,. 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,,.

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1 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, b {x R : a < x b} : R {x, y x, y R} R n {x, x,, x n x, x,, x n R} : n For all There exists s.t. such that so that i.e. id est, ex. i A, B s.t. C A, C B. ii lim n a n a ε > 0, N N s.t. n N a n a < ε ε, N, n N n a n a < ε iii lim n a n + M R, N N s.t. n N a n > M M, N, n N n a n > M iv lim n a n M R, N N s.t. n N a n < M M, N, n N n a n < M v lim x a fx fa ε > 0, δ > 0 s.t. x a < δ fx fa < ε ε, δ, x a < δ x fx fa < ε

2 ,. 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,, n + F x, y 0, y,, y n x y yx F x, y, y, y,, y n 0 n., n y φx, F x, φx, φ x, φ x,, φ n x 0, x I I, y φx. n, n.

3 . a. y sinx + a a. Proof y sinx + a, y cosx + a y y y + sin x + a + cos x + a y +, y sinx + a 4y + y 4 0, y ±. n, n., n n. a, b. x a + y b 4 a, b.. n {N t,..., N n t},,, n dn i dt f i N t,, N n t, tn i t, i,..., n.., f i i n +.,,, -., f i : i j : i j : i j : f i < 0, N j f i > 0, N j f i > 0, N j f j < 0 N i f j > 0 N i f j < 0 N i, f i N j j,..., n, Lotka-Volterra : Lotka-Volterra :.3 dn i dt ε i + a ij N j t N i t, i,..., n. j, SIR., susceptible, infected, recovered, t St, It, Rt., SIR : ds λn µs βcis + fi + hr, dt di µ + δ + f + gi + βcis, dt dr µ + hr + gi, dt β :, c :, µ :, δ :, f :, g :, h :, λ :, N S + I + R.

4 . 3. y fxgy., fx, gy. gy 0, x gy y fx i.e. dy gy dx dx fxdx dy gy dx fx i.e. gy dy, fxdx. gy 0 0 y 0, y y 0, y y 0,.. xy y x. Proof, y xy,. y ±, y x y dy xdx y dy x + c 0 c 0 : y + log y y + x + c 0 y y + cex c ±e c 0 : 0 y + cex ce x c 0 y c 0. y., : y + cex ce x c :, : y.. dn dt ε µntnt, N0 < a : ε µ. Proof N a, a NtNt t NN a dn µdt a N a dn µt + c 0 c 0 : N log N a N εt + c 0a a Nt ce εt c ±e c0a : 0 t 0 c a N0.3, a Nt N0 a + e εt N0 ε + µn0e εt.... ε > 0,. µ > 0,,.. N0,,, a ε/µ S.,. εe εt

5 4. y y h x.., h. y xzx, x y z + xz,. z + xz hz i.e. z hz z x, hz z dz dx log x + c c :. x.4 x + yy x y. Proof + y y y x x. z y x, y z + xz, + zz + xz z x + zz z z. z + z 0, z + z + z z x, x z + z + z dz x dx log z + z log x + c 0 c 0 : x z + z c c ±e c 0 : 0 z + z 0 c 0. y xz, : y x ± x + c c :..5 s R : k, l > 0 S.. dr ds k s, s > s 0 0, Rs 0 0. dr ds l R, s > 0, R0 0. s [ Rs k log s s 0, s s 0 Rs cs l, s 0 c > 0.6, k, s 0,. S., c, l., i l > Rt : ii l Rt : iii l < Rt :

6 . 5.3 y + P xy Qx, x I..3, P x, Qx I. e R P xdx y e R P xdx + P xe R P xdx y Qxe R P xdx y e R { P xdx + e R } R P xdx y Qxe P xdx { ye R } R P xdx Qxe P xdx ye R P xdx Qxe R P xdx dx y e R P xdx Qxe R P xdx dx.3..7 y y x, y0. Proof e x, y e x ye x xe x {ye x } xe x ye x xe x dx + xe x + c y ce x + x c :., y0 c, c. y e x x : y y y 0 + y y x ce x + + x Bernoulli y P xy + Qxy n, x I n 0,.4 Bernoulli., P x, Qx I. n 0,,. y 0. y 0, y n y n y P xy n + Qx { y n } n { P xy n + Qx }. z y n z + n P xz n Qx z e n R P xdx n Qxe n R P xdx dx. Bernoulli.4 y z R n e P xdx [ n Qxe n R P xdx n dx.

