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1 Ogawa SFC ( ). Fourier (SFC),. SFC Ogawa, [5], [], [6]. [5], [], SFC, [6], cross variaion., CONS Ogawa cross variaion. 2. (B ) [,] filer (Ω, F, (F ) [,], P ) Brown, (e n ) n N L 2 ([, ] ; C) CONS e n,. L 2 ([, ] ; R) CONS (φ m ) m N, sup M φ m φ m(s) ds. L L [,]< 2 ([, ] ; R) 2 M N m= CONS Ogawa ( Ogawa ), Iô, r i- parameer 2 Wiener Sobolev, Skorokhod d B, db, Lr,2 i, δb ([2] Definiion 2.,2.3,2.4 ). X, Y : [, ] L (Ω) cross variaion, X, Y, quadraic variaion X, X, [X]. a L ([, ] Ω ; C), b L (Ω ; L 2 ([, ] ; C)). ( Ogawa SFC) n N, e n a Ogawa. Ogawa d Y = a() d B + b() d, [, ] (e n ) n N Fourier ( Ogawa SFC,, SFC-O ) (e n, d Y ) : (e n, d Y ) := e n () d Y = e n ()a() d B + e n ()b() d., b =, (e n, d Y ) = (e n, a d B) a() SFC-O. 3. Ogawa SFC, (L ([, ] Ω) ) : A = {A L ([, ] Ω) Re A, Im A a.s. }, { } M = f db f L (Ω ; L 2 [, ]), f (F ) [,] -, { } { } W = f δb f L 2,2 + span T K f f L,2, sup K(, ) L 2 [,] <, [,], f L,2, K L 2 ([, ] 2 ), T K f() := K(, s)f(s) ds., A, W. L = A + M + W. su332@edu.osakafu-u.ac.jp

2 a L, : () v : [, ] R, va Ogawa. (2) L 2 - (v n ) n N, l.i.p. v na d φ B =., P((e n, d Y )) n N () := l.i.p. N N n=, d Y = a() d B + b() d, [, ]. n e n(s) ds (e n, d Y ) = Y, [, ],, P cross variaion, SFC-O L e, PC : L e, PC {a = L ([, ] Ω) P((e n, a d B)) n N = a d B, [ ] s } s, [, ] a d B = a(u) 2 du, a d B, B s = a(u) du. 2 A, M, W L e, PC. L e, PC. a() Re a, Im a L e, PC., d Y = a() d B +b() d, [, ] : () a() ((e n, d Y )) n N a() = d d P((en, d Y )) n N, B. (2) Re a, Im a, Re a Im a, (sgn a)a ((e n, d Y )) n N Y = P((e n, d Y )) n N : f, g L e, PC f(s)g(s) ds = f d B, g d B. ) (2) (B ) [,]. 2) ((e n, d Y )) n N SFC (e n, d Y ) (),(2). 3) a() b ((e n, d Y )) n N. 4) a L, a(). [] K. Hoshino, Idenificaion of finie variaion processes from he SFC. MSJ Auumn Meeing, absrac (27) [2] K. Hoshino, T. Kazumi, On he Ogawa inegrabiliy of noncausal Wiener funcionals, o appear, Sochasics (28) [3] D. Nualar, E. Pardoux, Sochasic calculus wih anicipaing inegrands. Probab. Th. Rel. Fields, 78, pp (988) [4] S. Ogawa, The sochasic inegral of noncausal ype as an exension of he symmeric inegrals. Japan J. Appl. Mah. 2, (985) [5] S. Ogawa, H. Uemura, On he idenificaion of noncausal funcions from he SFCs. RIMS Kôkyûroku. 952, (25-6) [6] S. Ogawa, H. Uemura, On he reconsrucion of random funcion from is SFCs defined by an arbirary CONS. Symposium on Probabiliy Theory, absrac (27) sgn z = if arg z < π, sgn z = o/w, z C (arg := ).

