Green

28 6/-3 2 ax, bx R, L = d2 ax 2 dx 2 + bx d dx 2 L X = {Xt, t }. σ y = inf{t > ; Xt = y} t Xt L y = sup{t > ; Xt = y} 2. Gauss Gamma 3. tree 4. t Black-Scholes Madan-Roynette-Yor [6] σ y E x [exp λσ y ], x = X, L Green L y X scale s s = speed pt, x, y P x L y dt = pt, x, ydt 2sy Bessel * 27 *2 i

......................................2 Green...................................... 3.3.................................... 6.4.................................. 7 2 Gamma GIG 3 2........................................ 3 2.2........................................ 5 2.3 tree GIG, Gamma.............................. 7 3 2 3. Black-Scholes formula................................ 2 3.2.................................. 24 A 2.5 27 3 ii

. S R R + =, S a : S R +, b : S R 2 L = d2 ax 2 dx 2 + bx d dx X = {Xt, t } W, P Wiener dxt = axtdwt + bxtdt, X = x, X = {Xt, t } W x 2 X Wiener P P x X Xwt = wt, w W x, P x E x f E x [fxt] fx lim = Lfx. t t {P x } x S X S = R ± S = R +.2. * W = {w : [, R; w = } *2 W x = {w : [, S; w = x}

c S s x, m x x s 2bz x = exp az dz, m x = 2 x ax exp c L = d2 ax 2 dx 2 + bx d dx = m x d d dx s x dx c 2bz az dz, m xdx speed s x sx scale.2 d d dm ds mdx sx m density s S = R + c s ξdξ c ξ m ηdη =, s ξdξ ξ c c m ηdη =, L L 2 S, m xdx Luxvxm xdx = uxlvxm xdx, S Ls = S u, v C S,. X y σ y σ y = inf{t > ; Xt = y} α, β S, α < x < β, P x σ β < σ α = sx sα sβ sα. σ = σ α σ β Ls = {sxt} {sxt σ} E x [sxt σ] = sx t p = P x σ β < σ α sx = E x [sxσ] = sα p + sβp.2 i X = {Xt} speed pt, x, y P x Xt dy = pt, x, ym ydy, t >, x, y S. ii pt, x, y u t = Lu u, x = φx ut, x = E x [φxt] = φypt, x, ym ydy S 2

φ {T t } t T t φx = E x [φxt] {T t } pt, x, y. Brown Brown {Bt, t } Brown k > B k t = Bt + kt B k = {B k t} Brown B k d 2 L k = 2 dx 2 + k d dx = 2e 2kx d d dx e 2kx dx R speed 2e 2kx dx scale e 2kx scale 2k e 2kx B = X x t = x expbt + kt {X x t} dx x t = X x tdbt + 2 + k X x tdt L = d2 x2 2 dx 2 + 2 + k x d dx = 2x 2k d dx x 2k d dx, speed 2x 2k dx scale x 2k+ scale 2k x 2k.2 Bessel δ {Bt, t } B Brown {Rt} drt = dbt + δ 2Rt dt {Rt} L = 2 dx 2 + δ d 2x dx = d 2 2x δ d d dx x δ+ dx {Rt} δ Bessel δ {Rt} δ Brown δ 2 δ > 2 scale sx x δ+2 sx <, lim x sx =.2 Green Green Green Bessel L S 2 X = {Xt} X speed m xdx pt, x, y α >.2 Green 3

