Forward Backward Stochastic Differential Equationsに関する一考察

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1 Forward Backward Sochasic Differenial Equaions Backward Sochasic Differenial EquaionsBSDE s Forward Backward Sochasic Differenial Equaions FBSDE s forward process backward process BSDE s FBSDE s BSDE sfbsde s Harrison and Pliska BSDE sfbsde s oshihiro.yoshida@boj.or.jp

2 Backward Sochasic Differenial EquaionsBSDE sforward Backward Sochasic Differenial EquaionsFBSDE sbismu Pardoux and PengMa and Yong Cvianić and KarazasDuffie, Ma and YongOaka and Yoshida Duffie and Epsein BSDE s El Karoui, Peng and Quenez Four Sep Scheme Ma, Proer and Yong BSDE sfbsde s Forward Backward Sochasic Differenial Equaions FBSDE s forward process FBSDE s

3 Forward Backward Sochasic Differenial Equaions FBSDE s T T Y =g(x T ) + h(s, X s,ys,zs)ds Z s. dws. T X = x+ b(s, X s,ys,zs) ds + (s, X s,y s ). dws, X = x. FBSDE s C ( C = ) = ( 1, 2,, n) V = i V = C +V dv = dc + r V d + (b r 1) d + dw = dc + r V d + [ dw + d] = dc + r V d + Z [ dw + d], b r 1=. =, 1, 2 (,, n). n i= + n s s i= ds i i u u Su i, T S ds = r S d, ds i S b i d i = d i, j dw j +, i = 1,, n. j=1

4 BSDE s Backward Sochasic Differenial Equaions filered probabiliy spaceω, F, P, F T my m Z m dbsde s. Y = Y T + T h(s, Y s, Z s ) ds T Z s dw s, Y T =, w dwiener BSDE ssolvabiliy of BSDE s BSDE s1 sandard parameer, h Y, Z L 2 T (Rm ) L 2 T (Rm )F T E ( X 2 ) h Lipschizh (.,, ) H 2 T (Rm ) : H 2 T (Rm ) E T 2 d : Ω [, T ] R m Pardoux and PengTheorem Local Lipschiz Condiion BSDE s1 h(, y 1, z 1 ) h(, y 2, z 2 ) (l)( y 1 y 2 + z 1 z 2 ), y 1, y 2 R m, z 1, z 2 R m d s.. y 1, y 2 l. E X 2 + T h (,, s) 2 ds F k 2 a.e., a.s.

5 Forward Backward Sochasic Differenial Equaions T [, T )T, T Y, Z X 4k a.s., a.e T T = 4C 1 1 log2, C 1 = 1+2 (4k) +2( (4k)) 2 PengCorollary Forward Backward Sochasic Differenial Equaions FBSDE snx X BSDE sy, Z X, Y, Z (.)n d X = x + b (s, X s, Y s, Z s ) ds + (s, X, X s, Y s ). dw s = x, Y = g(x T )+ T h (s, X s,y s, Z s ) ds Z s. dw s. T FBSDE ssolvabiliy of FBSDE s FBSDE s X, Y, Z b (.)h (.) (.)g (.)L (, 1)g C 2+ (R n ) (.)v ( y ) I (, x, y) (, x, y) vi. (, x, z) [, T ) R n R m d v b (, x,, ) + h (, x,, z ) v. Ma and YongChaper 4 Theorem FBSDE sx, Y, Z m = 1 Ma, Proer and YongTheorem

6 X = x + b (s, X s, Y s, Z s ) ds + (s, X s, Y s, Z s ). dw s, X = x, Y = g (X T )+ T h (s, X s, Y s, Z s ) ds T (s, X s, Y s, Z s ). dw s. Four Sep Scheme FBSDE s (.) Y = (, X ) z(.) z(, x, y, p)=p(, x, y). Sep 1 z(.) (, x) k k [ ] r xx ( )(, x, ) k + < b (, x,,z(, x,, x)), x > + h k (, x,, z (, x,, x )) =, 1 k m, (T, x) =g (x ). z(.) (, x) FSDE s X = x + b (s, X s ) ds +. (s, X s ). dw s, b (, x )=b(, x, (, x ), z(, x, (, x ), x (, x ))), (, x )= (, x, (, x )). X Y = (, X ), Z = z(, X, (, X ), x (, X )), Ma, Proer and Yong

7 Forward Backward Sochasic Differenial Equaions Exended Forward Backward Sochasic Differenial Equaions BSDE s(x, Y, Z (.) Exended FBSDE sefbsde s X = x + b( X s, Y s, z) s (dz ) ds + (X s, Y s, z) s (dz ). dw s, X U = x, U Y = g (X T )+ T h ( X s, Y s, z ) s (dz ) ds z s (dz ). dw s. U T U EFBSDE b (.)h (.) (.) xy L = {x, y} ( b (x, y, u) + h (x, y, u) + (x, y, u) ) L, (x, y) R n R m, u U. u UC b (,, u) + h (,, u) + (,, u) C. g Admissible Relaxed Conrol U R A = (Ω, F, P, F, W, )U R Ω, F, P, F W F M (Ω) F Z (du, ) = Z () (u) Ma and Yong (.,)Z () Dirac-