3 websie: hps://sies.google.com/sie/ryoichisuzukifinance/ December 9, 28. Locally risk-minimizing (LRM) is a well-known hedging mehod for coningen claims in a quadraic way (see e.g., [2] and [3]). By using Malliavin calculus, we can obain explici represenaions of LRM for incomplee marke models whose asse price process is described by a soluion o a sochasic differenial equaion (SDE) driven by a Lévy process ([]). On he oher hand, here is one imporan derivaive securiy describe by indicaor funcion called digial opion. A digial opion pays a fixed cash amoun if some condiion is realized. Mahemaical represenaion of digial (or binary) opions are given by { for ST K, [K, ) (S T ) = oherwise, where {S } [,T ] is a sock price process and K > is a consan number ha is fixed by he conrac. I is popular and imporan derivaive securiy. Therefore, o sudy digial opions, we consider Malliavin differeniabiliy of indicaor funcions ([4]). In his alk, we firs consider Malliavin differeniabiliy of indicaor funcions on canonical Lévy spaces. By using i, we obain explici represenaions of LRM for digial opions in markes driven by Lévy processes. References [] T. Arai and R. Suzuki. Local risk-minimizaion for Lévy markes. Inernaional Journal of Financial Engineering. vol.2 (25), no. 2, 555, 28 pp. [2] M. Schweizer. A guided our hrough quadraic hedging approaches. Handbooks in Mahemaical Finance: Opion Pricing, Ineres Raes and Risk Managemen. Cambridge Universiy Press, , 2. [3] M. Schweizer. Local risk-minimizaion for mulidimensional asses and paymen sreams. Banach Cener Publicaions. 83 (28), [4] R. Suzuki. Malliavin differeniabiliy of indicaor funcions on canonical Lévy spaces. Saisics and Probabiliy Leers. 37 (28), 83 9.

4 FBSDES ) () Forward-backward sochasic differenial equaions (FBSDEs) ( ) ( ) X() = X() + b s, ω, Θ(s) ds + σ s, ω, Θ(s) dw (s), Y () = φ (X(T )) + T ( ) f s, ω, Θ(s) ds T Z(s) dw (s), Θ (X, Y, Z) (X(), Y (), Z()) [,T ] R l R m R m d W d- Wiener b, f, σ, φ FBSDEs () Y (T ) = φ (X(T )) Sochasic differenial equaions (SDEs), [8] [4] Conracion mapping: T >. The Four Sep scheme:. The mehod of coninuaion: monooniciy. (X, Y, Z) (X n, Y n, Z n ) Newon : φ, b, f, σ ) ) X n+ () = X n+ () + b n (s, ω, Θ n+ (s) ds + σ n (s, ω, Θ n+ (s) dw (s), (2) Y n+ () = φ n (X n+ (T )) + T ) f n (s, ω, Θ n+ (s) ds T Z n+ (s) dw (s). φ n (x) = φ(x n (T )) + x φ(x n (T ))(x X n (T )), x R l b n, σ n, f n () θ = (x, y, z) R l R m R m d ) ( ) ( ) b n (s, ω, θ = b s, ω, Θ n (s) + θ b s, ω, Θ n (s) (θ Θ n (s)), (s, ω, θ) [, T ] Ω R l R m R m d, σ n, f n b n, σ n, f n Θ n (X n, Y n, Z n ) (X n (), Y n (), Z n ()) [,T ] well-defined FBSDEs () [4]. [6] X (Y, Z) decoupled FBSDEs, (3) X() = X() + Y () = φ (X(T )) + b(s, X(s)) ds + T σ(s, X(s)) dw (s), f(s, X(s), Y (s), Z(s)) ds T Z(s) dw (s). Theorem ([6]). b, σ, f, φ (s, ω)-a.e. ( ) T ( ) E X() 2 2 ( ) σ 2 ( ) f 2 + b s, ω, + s, ω, + s, ω,,, ds <,