G α φx = e αt E x [φxt]dt = e αt dt φypt, x, ym ydy S G α L Green Gx, y; α Gx, y; α = e αt pt, x, ydt L Green T t = exptl G α φ = e αt T t φdt = e α Lt φdt = α L φ.3 G α φ α Lu = φ Green Lu = αu.4 α > Lu = αu 2 u x; α u 2 x; α Lu = αu e, e 2 c S e c =, e 2 c =, de ds c s c de 2 c =, ds de c =, dx u, u 2 [], page 74,. hα, α >, x Wronskian hα = u s x x; αu 2 x; α u x; αu 2x; α..5 L speed Green Gx, y; α hαu x; αu 2 y; α, x y, Gx, y; α = hαu y; αu 2 x; α, x > y, φ C S gx = Gx, y; αφym ydy S g α Lg = φ g.3 Brown B k = {B k t = Bt + kt} P x Bt + kt dy = 2πt e y x kt2 /2t dy B k Lebesgue pt, x, y = e ky x y x2 exp k2 t. 2πt 2t 2 4

speed 2e 2ky dy y x2 2t pt, x, y = 2 2πt e ky+x exp k2 t 2 speed Green Gx, y; α = e αt pt, x, ydt 2 k 2 + 2α e k+ k2 +2αx e k+ k 2 +2αy, x y, α >, d 2 L = 2 dx + k d 2 dx Lu = αu u, u 2 u x; α = e k+ k 2 +2αx, u 2 x; α = e k+ k 2 +2αy, Wronskian Green.4 Bessel δ > 2 δ Bessel L = d 2 2 dx 2 + δ d 2x dx = d d 2x δ dx x δ. dx Green ν = δ 2/2 ν index Lu = 2 u x + 2 + ν x u x = αux Bessel I ν, K ν ν z z/2 2n I ν z = 2 n!γν + n + = 2πi K ν z = π 2 n= I ν z I ν z sin νπ = w z + z w z + ν2 wz = z 2 α > +πi πi. e z cosht coshνtdt. e z cosht νt dt, u x; α = x ν I ν 2αx, u 2 x; α = x ν K ν 2αx Lu = αu I νzk ν z I ν zk νz = z s x u x; αu 2x; α u x; αu 2 x; α = speed 2x δ dx = 2x 2ν+ dx Green Gx, y; α x y Gx, y; α = xy ν I ν 2αxK ν 2αy.2 Green 5

Bessel I ν xk ν x = 2 Gx, y; α = xy ν 2 e t/2 x2 +y 2 /2t I ν xy t e αt x2 +y 2 /2t xy dt I ν t t dt, < x y,. t δ Bessel speed p δ t, x, y p δ t, x, y = 2 2txy ν e x +y 2 /2t xy I ν t Lebesgue p δ t, x, y p δ t, x, y = y ν+ 2 t x ν e x +y 2 /2t xy I ν t.3 X = {Xt} S 2 L = d2 ax 2 dx 2 + bx d dx = d d m x dx s x dx c S σ c σ c = inf{t ; Xt = c}. { } = σ c = σ c Laplace L Green Lu = αu.6 c S α > E x [e ασ c ], x c, Ec [e ασ x ], x c, v x; α = Ec [e ασ x ] v 2 x; α =, x c, E x [e ασ c ], x c, S v, v 2 v i c = Lu = αu f x c fx = lim x fx = G α fc S X Markov G α fx [ ] [ ] G α fx = E x e αt fxtdt = E x e αt fxtdt [ ] [ = E x e ασc e αt fxσ c + tdt = E x [e ]E ασc c σ c ] e αt fxtdt. x c v x; α Lu = αu v c; α = x c S g x c gx = lim x gx = S = R g = S = R + G α gc v 2 6

.5 Brown {Bt} Brown k > B k = {B k t Bt + kt} c σ c c > x E x [e ασ c ] = e k+ k 2 +2αx c = P x σ c dt = c x 2πt 3 e c x kt2 /2t dt = e αt c x 2πt 3 e c x kt2 /2t dt c xec xk e c x2 /2t k 2 t/2 dt 2πt 3 Gauss x > c E x [e ασc ] = e k+ k 2 +2αx c α P x σ c < = P x inf B k t c = e 2kx c t α > 2 v + kv = αv, v c; α =,.6 Bessel δ > 2 X = {Xt} δ Bessel L u x; α = x ν I ν 2αx, u 2 x; α = x ν K ν 2αx, Lu = αu x c E x [e ασc ] = u ν x; α c u c; α = I ν 2αx x I ν 2αc, x c E x [e ασ c ] = u ν 2x; α c u 2 c; α = K ν 2αx x K ν 2αc. K ν z = 2 ν Γνx ν + o, x, α x c P x σ c < = P x inf Xt c = t 2ν c. x B 3 ξ R 3 3 Brown ξ c P inf B 3 t c = c t ξ..4 Brown δ Bessel δ > 2 S =, X = {Xt} σ = lim t Xt =.4 7