7 6.9 [n y P xy Qxy.. ε, µ P x, Qx. Proof z y y z y y z y z z P xz Qxz z + P xz Qx z e R P xdx Qxe R P xdx dx y z e R P xdx Qxe R P xdx dx.0 [n /3 w t 3βαwt /3 wt, w0 0. wt t.. Proof z w /3 w /3 w z 3, w 3z z 3z z 3βαz z 3 z + βz αβ zt α e βt z0 w0 0 wt α 3 e βt 3 Riccati y P xy + Qxy + Rx, x I Qx 0.5 Riccati., P x, Qx, Rx I. P, Q, R,.,,. ux Riccati.5. y u + z y u z z.5 u z z P xu + z + Qxu + z + Rx [P xu + Qxu + Rx + P xz + Qxuz + z u + P xz + Qxuz + z z + P x + uxqxz Qx,. z e R [P x+uxqxdx Qxe R [P x+uxqxdx dx y ux e R [P x+uxqxdx Qxe R [P x+uxqxdx dx Riccati.5.. x y + xy x 3 y y + x xx + yy y3x + y 4 y + y e x y 3

8 I P x, Qx, Rx, y + P xy + Qxy Rx, x I., y + P xy + Qxy 0, x I., P x, Qx.,. 3., y + py + qy Rx p, q : 3.., t + pt + q 0 3. α, β. 3. : y α + βy + αβy Rx 3.3 y αy βy αβy Rx y αy βy αy Rx { e βx y αy } e βx Rx y αy e βx e βx Rxdx 3.4 y βy αy αβy Rx y βy αy βy Rx { e αx y βy } e αx Rx y βy e αx e αx Rxdx 3.5 α, β,,,,. 4.. α β, i.e. p 4q β αy e βx e βx Rxdx e αx e αx Rxdx y eαx e αx Rxdx + eβx e βx Rxdx 3.6 α β β α. p 4q > 0, i.e. α, β, p 4q < 0, i.e. α, β 3.6 : : e iθ cos θ + i sin θ i.

9 8 α a + i b, β a i b a, b R, b y eax e i bx e ax e i bx Rxdx eax i bx e e ax e i bx Rxdx i b i b [e eax i bx e ax cos bx i sin bxrxdx e i bx e ax cos bx + i sin bxrxdx ib eax [ e i bx i bx e e ax cos bxrxdx ei bx i bx + e b i [ e ax sin bx e ax cos bxrxdx e ax cos bx b e ax sin bxrxdx e ax sin bxrxdx 3.7., Rx 0,, : 3.6 : y c e αx + c e βx c, c, 3.7 : y c e ax cos bx + c e ax sin bx c, c. α β, i.e. p 4q 0 3.4, { e αx y } e αx y αy e αx Rxdx e αx y e αx Rxdx dx x e αx Rxdx xe αx Rxdx y xe αx e αx Rxdx e αx xe αx Rxdx 3.8., Rx 0, Rx 0, y c e αx + c xe αx c, c α, β. y + py + qy yx yx c ux + c vx c, c. ux, vx, p 4q > 0 i.e. α, β, ux e αx, vx e βx. p 4q 0 i.e. α β, ux e αx, vx xe αx. 3 p 4q < 0 i.e. α a + i b, β a i b a, b R, b 0, ux e ax cos bx, vx e ax sin bx. ux, vx 3.9. ux, vx , α + β p 3.6 : y [ vx β α [ 3.8 : y vx [ vx 3.7 : y b, [. uxrxe px dx ux uxrxe px dx ux uxrxe px dx ux vxrxe px dx vxrxe px dx vxrxe px dx