5 decoupled FBSDEs (3) T b, σ, f C > X () = X() (X, Y, Z ) S 2 l S2 m H 2 (X X n+, Y Y n+, Z Z n+ ) C2 n, n N {}. [5, page 453] [2, Theorem XVI] Vidossich Chaplygin Banach [7, Theorem (.3)] SDEs [3]. decoupled FBSDEs Forward X [] Gronwall Backward (Y, Z) T > m N { [ ] } S 2 m = Y : Ω [, T ] R m coninuous, adaped : Y S 2 m = E sup Y (s) 2 <, s T { [ ] } T H 2 = Z : Ω [, T ] R d adaped : Z H 2 = E Z(s) 2 ds <, Banach S 2 m H 2 S 2 l (Y, Z) 2 = Y 2 S + 2 Z 2 m H 2, X 2 = X 2 S. 2 l α R, [ ] [ ] T (Y, Z) 2 α = E sup e αs Y (s) 2 + E e αs Z(s) 2 ds. s T Kanorovich n N [6] References. Kazuo Amano, A noe on Newon s mehod for sochasic differenial equaions and is error esimae., Proc. Japan Acad., Ser. A 85 (29), no. 3, L Kanorovich, The mehod of successive approximaions for funcional equaions, Aca Mah. 7 (939), Shigeoku Kawabaa and Toshio Yamada, On Newon s mehod for sochasic differenial equaions, Séminaire de probabiliés de Srasbourg, Lecure Noes in Mah., vol. 485, Springer, Berlin, 99, pp Jin Ma, Zhen Wu, Deao Zhang, and Jianfeng Zhang, On well-posedness of forward-backward SDEs a unified approach., Ann. Appl. Probab. 25 (25), no. 4, J M Orega and W C Rheinbold, Ieraive soluion of nonlinear equaions in several variables, Classics in Applied Mahemaics, vol. 3, Sociey for Indusrial and Applied Mahemaics (SIAM), Philadelphia, PA, Dai Taguchi and Takahiro Tsuchiya, Newon-Kanorovich mehod for decoupled forward-backward sochasic differenial equaions, (28). 7. Giovanni Vidossich, Chaplygin s mehod is Newon s mehod, Journal of Mahemaical Analysis and Applicaions 66 (978), no., Jianfeng Zhang, Backward sochasic differenial equaions. From linear o fully nonlinear heory., New York, NY: Springer, 27. QR

6 CONSERVATIVENESS AND FELLER PROPERTY OF DIFFUSION PROCESSES ON RIEMANNIAN MANIFOLDS WITH m-bakry-émery RICCI TENSOR FOR m (K. Kuwae). Laplacian Xiang-Dong Li Laplacian [5], Songzi Li [6]. (M, g) ϕ C 2 (M) C (M) L L := ϕ, µ := e ϕ vol g M ( Lf)gdµ = M f, g dµ =: E(f, g) f, g C (M) L Wien Laplacian Laplacian (E, F) (E, C (M)) L2 (M; µ) (E, F) X = (Ω, X, P x ) (E, F) L 2 (M; µ)- (ϕ C (M) [3, 7.5] ) m ], + ] m-bakry-émery Ric m,n(l) Ric m,n (L)(x) := Ric(x) + Hess ϕ(x) ϕ(x) ϕ(x) m n M K Ric m,n (L)(x) K(x), x M CD(K, m) m = n ϕ Ric m,n (L)(x) = Ric(x) m n K (M, d g, µ) RCD(K, m)- m Laplacian L ([5]) X Feller r p (x) := d g (x, p) m = κ [9]. (Laplacian [5], cf. [9]). x, p M, m Ric m,n (L)(x) (n m)κ(s p (x))e 4ϕ(x) n m Lr p (x) (n m)co κ (s p (x))e 2ϕ(x) n m s p (x) < δ κ κ [, + [ s p (x) { } rp (x) s p (x) = inf e 2ϕ(γ ) n m d γ γ = p, γ rp(x) = x co κ lim s s co κ (s) = Riccai (.) dco κ ds (s) = κ(s) + co κ(s) 2, [, δ κ [ δ κ κ δ κ = + δ κ < lim s δκ (δ κ s) co κ (s) = κ κ co κ (s) = κ cos( κs)/ sin( κs), δ κ = π/ κ +