L = d2 ax 2 dx 2 + bx d dx = m x X dxt = axtdbt + bxtdt a < x < b P x σ a < σ b = sb sx sb sa d d dx s x dx s+ =, s < a b s = X y, L y L y = sup{t ; Xt = y}. { } = L y =.7 Pitman-Yor scale pt, x, y X speed m ydy P x L y dt = pt, x, y dt, t >. sy Pitman-Yor [9] Revuz-Yor [] - - [4].8 x, y, u y x = P x σ y < = P x L y >, x y u y x = sx/sy, x > y. x y u y x = x > y M > x M P x σ y < σ M = sm sx sm sy M s = P x σ y < = sx/sy.3 u y x x.9 P x L y > t = E x [u y Xt], t >. x X t y F t = σ{xs, s t} Xt y P x L y > t F t = u y Xt.9 E x [u y Xt] u y 8

. y > t {Λ y t } Xt y = X y + Xt y + = X y + + Xt y = X y + sgnxs ydxs + Λ y t,, dλ y t {Λ y t } X y {Xs>y} dxs + 2 Λy t, {Xs y} dxs + 2 Λy t, {t; Xt = y}. t > ϕ ϕxsaxsds = ϕzλ z t dz. R +.2 R + f fxt = fx + f XsdXs + Λ z t f dz 2 R + f f f f 3 f C 2 f dy = f ydy. u y Xt u y.3 y > u yx = sy s x y, x, u yx = s y sy δ yx + s x sy y, x..7 u y Xt = u y X + u y Xt s y2sy Λ y t u yxs axsdbs + s y 2 sy Λy t 4 E x [u y Xt] = u y x + s y 2sy E x[λ y t ] φ C R +. [ ] [ ] E x φxsaxsds = E x φyλ y t dy = φye x [Λ y t ]dy R + R + *3 f f *4 Pitman-Yor [9], p.327, Revuz-Yor [], p.32, 4.6 Exercise.4 9

E x [ ] φxsaxsds = ds φyayps, x, ym ydy R + = φyaym ydy R + ps, x, yds E x [Λ y t ] = aym y.9 ps, x, yds P x L y > t = E x [u y Xt] = u y x + s y 2sy E x[λ y t ] = u y x + s y 2sy aym y ps, x, yds s ym y = 2/ay t.7 Brown B = {Bt} Brown k > L c = sup{t > ; Bt + kt = c} c > x P x L c dt = k e kc x exp c x 2 2πt 2 t + k 2 t dt L c P x Gauss GIGc x 2, k 2 ; /2 c = x P x L c dt = k 2πt e k2 t/2 dt 2k 2 L c /2 Gamma Brown Xt = expbt + kt.7 B = x c σ x c L x c Brown Markov σ x c L c c L x c 2.3 c < x P x L c dt = e kc x c x = e 2kx c..5 P x L c < = P x σ c < = P x inf B k t c = e 2kx c. t.8 Bessel δ > 2 L y δ Bessel y > ν = δ 2/2 ν y P x L y dt = e x2 +y 2 /2t xy I ν. 2δ 2t x t Bessel. α >

E x [e αl y ] = σ y ν E x [e ασ y y I ν 2αx ] = x I ν 2αy e αt P x L y dt = ν y I ν 2αxK ν 2αy δ 2 x Bessel Markov E x [e αl y ] = E x [e ασ y ]E y [e αl y ].4

