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2 1 2, (Theorem 1) (Theorem 2)
3 1 2, (Theorem 1) (Theorem 2)
4 . (c.f. De Donno and Praelli, 24) (Ω, F, (F ) [,T ], P):, usual condiions, T < P : predicable σ-field on Ω [, T ] H 2 = { } X : (index se) C = C(X ), M = C (Radon measures on X ) def µ, ϕ M,C = X ϕ(x) µ(dx), µ M, ϕ C. M = {(M x ) [,T ] } x X, M x = (M x ) [,T ] H 2, x X
5 Assumpion A: sricly increasing, bounded, coninuous, adaped process, Q: Ω [, T ] X X R, P B(X ) B(X )-measurable s.. (ω, ) Ω [, T ], X X (x, y) Q ω, (x, y) R coninuous, symmeric, nonnegaive-definie. X X (x, y) E [ T Q s(x, y) da s ] R coninuous, symmeric, nonnegaive-definie. M x, M y = Q s(x, y) da s, x, y X.
6 Simple inegrand Simple inegrand: H = n h i δ x i i=1 (h i : R-valued bounded predicable process, x i X, i = 1,..., n) H M H s dm s def = n i=1 h i s dm x i s.
7 Simple inegrand H s dm s H 2 [ ( T ) 2 ] E H s dm s = E = E T T n hsh i s j d M x i, M x j s i,j=1 n hsh i sq j s (x i, x j ) da s i,j=1 [ T ] = E H s, Q s H s M,C da s., coni. symm. nonnegaive-def. Q: X X R, linear map Q: M C ; (Qµ)( ) def = Q(, y) µ(dy), µ M. X
8 Measure-valued inegrand inegrand ; H is an M-valued predicable process s.. L 2 (M, M) = H [ T ] 1/2 E H s, Q s H s M,C da s <. H L 2 (M, M), H s dm s H 2. quadraic variaion: H s dm s = H s, Q s H s M,C da s Measure-valued inegrand, Finance., L 2 (M, M) [ ] T 1/2 H E H s, Q s H s M,C da s.
9 Generalized inegrand Coni. symm. nonnegaive-def. Q: X X R linear map Q: M C, M/KerQ ) def (µ, ν) UQ = µ, Qν M,C (= Q(x, y) ν(dy)µ(dx) X X. M/KerQ compleion U Q. U Q separable Hilber space., M = {M x } x X covariance operaor Q,ω (x, y), (, ω) [, T ] Ω Hilber space U,ω = U Q,ω. (covariance spaces) Process H = (H ) [,T ] (, ω) [, T ] Ω H,ω U,ω, U-valued.
10 Generalized inegrand inegrand ; H is a U-valued predicable process s.. L 2 (M, U) = H [ T ] 1/2 def H M = E H s 2 U s da s <. (L 2 (M, U), M ) Hilber space. H L 2 (M, U), ( ) H s dm s H 2. H s dm H s 2 = H M. L 2 (M, U) Finance.. BSDE
11 1 2, (Theorem 1) (Theorem 2)
12 , (Theorem 1) M = {M x } x X (given). f ξ, BSDE(f, ξ); T T T Y = ξ + f (s, Y s, H s ) da s H s dm s dn s, [, T ] riple (Y, H, N). BSDE(f, ξ) Finance ξ : (F T - ) f : Y : ξ (adaped process) H: ( ) N : cos process.
13 , (Theorem 1) BSDE(f, ξ); Y = ξ + T Assumpion f (s, Y s, H s ) da s T H s dm s T dn s, [, T ] ξ L 2 (F T, P) f {(ω, s, y, h) ω Ω, s [, T ], y R, h U s,ω } ( ), ; [ ] T 1 E f (ω,,, )2 da (ω) <. 2 η, θ : posiive, bounded, predicable processes s.. f (, y 1, K 1 ) f (, y 2, K 2 ) η y 1 y 2 + θ K 1 K 2 U for any y 1, y 2 R and U-valued processes K 1, K 2. (f, ξ) sandard daa.
14 , (Theorem 1) BSDE(f, ξ); Y = ξ + T Definiion f (s, Y s, H s ) da s T H s dm s T dn s, [, T ] (f, ξ): sandard daa. riple (Y, H, N) BSDE(f, ξ) ; [ ] T Y : R-valued càdlàg adaped, E Y 2 da <. H L 2 (M, U) N H 2, srongly orhogonal o M = {M x } x X. (i.e. N, M x, x X ) (Y, H, N) (f, ξ).
15 , (Theorem 1) BSDE(f, ξ); Y = ξ + T f (s, Y s, H s ) da s T Theorem 1 (, H. 218) H s dm s T dn s, [, T ] (f, ξ): sandard daa, BSDE(f, ξ) (Y, H, N). Remark (Economic inerpreaion) H L 2 (M, U) ξ,., Finance, H.
16 (Theorem 2) BSDE,. {x 1, x 2,... } X. n N, n. M n def = (M x 1,..., M xn ) M n - (n ) L 2 (M n, R n ). H L 2 (M n, R n ), H s dm n s H 2.
17 (Theorem 2) Lemma n N: fix. H = (H 1,..., H n ) L 2 (M n, R n ), measure-valued inegrand H = n i=1 Hi δ xi, H L 2 (M, U), H s dm n s = H s dm s., L 2 (M n, R n ) L 2 (M, U).