10 y 3y + y x 3 + x. Proof t 3t + 0 t /,, ux e x, vx e x., 3.6, y e x/ e x/ x3 + x dx + e x e x x3 + x dx e x/[ x 3 + 6x + 5x + 50e x/ + c + e x[ x 3 + 3x + 7x + 7e x + c c e x/ + c e x + x 3 + 9x + 43x + 93 c 0, c : 3.3, zx x 3 + 9x + 43x + 93,, y x c ux + c vx y 3y + y 0., ,.,,. 3.4 y y + 5y e 3x. Proof, zx ae 3x., zx e 3x /8. z z + 5z 9a 6a + 5ae 3x 8ae 3x e 3x, t t t ±, ux e x cos x, vx e x sin x., y c e x cos x + c e x sin x + 8 e3x c, c : 3.5 m, k.,, µ., t xt., x 0., g,. mx t + µmgx t + k xt 0 x y y y 0 y + 4y + 4y 0 3 y y + 0y 0 5 y 4y + 4y 4x 4 y y e x + e x 6 y 4y + 5y cos x

11 0 Euler x y + axy + by 0. x > 0, x e t, zt ye t z t e t y e t xy x, z t e t y e t + e t y e t x y x + z t,,, : z t + a z t + bzt x < 0, x e t, zt y e t 3.3., 3.3 z { } zlog x, x > 0 yx zlog x, x 0 zlog x, x < 0 Euler. y, 0, 0,,, a, b, x 0. x 0 y 0.,. 3. I P x, Qx, Rx, y + P xy + Qxy Rx, x I 3.4, y + P xy + Qxy 0, x I 3.5. yx 3.4, ux 0 3.5, i.e. u + P xu + Qxu 0, x I u 3.6 y,, y u yu + P xy u yu ur y u yu + P xy u yu ur { e R } P xdx y u yu R ure P xdx y u yu e R P xdx ure R P xdx dx y y u yu u u u e R P xdx ure R P xdx dx vx ux ux e R P xdx dx,, yx ux ux e R P xdx uxrxe R P xdx dx dx ux ux e R P xdx dx uxrxe R P xdx dx ux ux e R P xdx dx uxrxe R P xdx dx vx uxrxe R P xdx dx ux vxrxe R P xdx dx

12 3.., Rx 0, y c ux + c vx c, c. 3.6 ux 0 3.5, vx ux ux e R P xdx dx, vx 3.5,, : 3.5 y 0 x y 0 x c ux + c vx c, c yx yx vx uxrxe R P xdx dx ux vxrxe R P xdx dx. 3.8 Proof,. vx 3.5, c 0, c. 3.7, 3.8 vxrxe R P xdx dx V x c, uxrxe R P xdx dx Ux + c c, c, 3.8 yx c ux + c vx + { vxux uxv x } y 0 x + { vxux uxv x }. zx vxux uxv x 3.4, ,, 3.4,.

13 4 I fx, N N y N + p k y k fx, x I. 4.,.,., i, I. 4. ux, vx I. fx ux + i vx, ux, vx I, fx f x f x u x + i v x. fx fxdx uxdx + i vxdx., I ux, vx, fx ux + i vx I fx CI C., ux, vx I fx C I C.. 4. a 0, a, a C., C 0 c 0 + i c C. f, f C I C a b c d a f x + a f x a f x + a f x f xdx f x + C 0 f xf x f xf x + f xf x f xf xdx f xf x f xf xdx f, f CI C a a f x + a f xdx a f xdx + a b f xdx f x 3 e ax ae ax, e ax dx a eax + C 0 f xdx

14 4. 3 Proof a k b k + i t k, b k, t k R, f k x u k x + i v k x k,. -a & -a : a k f k b k u k t k v k + ib k v k + t k u k, a k f k x b k u k x t k v k x + i b k v k x + t k u k x k k b k u kx t k v kx + i k k b k v kx + t k u kx k b k + i t k u kx + i v kx k a k f kx k a k f k x dx k -b & -b : k b k u k x t k v k x dx + i k b k v k x + t k u k x dx k b k u k x dx t k v k x dx + i [b k k a k f k xdx k f x u x + i v x u k x dx + i b k v k x dx + t k k v k x dx + i t k f xdx u k x dx + i u x dx + i u k x dx v k x dx v x dx f x dx u x dx + i v x dx u x + c 0 + i v x + c f x + C 0 f x dx u x dx + i v x dx u x + i v x -c & -d : f f u u v v + iu v + v u, f f u u v v + iu v + v u u u + u u v v v v + iu v + u v + v u + v u u u + i v + u + i v u + v v + i u + v + i u v u f + f u + i v f + i v f u + i v f + f u + i v f f + f f -a, -c, -b f xf x dx + f xf x dx f xf x + f xf xdx f xf x dx f xf x + C 0 3 : a b + i t b, t R,, e i tx cos tx + i sin tx cos tx + i sin tx t sin tx + i cos tx i te i tx,-c e ax e bx e i tx e bx e i tx + e bx e i tx be ax + i te ax ae ax