2 Gamma GIG 2. 2. µ > Ω, F, P γ µ P γ µ dy = Γµ yµ e y dy, y >, γ µ µ Gamma 2.2 ν R, a, b > Ω, F, P I ν ν/2 P I ν b dx = x ν a 2K ν ab exp a 2 x + bx dx, x >, I ν Gauss GIG GIGa, b; ν K ν Bessel 2. B k = {Bt + kt} x k > Brown σ c B k c σ c GIGc x 2, k 2 ; /2 2. ν = /2 Gauss GIGa, b; /2 Gauss N Gauss Nm, σ 2 k N µ log E[expµN] = mµ + σ 2 µ 2 /2 k I λ log E[expλI /2 ] K /2 z = K /2 z = π/2z /2 e z k I λ = ab ab 2λ m < a = /σ 2, b = m 2 /σ 2 k N k I λ = λ 2. X GIGa, b; ν X GIGb, a; ν I ν E[I ν α ] = α/2 a K ν+α ab b K ν ab GIG I ν Iν b,c Iν c,b Iν ] ] E [ exp α log a Iν Iν b,c [ = E exp α log c Iν c,b Iν b,a α > b,a 3

2.2 I ν Iν b,c Iν c,b Iν b,a a Iν Iν law b,c = c Iν c,b Iν b,a. Gamma GIG 2.3 µ > I µ γ µ I µ + 2b γ µ I µ Brown GIGc x 2, k 2 ; /2 k = /2 Brown 2.3 GIG Letac-Seshadri [5] 2.4 i X, Y a > Y law = 2a γ µ X law = X + Y X GIGa, a; µ law ii X, Y, Y 2 a, b > Y = 2a law γ µ Y 2 = 2b γ µ X law = Y + Y 2 + X X GIGa, b; µ X GIG X X + Y Y + X + Y 2 2. 2.3 {y n } n= [y,..., y n ], n =, 2,..., [y ] = y, [y,..., y n ] = y + [y,..., y n ]. 2.5 d X, Y, Y 2,..., r, m N law Y md+r = Y r i Z n [Y,..., Y n ] n ii {X m } m= = [Y md+, Y md+2,..., Y m+d, ], m =,, 2,..., 2. X m+ X m i Z X X m m Z iii X law = [Y, Y 2,..., Y d, X ] X 2.5 appendix 2.4 i 2.5 d =, Y i Y +/X law = Z law = 2a law γ µ X = = Y + X X Z GIGa, a; µ X GIGa, a; µ law ii 2.5 d = 2 a, b > Y 2i+ = 2a law γ µ Y 2i = 2b γ µ law X = [Y, Y 2, X ] X Z 4 2 Gamma GIG

X GIGa, b; µ X law = [Y, Y 2, X ] X GIGa, b; µ 2.6 i {Y i } i= 2a γ µ IID [Y, Y 2,..., Y n ] n GIGa, a; µ ii {Y i } i= a, b > Y 2i+ law = 2a γ µ Y 2i law = 2b γ µ [Y, Y 2,..., Y n ] n GIGb, a; µ 2.2 2.3 Gamma GIG 2.7 γ µ µ Gamma I µ, I µ γ µ GIGa, b; µ, GIGa, b; µ law, =, 2 I µ + 2b γ µ I µ I µ + 2b γ µ I µ a γ µ. 2.3 Gamma GIG 2.8 X, Y Y X + Y X law a, b, µ > X = I µ, Y law = 2b γ µ. X + Y 2.7 M = {k, k 2, 2 k ; }, k 2 ψ 2 k, k 2 = k k 2, k 2 : M, 2, ψ 2 2 k, k 2 = k, k 2 k : M, 2 ψ 2 ψ 2 2 x, y = x, y + x y + x. 2.2 µ >, a, b > K, K 2 M- P K dk, K 2 dk 2 = C k k 2 µ exp 2 ak + bk 2 dk dk 2 2.9 I ±µ, I µ b,a γ µ ψ 2 K, K 2 law = 2 a γ µ, I µ, ψ 2 2 K, K 2 law = I µ b,a, 2 b γ law µ =, 2 I µ b γ µ. 2.2 5