18 (Theorem 2) n N: fix. Sandard daa (f, ξ), BSDE n (f, ξ) ; Y n = ξ + Definiion T f (s, Y n s, H n s ) da s T H n s dm n s T n N, (f, ξ): sandard daa. riple (Y n, H n, N n ) BSDE n (f, ξ) ; [ ] Y n T : R-valued càdlàg adaped, E (Y n ) 2 da <. H n L 2 (M n, R n ) dn n s, [, T ] N n H 2, srongly orhogonal o M n = (M x 1,..., M xn ). (i.e. N, M x i, i = 1,..., n) (Y n, H n, N n ) n (f, ξ)., (c.f. Carbone, Ferrario, and Sanacroce, 27).
19 (Theorem 2) Theorem 2 (, H. 218) (f, ξ): sandard daa. (Y, H, N) BSDE(f, ξ), n N (Y n, H n, N n ) BSDE n (f, ξ). ; lim = Y n in L 2 (da dp), lim n Hn = H in L 2 (M, U), lim n Nn = N in H 2., γ >, n N. Y Y n 2 L 2 (da dp) + H Hn 2 M γ N Nn 2 H 2
20 (Theorem 2) Remark (Economic inerpreaion) n N, M n = (M x 1,..., M xn ) n-h small marke ( M ), L 2 (M n, R n ). BSDE n (f, ξ) (Y n, H n, N n ), H n n-h small marke ( ) (c.f. Schweizer, 28). Theorem 2,, small marke ( ).
21 (Theorem 2), 1/5 α 2 = 1 + η + θ 2, K = α2 s da s, β > ( ), L 2 (da dp) [ T ] 1/2 def X T,β = E e βk X 2 da. M,β, H 2,β. δ n Y = Y Y n, δ n H = H H n, δ n N = N N n. α, K,. αδ n Y 2 T,β + δ nh 2 M,β + δ nn 2 H 2,β
22 (Theorem 2), 2/5 e βk δ n Y 2 ; E [e βk T δ n Y T 2] E [ δ n Y 2] [ T ] = E e βk (βα 2 δ n Y 2 2δ n Y (f (, Y, H ) f (, Y n, H n ))) da [ T )] + E e ( δ βk n H 2 U da + d δ n N + 2d δ n H s dm s, δ n N β αδ n Y 2 T,β + δ nh 2 M,β + δ nn 2 H 2,β [ T ] E e βk 2 δ n Y f (, Y, H ) f (, Y n, H n ) da [ T ] + 2E e βk d δ n H s dm s, δ n N 2 1
23 (Theorem 2), 3/5 1 [ ] T E eβk 2 δ n Y f (, Y, H ) f (, Y n, H n ) da f Lipschiz 2ab µ 2 a 2 + b2 for all a, b R and µ > µ 2, µ > 1 (2 + µ 2 ) αδ n Y 2 T,β + 1 µ 2 δ nh 2 M,β.
24 (Theorem 2), 4/5 2 [ T 2E eβk d δ nh s dm s, δ n N ] Ĥ n s,ω H s,ω U s,ω (span{δ x1,..., δ xn }). Ĥn L 2 (M, U), H Ĥn L 2 (M n, R n ) δ n N = N N n M n = (M x 1,..., M xn ) srongly orhogonal, δ n H s dm s, δ n N = Ĥ n s dm s, δ n N. Kunia Waanabe ineq., [ T ] 2 2E e βk Ĥn 2 U da δ nn 2 H 2,β.
25 (Theorem 2), 5/5 β, µ > n N, ( β 2 µ 2 ) αδ n Y 2 T,β + ( 1 1 µ 2 ) δ n H 2 M,β δ nn 2 H 2,β [ T 2E e βk Ĥn 2 U da ]. (β > 3, µ >, >.) lim n Ĥn,ω U,ω =,, ω, e βk Ĥn 2 U e βk H 2 U L 1 (da dp), n N, [ T ] E e βk Ĥn 2 n U da.
26 R. Carbone, B. Ferrario, and M. Sanacroce. Backward sochasic differenial equaions driven by càdlàg maringales. Teor. Veroyan. Primen., 52(2): , 27. M. De Donno and M. Praelli. On he use of measure-valued sraegies in bond markes. Finance Soch., 8(1):87 19, 24. Y. Hamaguchi. BSDEs driven by cylindrical maringales wih applicaion o approximae hedging in bond markes. ArXiv e-prins, June 218. M. Schweizer. Local risk-minimizaion for mulidimensional asses and paymen sreams. In Advances in mahemaics of finance, volume 83 of Banach Cener Publ., pages Polish Acad. Sci. Ins. Mah., Warsaw, 28.
O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0
9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
第10章 アイソパラメトリック要素
June 5, 2019 1 / 26 10.1 ( ) 2 / 26 10.2 8 2 3 4 3 4 6 10.1 4 2 3 4 3 (a) 4 (b) 2 3 (c) 2 4 10.1: 3 / 26 8.3 3 5.1 4 10.4 Gauss 10.1 Ω i 2 3 4 Ξ 3 4 6 Ξ ( ) Ξ 5.1 Gauss ˆx : Ξ Ω i ˆx h u 4 / 26 10.2.1
,. 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,
(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
December 28, 2018
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第1章 微分方程式と近似解法
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
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20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = , ( ) : = F 1 + F 2 + F 3 + ( ) : = i Fj j=1 2
6 2 6.1 2 2, 2 5.2 R 2, 2 (R 2, B, µ)., R 2,,., 1, 2, 3,., 1, 2, 3,,. () : = 1 + 2 + 3 + (6.1.1).,,, 1 ,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = 1 + 2 + 3 +,