15 4, -a, -b a e ax dx e ax dx e ax + C 0 4.3,,,., i,. 4.4 a b + i t C b, t R,, : e ax e ax e bx cos tx + i sin tx e bx cos tx i sin tx cos tx + sin tx CI C CI C F a. a C fx CI C F a [f : e ax e ax fxdx 4., 4.5 a 0, b, b C f x, f x CI C, : F a [b f + b f b F a [f + b F a [f Proof 4.-a, F a [b f + b f e ax e ax b f x + b f x dx e ax b e ax f x + b e ax f x dx e ax b e ax f xdx + b e ax f xdx b F a [f + b F a [f F a CI C CI C. CI C., F a. 4.6 a 0, b, b j C, t 0 R, n N {0}. c 0 R, C 0 C. F a [0 c 0 e ax C 0 e ax + F a [e bx b a ebx b a c 0 e ax + xe ax b a 3 F a [cos tx C 0 e ax + 4 F a [sin tx C 0 e ax a cos tx + t sin tx a + t a ±i t xeax 4a e ax a ±i t a sin tx + t cos tx a + t a ±i t a t xeax + 4t e ax a ±i t

16 F a [x n C 0 e ax n! a n+ 6 F a b j x j C 0 e ax j0 a k k! xk j j0 b j j! k! ak j+ x k Proof :. : 4., e ax F a [e bx C 0 + e b ax dx b a eb ax b a c 0 + x b a 3:, cos tx ei tx i tx + e [ e i tx + e F a [cos tx F a i tx. a ±i t, 4.5, F a [e i tx + F a [e i tx C 0 e ax C 0 e ax ae i tx + e i tx + i te i tx e i tx a + t C 0 e ax a e i tx + e i tx a + t a ±i t, F a [cos tx C 0 e ax + 4:, sin tx ei tx i tx e i [ e i tx e F a [sin tx F a i i tx i e i tx a i t + e i tx a + i t t e i tx e i tx a + t C 0 e ax a cos tx t sin tx i a + t xe ax e ax a. a ±i t, 4.5, F a [e i tx F a [e i tx C 0 e ax i C 0 e ax ae i tx e i tx + i te i tx + e i tx i a + t C 0 e ax a e i tx e i tx a + t + i e i tx a i t e i tx a + i t t e i tx + e i tx a + t C 0 e ax a sin tx + t cos tx a + t a ±i t, F a [sin tx C 0 e ax a xe ax + e ax t a 5:. a b + i t b, t R. n 0, 4.5, F a [ e ax e ax dx C 0 e ax a,. n. i.e. e ax F a [x j ax j! C j e a j+ j a k k! xk, j 0,,..., n

17 6, 4.-d, e ax F a [x n+ e ax x n xdx x e ax x n dx xe ax F a [x n e ax F a [x n dx x ax n! C n e a n+ ax n! e a n+ ax n! e a n+ ax n! e a n+ ax n! e a n+ ax n! C e a n+ e ax x n dx dx a k k! xk ax n! C n e a n+ a k+ x k+ + n! k! a n+ a k+ x k+ + n! k! a n+ a k+ x k+ + n! k! a n+ a k+ k! [ ax n! n C e a n+ [ n ax n! C e a n+ ax n! C e a n+ [ a n+ n! j0 j0 j0 a j j! a j x k+ ax n! + C e a n+ a k+ k! a k+ k! n+ ax n +! a k C e a n+ k! xk, n +. 6: 5 4.5, b j x j [ b j F a x j F a j0 j0 x k+ + x k+ + b j C j e ax j! a j+ j0 e ax x j dx a j j! xj j0 j! e ax F a [x j a j ax j! C j e j! a j+ a k k! xk jk j j0 a k k! xk n + k a k k! xk j a k+ x k+ a k k! k! xk + n + k x n+ a k + n + k! xk j a k k! xk C 0 e ax j j0 dx a k k! xk a k k! xk b j j! k! ak j+ x k