Laplace 2.7 2.2 ψ 2 ψ 2 2, 2 I µ b γ µ = I µ 2b γ µ + I µ, 2b γ µ + I µ. 2.9 ψ 2 ψ 2 2, 2 I µ b γ µ law = 2a γ µ, I µ. 2.8 U = X + Y, V = X X + Y α, θ >, σ >, A = exp σx θ, B = exp σu X θu V U = Y X 2. E[Y α e σy ]E[A/X α ] = E[V α e θv ]E[B/U α ]. U + V = X V α e θv B U α = V U α e θu+v σ/u = α Y e θ/x σx+y = Y α e σy A X X α. X Y U V log E[Y α e σy ] + log E[X α e σx θ/x ] = log E[V α e θv ] + log E[U α e θu σ/u ]. θ E[X α e σx θ/x ] E[X α e σx θ/x ] = E[V α+ e θv ] E[V α e θv ] σ + E[X α e σx θ/x ]E[X α+ e σx θ/x ] E[X α e σx θ/x ] 2 + E[U α+ e θu σ/u ] E[U α e θu σ/u ] = + E[U α+ e θu σ/u ]E[U α e θu σ/u ] E[U α e θu σ/u ] 2 α = E[X 2 A]E[A] E[X A] 2 = E[B]E[U 2 B] E[U B] 2 2.3 α =,, 2 E[e σy ]E[A] = E[e θv ]E[B], E[Y e σy ]E[X A] = E[V e θv ]E[U B], E[Y 2 e σy ]E[X 2 A] = E[V 2 e θv ]E[U 2 B]. 6 2 Gamma GIG

E[Y 2 e σy ]E[e σy ] E[Y e σy ] 2 = E[V 2 e θv ]E[U 2 B] E[X 2 A] = E[V 2 e θv ]E[e θv ] E[V e θv ] 2 2.3 E[e θv ]E[B] E[A] E[X A] E[V e θv ]E[U B] σ θ Y E[Y e σy ] 2 E[Y 2 e σy ]E[e σy ] φσ = E[e σy ] φσφ σ = + p φ σ 2 p > log φ σ = + plog φσ log φ σ Y φσ +p φ σ = mφσ +p, m = E[Y ], φσ = E[e σy ] = + pmσ /p Gamma Laplace β > E[e βσγ µ ] = Γµ xµ e +βσx dx = + βσ µ. /p = µ Y law = 2b γ µ b > V a > V law = 2a γ µ U, V γ µ, γ µ 2 X = U + V law = U + 2a γ µ = 2a γ µ + 2b γ µ +X 2.4 ii X GIGa, b; µ 2.3 tree GIG, Gamma tree 2.7 2.9 2.9 ψ, ψ 2 2 {, 2} tree, 2 k, k 2 2 leaf 2 k 2 root k k /k 2 ψ M ψ Gamma GIG 2.9 3 tree V = {, 2, 3}, E = {, 2, 2, 3} 2 3 tree 2 i, j E 2.3 tree GIG, Gamma 7

α 2 = α 2 = α, α 23 = α 32 = β. k α M α,β = {k = k, k 2, k 3, 3 ; Kk = α k 2 β }. β k 3 k = k, k 2, k 3 k tree root root 2 3 leaf 3 k3 = k 3 leaf 2 k 2 = k 2 β 2 /k 3 k = k α 2 / k 2 leaf ψ 3 k, k 2, k 3 = k, k 2, k 3 : M, 3, diffeomorphism ψ 3 2 root 2 3 leaf, 3 k = k, k 3 = k 3 leaf 2 root k 2 = k 2 α 2 /k β 2 /k 3 2 ψ 3 2 k, k 2, k 3 = k, k 2, k 3 : M, 3, diffeomorphism ψ 3 2 M α,β - K, K 2, K 3 µ >, a, 3 P K dk, K 2 dk 2, K 3 dk 3 = C detkk µ e a,k /2 dk dk 2 dk 3. 2.2 Wishart 2. i ψ 3 K, K 2, K 3 law = ii ψ 3 2 K, K 2, K 3 law = 2 a γ µ, I µ α 2 a,a 2, I µ β 2 a 2,a 3. I µ α 2 a 2,a, 2 a 2 γ µ, I µ β 2 a 2,a 3. i detkk = k k 2 k 3 α 2 k 3 β 2 k k k2 k3 = k α2 k2 k 3 = k k2 β2 k 3 α 2 k 3 = det Kk. k2 k 3 a, k = a k + a 2 k 2 + a 3 k 3 = a k α2 k2 + = a k + a 2 k2 + a α 2 p, 3 k2 + a3 k3 + a 2 β 2 k3 a 2 k2 β2 α 2 + a + k 3 k2 β a 2 3 k 3 + a 2 k 3 8 2 Gamma GIG