8 FBSDE s Γ= {(x, g(x)) x R n }(X (T ), Y (T )) Γ ZY (T ) = g (X (T )) J (, x, y; Z(.))=Ef (X (T;, x, y, Z ()),. Y(T ;, x, y, Z (.))). Viscosiy Soluion-Hamilon-Jacobi-Bellman Equaion-Value Funcion FBSDE sefbsde sv(, x, y) V (, x, y ) = inf, P Z (). Z [, T ] J (, x, y; Z ())=J. (, x, y; Z(.)), V (T, x, y) = f (x, y). V(, x, y) viscosiy soluion Z (.) Hamilon-Jacobi-BellmanHJB Value Funcion V (, x, y)+ H (, x, y, DV (, x, y), D 2 V (, x, y)) =, V (T, x, y) = f (x, y), DV = V x D 2, V = V y V xx V T V xy. xy V yy H(.) Hamilonian b(, x, y, z) H(, x, y, q, Q, z) = q, h (, x, y, z) (, x, y, z ) r Q z (, x, y, z ) z T, H (, x, y, q, Q ) = inf H (, x, y, q, Q, z), z R n m R Mayer ype EFBSDE s J = (,, A) = E P [G( T (,, A))], G A

9 Forward Backward Sochasic Differenial Equaions EFBSDE s Solvabiliy of EFBSDE s HJB V (.) nodal se N (.) (, x, y) Ma and YongTheorem X Y Z FBSDE s X dx = r X d, dx i X i b i d d ij dw j = +, i = 1,, n. j=1 X X i (i = 1,, n) Y 1 = ( 1, 2,, n ) = Y 1 n i =1 i Y 1 = Y 1 s n + s i = i dx u i u X u i, dy 1 = r Y 1 d + (b r 1) d +. dw = r Y 1 d +. [dw + d], b r 1 =. risk premium N (V ) = {(, x, y) V (, x, y) =

10 Self-Financing Sraegy (Y 1, ) ( T 2 d < )Y 1, [, T ] feasible Self-Financing Supersraegy cumulaive consumpion processy (Y = ) dy 1 = r Y 1 d dy +. [dw + d ], Y =(Y, Y 1 ) Y, Meron c (Y = c s ds) J (, x, y, Y( ), Z ()) T E U(s, c s) ds B (T, Y 1 T ),.. = + Y, ZU(.) B(.) Harrison and Pliska arbirage opporuniy Hedging Sraegy X 1 T = g(x T ) Y 1, Complee Marke aainable X W n = d Dohan

11 Forward Backward Sochasic Differenial Equaions Complee Marke - Equivalen Maringale Measure Equivalen Maringale MeasureQ Radon-Nikodym derivaive = dq /dp Harrison and PliskaProposiion Corollary Proposiion dq dp T ( W ) exp T s dw 1 s s 2 ds T =. = 2, (.)exponenial maringale PQ E P [ X] = X dp = XdQ = E Q [X]. QdW Q = dw + dw Q Likelihood Raio Process z = E P (dq /dp F ) {x } E P [z x F s ] E P [z F s ] = E Q [x F s ], DohanTheorem Price of Coningen Claim-Hedging Porfolio Coningen Claim Y 1 Harrison and PliskaProposiion Proposiion SDE g (x) = max{x K, }

12 Binomial Model rr = 1+ rd < R< u C B S + B us, p, S ds, 1 p. C u = max[ us K, ],p, C C d = max[ ds K, ], 1 p, us + RB,p, S + B ds + RB, 1 p. C u C d (u d ) S, uc u dc d = B =. (u d) R C = S + B 1 R d u R = C u C d R ( u d) + ( u d). q = R d u d, 1 q u R = u d, 1 C = R [qc u + (1 q ) C d ], < q < 1q qu + (1 q) d = R

13 Forward Backward Sochasic Differenial Equaions n C (n, S, K)recursive formula C (n, S, K) = R 1 [qc (n 1, us, K) + (1 q)c (n 1, ds, K)], Price of American Coningen Claim - Superhedging Porfolio American Coningen Claim Y 1 Z = Y obsacle X dy 1 = h (, Y, Z ) d + dy Z. dw, Y 1 T = g (X T ), X, (Y 1 X ) dy =, X = l (, X ), Y 1 T Y, Z l(.) l : [, T ] R n R l (, x ) K (1+ x p ), [, T ], x R n, l (T, x ) g(x), sopping imed = inf { s T ; Y s1 = X s } Γ = ess sup v Y 1 (v, X v )=Y 1 Γ (D, X D ), X = g(x T )1 = T + X 1 < T. Y 1 El Karoui, Pardoux and QuenezProposiion Y early exercise premium h (, y, z) = r y Y 1 = u (, X ) u min u (, x ) l(, x ), (, x ) L ( u (, x ) h (, x, u (, x ), ( u ) (, x ))) =, u (T, x )= g (x ),