18 I fx {p k } N N y N + p k y k fx, N t N + p k t k 0, {a j } N j C., y C N x I x I yx F a F a F an [f 4.. G F G F [f G[F [f.,, F k F ak. Proof.. N,,. y + p 0 y f y a y f e a x y e a x f e ax y e ax fxdx y e a x e ax fxdx F [f. N n. N n +, n t n+ + p k t k t a t a t a n t a n+ t n + q k t k t a n+ {q k } n C. n p k t k t n + q k t k t a n+ t n+, n n q k t k+ a n+ t n a n+ q k t k n q k t k a n+ t n a n+ q k t k k n [q k q k a n+ t k + [q n a n+ t n q 0 a n+ k p 0 q 0 a n+, p n q n a n+, p k q k q k a n+ k,,..., n., fx y n+ + y n+ + k n p k y k y n+ + [q n a n+ y n + [q k q k a n+ y k q 0 a n+ y n q k y k a n+ y n + q k y k n n y n+ + q k y k+ a n+ y n + q k y k n n y n + q k y k a n+ y n + q k y k e an+x [ n e y an+x n + q k y k k

19 8, n e y an+x n + q k y k, N n n e an+x fxdx y n + q k y k F n+ [f y F F F n [F n+ [f F F F n F n+ [f, N n +., N 4. N. 4.7 n. {a k } n k C F F F n [f k F k [f A k, A k a k a a k a k a k a k+ a k a n.,. F n a : n {}}{ F a F a F a e ax F n a [f n, b k n k! xn k e ax x k fxdx, {b k } R, : n b k x n k n k! e ax F a [x k f b 0, k b j b k k j! j0 k,,... Proof :.. n,, F F [f e ax e ax F [fdx e ax,. e a ax e ax fxdx dx [ e a x e a a x e e ax a a x fxdx e ax fxdx a a a a [e ax e ax fxdx e ax e ax fxdx a a. n. F F 3 F n [f k F k [f B k k F [f F [f a a, B k a k a a k a k a k a k+ a k a n., [ n F k [f F F F n [f F [F F n [f F F F k [f B k B k k k B k F k [f F [f a k a F k [f F [f A k k F k [f A k F [f k A k k

20 4. 9, n. k , A k n A k,n a k a a k a k a k a k+ a k a n + A, A, a a + a a 0.,. n j. i.e. j k A k,j 0, {b k } b j a j+, b k a k k,,..., j D k b k b b k b k b k b k+ b k b j., j 4.3, j k D k 0 A j+,j+ a j+ a a j+ a j a j+ a j b j b b j b j a j+ a j a j a j+ D j A j,j+ a j a a j a j a j a j+ a j a j+ A j,j A k,j+ a k a a k a k a k a k+ a k a j a k a j+ [ a j a j+ a k a a k a k a k a k+ a k a j a k a j a k a a k a k a k a k+ a k a j a k a j+ [ a j a j+ A k,j b k b b k b k b k b k+ b k b j b k b j [ a j a j+ A k,j D k k,,..., j j+ k A k,j+ a j a j+ j k [ A k,j j D k a j a j+ A k,j k j D k k 0

21 0, n j +. :.. n,, e ax F a F a [f e ax F a [fdx e ax fxdx dx x e ax fxdx xe ax fxdx b 0 x e ax fxdx + b e ax xfxdx,.. n. i.e. n e ax Fa n b k [f n k! xn k e ax x k fxdx,, e ax F n a [f e ax F a [F n a [f n n n n n b k n k! e ax Fa n [f dx x n k b k [ x n k n k! n k b k n k! xn k n b k n k! b k n k! xn k e ax x k fxdx dx e ax x k fxdx e ax x k fxdx x n e ax fxdx b k n k! xn k e ax x k fxdx + b n e ax x k fxdx x n k n k e ax x k fxdx e ax x n fxdx, n.

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

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