E[e p,ψk,k2,k3 ] = e p,ψk,k2,k3 C det Kk µ e a,k /2 dk M α,β = e p,e k C k k2 k3 µ exp, 2 a k α 2 a2 k2 + a 2 3 2.3 k2 2 β 2 a3 k3 + a 2 d k k3 ψ ψ2 x, y, z = u, v, w xyz = uvw 2.2 V = {, 2,..., n} E tree i, j E α ij = α ji k = k, k 2,..., k n, n n Kk = k ij k i, i = j, k ij = α ij, i j, i, j E,, i j, i, j E, M M = {k, n ; Kk } µ >, a, n, P K dk = C det Kk µ e a,k /2 dk M- K r V root ψ r : M, n k = k, k 2,..., k n M i V leaf k i = k i leaf j j pj kj = k j i pj α ij 2 k i k j root k l ψ r n k = k,..., k n 2.2 r V ψ r n K i V \ {r} r i j i GIGa j α 2 ij, a i; µ r 2 a r γ µ tree leaf 2.3 tree GIG, Gamma 9

3 3. Black-Scholes formula M = {M t } Ω, F, {F t }, P lim M t = t Brown {e σb t σ 2 t/2 } L b = sup{s; M s = b} { } = L b = Brown Madan- Roynette-Yor [6] b > E[M t b + ], E[b M t + ] L b European option Black-Scholes 3. M = sup t M t M < x P M x F = P < L x < F = M x P M < x = P L x = = M x. 3. x M P M < x = x > P M < x = P L x = = M x. + σ x = inf{t > ; M t = x} {M t σx } E[M σx F ] = M E[M σx F ] = xp σ x < F = xp M x F 2

3.2 b > F t - F t E[F t b M t + ] = be[f t {Lb t}] E[b M t + ] = bp L b t P L b t F t = b b M t +. L b t t b P L b t F t = P sup u t M u < b F t K European put option t = Black-Scholes 3. {B t } B = Brown σ >, S >, K > E[ K S expσb t σ 2 t/2 ] = KP Lσ + K t. L σ K = sup{u; S expσb u σ 2 u/2 = K}. E[ K S expσb t σ 2 t/2 sup {u; + ] = KP B u 2 } u = log KS σ 2 t. Black-Scholes σ, t call option E[M t b + ] lim t M t = M, M t > t >. P M t Q Q = M t P. Ft Ft 3.3 {M t } Q F s F s - t > s [ ] [ ] [ E Q F s = E P [F s ] = E P F s M s = E Q F s ]. M t M s M s E Q [M t F s ] = M s, a.s. 3.4 QM t, t =. c > σ c = inf{t; M t = c} Qσ c < n = cp σ c < n, n >, n 3. Qσ c < = cp σ c < = cp sup M t c = c > Qτ c < = {M t } Q t 22 3