7 2. ϕ p (r) := inf Br (p) ϕ K [, + [ (K) : (K) : r o dr ( ) = + for some r o >. K e 2ϕ p(r) n m r e 2ϕ p(r) n m (K) p M 2. (X ). p M (K) (2.) Ric m,n (L)(x) K(s p (x))e 4ϕ(x) n m for any x M m ], ] X 2.2 (X Feller ). (K) (2.2) Ric m,n (L)(z) K(s q (z))e 4ϕ(z) n m for any z, q M m ], ] X Feller 2. Grigor yan [2] 2.2 Azenco [] Feller ([4] Laplacian ) m n Laplacian Ric m,n (L)(x) (m )κ(r p (x)) Lr p (x) (m )co κ (r p (x)), r p (x) < δ κ, X Feller (K) r o dr K (r) = + for some r o > ([8, Theorems.4 and.5]) References [] R. Azenco, Behavior of diffusion semi-groups a infiniy, Bull. Soc. Mah. France 2 (974), [2] A. Grigor yan, On sochasically complee manifolds, Dokl. Akad. Nauk. SSSR 29 (986), [3] A. Grigor yan, Hea kernel and analysis on manifolds, AMS/IP Sudies in Advanced Mahemaics, 47. American Mahemaical Sociey, Providence, RI; Inernaional Press, Boson, MA, 29. [4] E. P. Hsu, Sochasic analysis on manifolds, Graduae Sudies in Mahemaics, 38. American Mahemaical Sociey, Providence, RI, 22. [5] K. Kuwae and X.-D. Li, Laplacian comparison heorem on Riemannian manifolds wih CD(K, m)-condiion for m, preprin, 28. [6] K. Kuwae, S.-Z. Li and X.-D. Li, Conservaiveness and Feller propery of diffusion processes on Riemannian manifolds wih m-bakry-émery Ricci ensor for m, 28, preprin. [7] K. Kuwae, S.-Z. Li and X.-D. Li, Liouville heorems and Harnack inequaliies on Riemannian manifolds wih CD(K, m)-condiion for m <, 28, in prepraion. [8] X.-D. Li, Liouville heorems for symmeric diffusion operaors on complee Riemannian manifolds, J. Mah. Pures Appl. (9) 84 (25), no., [9] W. Wylie and D. Yeroshkin, On he geomery of Riemannian manifolds wih densiy, preprin, 26.

8 Geomery of he random walk range condiioned on survival among Bernoulli obsacles Jian Ding (Universiy of Pennsylvania), Rongfeng Sun (Naional Universiy of Singapore), Changji Xu (Universiy of Chicago) Z d Bernoulli (annealed 2 ) (ω, P p ) Z d Bernoulli( p) ({S n } n, P ) d ω = (ω x ) x Z d obsacles O(ω) := {x Z d : ω x = } τ O µ N ( ) := P p P ( τ O(ω) > N) annealed pah measure d 2 Confinemen propery. (Szniman [6], Bolhausen [2] for d = 2, Povel [5] for d 3) d 2, p (, ) ϱ (d, p) > x N (ω) Z d ϵ > ( ) µ N S [,N] B(x N (ω), (ϱ + ϵ)n d+2 ), N. (confinemen) [4] S [,N] B(ω) ( ) N ϵ >, µ d N d+2 S[,N] B(, ϱ ) > ϵ, N. [2] Theorem. Confinemen propery x N ϵ > ( ) µ N S [,N] B(x N (ω), (ϱ ϵ)n d+2 ), N. (covering) Remark. Beresycki Cerf [] (confinemen) (covering) Bolhausen [2] (covering) (confinemen) [] d 3 ryoki@kurims.kyoo-u.ac.jp 2 quenched

9 (confinemen) (covering) log N B(x N (ω), ϱ N d+2 ) Theorem 2. a > ( ) µ N S [,N] N d d+2 (log N) a, N. S [,N] B(x N (ω), ϱ N d+2 ) Hausdorff 99 Szniman References [] N. Beresycki and R. Cerf. The random walk penalised by is range in dimensions d 3. arxiv:8.47. [2] E. Bolhausen. Localizaion of a wo-dimensional random walk wih an aracive pah ineracion. Ann. Probab. 22, , 994. [3] J. Ding, R. Fukushima, R. Sun and C. Xu. Geomery of he random walk range condiioned on survival among Bernoulli obsacles. arxiv: [4] R. Fukushima. Asympoics for he Wiener sausage among Poissonian obsacles. J. Sa. Phys. 33, , 28. [5] T. Povel. Confinemen of Brownian moion among Poissonian obsacles in R d, d 3. Probab. Theory Relaed Fields 4, 77 25, 999. [6] A.-S. Szniman. On he confinemen propery of wo-dimensional Brownian moion among Poissonian obsacles. Comm. Pure Appl. Mah. 44, 37 7, 99. 2

