x j min x j s.t. n c j x j j1 n a ij x j b i, i 1,, m j1 x j, j 1,, n 3 i -1 x i max x i s.t. n c i x i n w i x i G x i, 1, i 1,, n 4 x ijk : i j k -1 max x ijk j1 k1 7.A 8

9 s.t. 9 x ijk 1, j, k 1,, 9, 9 x ijk 1, i, k 1,, 9, j1 9 x ijk 1, i, j 1,, 9, k1 3s+3 3t+3 i3s+1 j3t+1 7.B 7.C 7.D x ijk 1, k 1,, 9, s, t, 1, 2, 7.E x ijk, 1, i, j, k 1,, 9, 7.F x 117 1, x 142 1,. 7.G 7.A 7.B 7.C 7.D 1,, 9 7.E 7.G 2 y 1, y 2, y 3, y 4 min 2y 1 + 3y 2 + 5y 3 + 1y 4 s.t. 2y 1 + 3y 2 + y 4 2 8y 1 2y 2 + y 3 2y 4 1 3y 1 3y 2 + 3y 3 3y 4 4 y 1, y 2, y 3 3 f/ x 3x 2 + 3y, f/ y 3x + 3y 2 x 1,, x 2 1, 1 x 2 z 1, z 2 z 1, z 2 2 fx 2 z1 z 2 6z 2 1 z 1 z 2 + z 2 2 fx p 1,, p m m p i 1, m m Lp 1,, p m ; λ p i log p i + λ p i 1 L p i log p i 1 + λ, i 1,, m p i i 1,, m 9

10 p i 1/m p i 2 f/ p i p j i j 1/p i i j x 1,, x m x 1,, x m 2 fp x 1,, x m t 5 1 n 1 2 k 1 k 1 f k 1 v k 1 j1 2 vk 1 v2 k 1 k 1 k 1 7.H 3 f k v k max f k 1 v k x k + x 2 k x k v k 7.H vk x k 2 f k v k min + x 2 k min x k v k k 1 k min x k v 2 k + v2 k x k v k k 1 k k x k v k v2 k k kx 2 k 2v kx k + v 2 k k 1 x k q/k x x k 1 v k x k k 1v k /k f k 1 v k x k x 1,, x k 1 x 1 x k 1 v k /k k 6 a n f n a max xa x x a x a/2 1 n 2 2 k 1 f k v v/k k f k+1 v max x v [xf kv x] max x v [ ] v x k x k gx x v x/k k v x k 1 g v k + 1x x k k gx x [, v] x v/k + 1 gx v/k + 1 k+1 v k + 1 1

11 7 DP k z SAM f k z f 1 m k SAM z y k + 1 SAM z y f k z max [v k1 1 p k y + f k+1 z y], z, 1,, m y,,z 7.I y SAM SAM f n z v n 1 1 p n z, z, 1, 2,, m 7.J 7.J 7.I k n, n 1,, 1 k z, 1,, m f 1 m k z SAM 7.I y k z 1 m n n fx, y, y n y f y n 1y const 1 + y 2 y yx nx, y ny fx, y, y ny 1 + y f y f y ny 1 + y 2 nyy y 1 + y 2 ny 1 + y 2 const 7.K n 1 y > n 2 y y const y tan θ 1 y tan θ 2 7.K n tan 2 θ 1 n 1 cos θ 1 const n tan 2 θ 2 n 2 cos θ 2 9 I[y] b a fx, y, y,, y n dx y x yx y x + εηx ε di b dε d a dε fx, y, y,, y n dx b a f y η + f y η + + f ηn dx y n 7.L 11

12 k b a f y k ηk dx [ f ηk 1 y k [ d dx ] b f y k 1 k b 7.L b d f dx a a ] b η k 2 + a a d k f dx k y k y k b a b η k 1 dx d 2 dx 2 ηdx di b f dε a y d f dx y n dn f dx n y n ηxdx a f y k η k 2 dx d f dx y k η k 1 dx pr, θ θ pr r, θ max φr 2π pr 1 exp αrφr rdr s.t. 2π φrrdr Φ, φr 8.3 φ r 1 [ log αrpr αr λ ] + 8.A 8.A pr αr α φ r 1 α [ log ] + α 2πσ 2 λ r2 2σ 2 8.B R 2 2σ 2 log α 2πσ 2 λ 8.C R 8.B φ r 1 [ 2ασ 2 R 2 r 2] + R 2 r 2, r R,, r > R 8.D R λ Φ 2π R 1 2ασ 2 R 2 r 2 rdrdθ π [ ] R ασ 2 R 2 r2 2 r4 π 4 4ασ 2 R4 12