Q lim M t = < Qsup M t < >, t t c > Qsup M t < c > Qτ c < =, c >, call option 3.5 E P [M t b + ] = QL b t. Q E P [M t b + ] = E P [ b M t + M t ] = E Q [ b M t {b/m t } Q- t Q 3.2 + ]. European call option 3.2 σ, r >, S >, K {B t } B = Brown T > e rt E P [ S e σb T +r σ 2 /2T K + ] = EP [ S e σb T σ 2 T/2 Ke rt + ] [2] P {B t } t < T E P [ S e σb t σ 2 t/2 Ke rt + ] = S E P [ S e σb t σ 2 t/2 Ke rt /S + ] t t T {M t = expσb t σ 2 t/2} Q Q = M t P. Ft Ft 3.5 Cameron-Martin {B σ t B t σt} Q Brown E P [ S e σbt σ2t/2 Ke rt { + ] = S Q sup u; e σbσ u +σ2u/2 = K } e rt t S = S P sup {s; B s + 2 } s = log e KS rt σ 2 t Gauss t T E P [ S e σb t σ 2 t/2 Ke rt + ] = σ 2 T S Ke rt/2 2 2πu exp 2 u 4 + logke rt /S 2 du. u Black-Scholes Φ 3. Black-Scholes formula 23

e rt E P [ S e σb T +r σ 2 /2T K + ] = S Φd Ke rt Φd 2. d, d 2 { S log + d = σ T d 2 = σ T { log K S K + r + σ2 2 r σ2 2 } T, } T = d σ T. 3.2 3.6 Madan-Roynette-Yor {M t } lim t M t = i t M t φ t x φ t x t, x ii d M t = u 2 t dt u t E[u 2 t M t = x] t, x. M non-random P < L b t = 2b E[u2 s M s = b]φ s bds. 3.2 P L b = = P sup M s < b = b M + {Λ b t} M b b M t + = b M + {Ms<b}dM s + 2 Λb t E[b M t + ] = b M + + 2 E[Λb t]. 3.2 E[b M t + ] = bp L b t b M + = bp L b = P < L b t = 2b E[Λb t] ϕ ϕm s d M s = ϕbλ b t db Fubini [ ] E ϕm s u 2 s = ϕbe[λ b t] db E[ϕM s u 2 s]ds = E[ϕM s E[u 2 s M s ]]ds = ϕbdb 24 3 E[u 2 s M s = b]φ s bds

P < L b t = 2b E[Λb t] = 2b E[u 2 s M s = b] φ s b ds. 3.3 σ {B t } Brown M t = M expσb t σ 2 t/2 dm t = σm t db t, d M t = σ 2 M 2 t dt, P M t db = φ t b db, φ t b = σb 2πt exp P L b dt = σ 2 2πt exp σ logb/m + σt/2 2 dt 2t L b Gauss σ logb/m + σt/2 2 2t, 3.2 25

A 2.5 {y n } n= [y,..., y n ] [y ] = y, [y,..., y n ] = y + [y 2,..., y n ], n = 2, 3 [y, y 2 ] = y + = y y 2 +, [y, y 2, y 3 ] = y +. y 2 y 2 y 2 + /y 3 {p n }, {q n } p = y, p 2 = y y 2 +, p n = y n p n + p n 2, n = 3, 4,... q =, q 2 = y 2, q n = y n q n + q n 2, n = 3, 4,... A. [y,..., y n ] = p n q n, n =, 2,..., ; p n q n+ q n p n+ = n, A. p n q n p n+ q n+ = n q n q n+, n =, 2,..., ; A.2 [y,..., y n, k] = kp n + p n kq n + q n, n = 2, 3,... A.3 A.3 n = 2 [y, y 2, k] = y + n = 2 A.3 n y 2 + /k = y ky 2 + + k = ky y 2 + + y ky 2 + ky 2 + [y,..., y n, y n+, k] = [y,..., y n, y n+ + /k] = y n+ + /kp n + p n y n+ + /kq n + q n = ky n+p n + p n + p n ky n+ q n + q n + q n = kp n+ + p n kq n+ + q n {p n }, {q n } n + A.3 A.3 n k = y n 27