10 A limi heorem for persisence diagrams of random complexes buil over marked poin processes * * 2 Euclid birh deah q = {(x, y) [, ] 2 : x < y } q = = =2 =3 deah 3 birh 2 Čech 2 S F(S) S M Polish κ : F(R d M) [, ] (K), (K2) (K) A B κ(a) κ(b). (K2) ρ : [, ] [, ] < ρ() < (x, m), (y, n) R d M x y ρ(κ({(x, m), (y, n)})) κ (T) x R d κ(t x A) = κ(a). T x F(R d M) T x A = {(y + x, m) : (y, m) A}. : F(R d M) (R) U O(d) κ(r U A) = κ(a). O(d) d R U : F(R d M) F(R d M) R U A = {(Ux, m) : (x, m) A}. π : R M R Ξ F(R d M) π Ξ Ξ = π( Ξ) Ξ F(R d M) π F( Ξ) σ σ F(Ξ) σ = {(x, m ), (x, m ),..., (x q, m q )} R d σ = {x, x,..., x q } {m, m,..., m q } Ξ F(R d M) K( Ξ) = {K( Ξ, )} K( Ξ, ) = {σ Ξ : κ( σ) } * JST CREST JPMJCR5D3 *2 k-suzaki@imi.kyushu-u.ac.jp

11 κ( σ) K( Ξ) σ. K( Ξ) κ- κ ι s r q H q (K( Ξ, )) K( Ξ, ) q r s : H q (K( Ξ, r)) H q (K( Ξ, s)) K( Ξ, r) K( Ξ, s) H q (K( Ξ)) = ({H q (K( Ξ, ))}, {ι s r} r s ) q H q (K( Ξ)) n q H q (K( Ξ)) I(b i, d i ) ([2]) I(b i, d i ) q K( Ξ) = b i b i i= < d i = d i D q ( Ξ) = {(b i, d i ) : i =, 2,..., n q } q D q ( Ξ) ξ q ( Ξ) = (b i,d i ) D q (Ξ) δ (bi,d i ) Radon Φ R d M R d Φ( ) = Φ( M) R d M R d {T x } x R d {R U } U O(d) R d M Φ Φ R d Borel A p E[Φ(A) p ] < L > Λ L = [ L/2, L/2) d M Φ κ- K( Φ ΛL ) = {K( Φ ΛL, )} q ξ q ( Φ ΛL ) ξ q,l. κ (T) Φ R d Φ. q Radon ν q L E[ξ q,l ]/L d v ν q v κ (R) Φ, L L d ξ v q,l ν q [] [] Y. Hiraoka, T. Shirai and K. D. Trinh, Limi heorems for persisence diagrams, Ann. Appl. Probab. 28 (28) [2] A. Zomorodian and G. Carlsson, Compuing persisen homology, Discree Compu. Geom. 33 (25)

12 Flucuaion resuls in Firs-passage percolaion 3 Firs-passage percolaion, opimizaion problem, random environmen Firs-passage percolaion Hammersley Welsh 965 Z d E(Z d ) e E(Z d ) τ e Z d e e k π (π) = k i= τ e i x, y Z d : T (x, y) := inf{(π) : π x y }. τ e e T (, x) x B() = {x R d T (, [x]) } B() Definiion. τ : { pc (d) if τ =, P(τ e = τ) < p c (d) oherwise, τ τ e suppor p c (d), p c (d) d percolaion d oriened percolaion B() Definiion 2. l > Γ R d Γ l = {v Γ d(v, Γ c ) l} and Γ + l = {v R d d(v, Γ) l}, d A, B R d B A : F (A, B) = inf{δ > B δ A B+ δ }. Theorem. F c, C > > Γ R d P(F (B(), Γ) c log ) C exp ( c ). [] Shua Nakajima Divergence of shape flucuaion for general disribuions in firs passage percolaion arxiv: njima@kurims.kyoo-u.ac.jp

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○松本委員 CIRJE-J-100 2003 11 CIRJE hp://www.e.u-okyo.ac.jp/cirje/research/03research02dp_j.hml Credi Risk Modeling Approaches 2003 11 17 Absrac This aricle originaes from a speech given by he auhor in he seminar