13 R 2 4ασ 2 Φ π φ r 2π R pr 1 exp αφ r rdrdθ R2 2σ 2 exp R2 2σ 2 αφ αφ πσ 2 exp πσ 2 2 ψx cxφx px 1 exp αx cx ψx dx ψx C, ψx K c αx αx/cx 8.3 ψ x cx/αx [αxpx/cxλ] + φ x φ x 1 [ log αxpx αx cxλ λ ] + ψ xdx x αxpx cxλ cx αxpx log αx cxλ Φ K c φ i 1 [ log α ip i α i c i λ ] + λ i α i p i c i λ c i α i log α ip i c i λ C 13

14 3 R T φ i i i [ T p i 1 D i tr i Ct dp i tφ dt CT dt [ [ ] T T p i 1 D i tr i Ct Pi t φ + CT ] 1 Pi T φ p i [ 1 D i T R i P T i φ + T ] 1 Pi T φ d i tr i + C P t i φdt d i tr i + C P t i φdt ] CT 8.E [ ] t R i [, T ] CT φ arg max φ R T φ φ δφ δr T φ [ ] T p i 1 D i T R i δpi T φ + δ d i tr i + C Pi t φ dt i δp T i φ δ T d itr i + C P t i φ dt δr T φ i p i α i T [1 D i T R i 1 Pi T φ ] T + R i dτ + C1 Pi τ φ dτ δφ i, tdt t φ i, t > i A T it φ λt 2 φ i, t i A T it φ λt A T it φ A T itφ p iα i c i [ 1 D i T R i 1 P T i φ T 8.E δ T R T φ i ] T + R i dτ + C1 Pi τ φ dτ t [ ] T p i δ T 1 D i T R i Pi T φ + δ T d i tr i + C Pi t φdt CδT 8.38 δ T P T i φ δ T P t i φ dt δ T 1 D i T R i Pi T φ 1 D i T R i α i 1 P T i φ 1 D i T R i δ T Pi T φ d i T R i Pi T φ δt T δφ i, tdt +1 D i T R i φ i, T 1 P T i φ δt d i T R i P T i φ δt 14

15 δ T T d i tr i + C P t i φdt R i d i T + CP T i φ δt + T T t α i R i d i τ + C1 P τ i φ dτ δφ i, tdt δ T R T φ i [ + i T i T p i α i c i 1 D i T R i 1 P T i φ + T p i α i 1 D i T R i 1 Pi T φ c i φ i, T + c i i [ A T itφ c i δφ i, tdt + i t R i d i τ + C1 P τ i φ dτ Cp i P T i φ A T it φ c i φ i, T C 1 P T φ ] δt CδT ] δt c i δφ i, tdt 8.38 φ i, T > A T it φ λt δ T R T φ C λt 1 P T φ δt 8.F 8.F 8.4 R i α i c i 1 D i T p i1 Pi T φ 1 P T φ 1 T c i min fx, y 2 y 2 xy2 + x + 1y + 3 fx, y x y + x x2 1x + 1 2x 4x 1 x > y x + 1/2x 2 x 1/3 i 1 3 x y y fx, y x2 1x + 1 4x ii < x < 1 3 y < 2 y 2 fx, 2 2x x f, y y + 3 y

16 3 x < x 1 y x + 1/2x y 2 fx, 2 2x x < 1/3 min y fx, y 2x + 1 1/3 x min y fx, y gx x 2 1x + 1/4x y 2x + 1 y gx 1/3 x 1 max x min y fx, y x 1 g1 2 2 x 1, y i x < y B A vx, y 1 f B y ii x > y A B A vx, y f A x 2 y max x vx, y max1 f B y, f A y 1 1-f B y f A x y x 3 y max1 f B y, f A y min y max x vx, y 1 f B y f A y B f A y + f B y 1 y f A y 1 y* 1-f B y f A y y 4 min y vx, y minf A x, 1 f B x f A x + f B x 1 x f A x A B x y f i x 1 exp α i x fi 1 µ 1/α i log1 µ f i x α i exp α i x 11.1 i N c i fi 1 µ i c i α i log1 µ C 16

17 µ 1 exp C i c i/α i 11.A φ i fi 1 µ 1 log1 µ C/α i α i j c j/α j p i c i/f i φ i j c j/f j φ j c i/α i j c j/α j p i f i φ i µ 1 exp C i c i/α i i 17

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,