[y,..., y n, y n ] = y np n + p n 2 y n q n + q n 2 = p n q n A. A.2 n = p q 2 q p 2 = y y 2 y y 2 + = n {p n }, {q n } p n q n+ q n p n+ = p n y n+ q n + q n q n y n+ p n + p n = p n q n q n p n A.2 2.5 i {P n }, {Q n } P = Y, P 2 = Y Y 2 +, P n = Y n P n + P n 2, n = 3, 4,... Q =, Q 2 = Y 2, Q n = Y n Q n + Q n 2, n = 3, 4,... Z n [Y,..., Y n ] = P n /Q n Claim n {Z 2n } {Z n } Z 2n+ < Z 2n A., A.2 P 2n+ P 2n = Y 2n+P 2n + P 2n P 2n Q 2n+ Q 2n Y 2n+ Q 2n + Q 2n Q 2n = Y 2n+P 2n Q 2n Q 2n P 2n Y 2n+ 2n = >. Y 2n+ Q 2n + Q 2n Q 2n Y 2n+ Q 2n + Q 2n Q 2n P 2n P 2n+2 = P 2n Y 2n+2P 2n+ + P 2n = Y 2n+2P 2n Q 2n+ Q 2n P 2n+ Q 2n Q 2n+2 Q 2n Y 2n+2 Q 2n+ + Q 2n Q 2n Y 2n+2 Q 2n+ + Q 2n = Y 2n+2 2n Q 2n Y 2n+2 Q 2n+ + Q 2n >. P 2n Q 2n P 2n+ Q 2n+ = P 2nQ 2n+ Q 2n P 2n+ Q 2n Q 2n+ = 2n Q 2n Q 2n+ >. A.4 A.4 Q n Q n+, a.s. Z n Q 2k Q 2 = Y 2 Q 2n+ = + n Q 2k+ Q 2k = + k= n Y 2k+ Q 2k + k= n Y 2k+ Y 2, a.s. i ii {X m } = [Y m d+, Y m d+2,..., Y md X m }{{}, Y m 2d+,..., Y m d }{{},..., Y,..., Y }{{ d }, ] X law = [Y,..., Y d, Y d+,..., Y 2d,..., Y m d+,..., Y md, ] X A.3 law = X P md + P md X X Q. md + Q md k= 28 A 2.5

a b < c d a b < a + c b + d < c d {P 2n+ /Q 2n+ } {P 2n /Q 2n } md P md /Q md < X P md/x Q md Z md = P md < P md + X Q md Q md + X md P md Q md Z md < P md + X P md Q md + X Q < Z md md < X P md X Q = Z md. md i Z md, Z md Z X m Z iii X law = [Y,..., Y d, X ] [Y,..., Y d, ] law =. X X X law law X = X m X m = X ii X m Z law X = Z Z law = X 2. m Z law = [Y,..., Y d, Z] X law = [Y,..., Y d, X ] 29

[] I, II,, 957 27 [2] K. Itô and H.P. McKean, Jr., Diffusion Processes and their Sample Paths, Springer, 974. [3],,,, Seminar on Probability 3, 96. [4] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd. Ed., Springer, 99. [5] G. Letac and V. Seshadri, A characterization of the generalized inverse Gaussian distribution by continuous fractions, Z. Wahr., 62 983, 485 489. [6] D.Madan, B.Roynette and M.Yor, From Black-Scholes formula, to local times and last passage times for certain submartingales, preprint. [7] - 24. [8] H.P. McKean, Jr., Elementary solutions for certain parabolic partial differential equations, Trans. AMS, 82 956, 59 548. [9] J.W. Pitman and M. Yor, Bessel processes and infinitely divisible laws, in Stochastic Integrals, ed. by D.Williams, Lecture Notes in Math., 85, 285 37, Springer, 98. [] D. Revuz and M. Yor, Continuous Martingales and Brownian motion, 3rd ed., Springer, 999. [] L.C.G. Rogers and D. Williams, Diffusion, Markov Processes and Martingales, Vol.2: Itô calculus, Wiley and Sons, 987. [2] 27. [3] V. Seshadri, The Inverse Gaussian Distributions, Oxford Univ. Press, 993. [4] - Riesz path Seminar on Probability, 3, 962. 3