14 Iwaki, Kijima and Yoshida = < 1 < < n = TTKT i = T i (i =,, n) S i = S e Xi{X i, i =,, n} X = A(S, T ) Q d Dynamic Programming A (S i, T i = max K S i, e r ( i +1 i ) E Q d [A (S i +1, T i +1 ) S i ], i =,, n 1, s.. A (S, ) = max [K S, ]. ) [ ] G = (G 1,, G n )boundary value G n = K ; e r ( i +1 i ) E Q d (G i e X i +1 [A X i, T i +1 )] = K G i, i = n 1,, 1, G i = G i /SA(S, T )indicaor funcioni {.} A (S, T )=p (S, T )+e(s, T ), p (S, T )=e rt E Q d (K Se X n )I {X n ln G n }, n 1 e (S, T )= e E Q d (K Se X i r i p (Se X i, T i )) I { X 1 > ln G 1,..., X i 1 > ln G i 1, X i ln G i }. i =1 p(s, T )e (S, T ) Incomplee Marke X W n d X j ( j n)primary asse 1 =, 1 2 =( ik ) 2 1 i j, 1 k d =( ik, ) j +1 i n, 1 k d, Z Iwaki, Kijima and YoshidaProposiion

15 Forward Backward Sochasic Differenial Equaions Z = = (1 ) 1 + ( 2 ) 2, admissible sraegy 2 = dy 1 = r Y 1 d + ( 1 ). (dw + d ), Föllmer and Schweizer Hedging Sraegy dy 1 = r Y 1 d + dy +( 1 ) 1. (dw + d ), Y 1 T = g(x T ),. Y Y s 1 dw s Y, 1 Minimal Maringale Measure minimal risk premium 1 Moore-Penrose 1 = ( 1 ) [( 1 )( 1 ) ] 1 1, 1 Ker ( 1 ) Hofmann, Plaen and Schweizer 1 relaive enropy log( dq H (Q P )= )Q (d ), if Q P and dq << L dp dp 1 ( Ω, F, P ), +, oherwise, mean-variance rade-off Delbaen e al.

16 Cvianić and Ma XY, FBSDE s dx = r (, Y, ) X d, dx i = b i (, X, Y, ) d + d ij (, X, Y ) dw j,, i =1,, n. j =1 Y = Four Sep Scheme Y 1 generaor f (.) dy 1 = f (, c, Y 1 ) d Z. dw, Y 1 T = Y 1, recursive uiliy Sandard Addiive Uiliy : f (c, y) =u(c) y. Uzawa Uiliy : f (c, y) = u (c) (c)y. Kreps - Poreus Uiliy : f (c, y) = c y y 1. dy 1 = f (, c, Y 1, ) d Z. dw, Y T 1 = Y 1. Z FBSDE s El Karoui, Peng and Quenez

17 Forward Backward Sochasic Differenial Equaions sochasic volailiykijima and Yoshida S =(1, 2,, m){ n, n =, 1, } F =(f ij ), f ij = Pr [ n+1 = j n = i ] i {X n } X n = u i X n 1 : q i, d i X n 1 : 1 q i, q i = R d i ; q u i d i = u i R 1, i S. i u i d i u i R d i Kn m C i (n, X, K )=R 1 f ij [q i C j (n 1, u i X, K )+(1 q i ) C j (n 1, d i X, K)]. j =1 C i (, X, K )=max {X K, } dx = µ( ) d + ( ) dw, X { } infiniesimal generaorq =(q ij ), q ii = j i q ij i C i (, X, K) m j =1 q ij C j + 2 C i C i i 2 X 2 X + rx C i rc 2 i =, X

18 Kijima and Yoshida Kijima and Yoshida (.) Q =(q ij ) FBSDE sefbsde s Oaka and YoshidaFBSDE sx 1 X 2 FBSDE dx 1 = X 1 (X 1, X 2, ) d X 2 dw 1,, b + dx 2 = X 2 (X 1, X 2, )d (X 1, X 2, ) dw 1, (X 1, X 2, ) dw 2,. W 1, W 2, = (b(x 1, X 2, ) r )/ X 2 v Q v dq v dp T 1 exp s dw 1, s ( s 2 ) ds. + T T = v s dw 2, s 2 + v s 2 dx 1 X 1 r d + X 2 v = dw 1,, dx 2 X 2 = (X 1, X 2, ) 1 2 ) d + (X 1, X 2, ) ( + v v + (X 1, X 2, ) dw 1, (X 1, X 2, ) v dw 2,, Theorem

19 Forward Backward Sochasic Differenial Equaions TK European Call OpionC = C (, x, K, T) Y = Y 1 + X 1 + Y = dy = dy 1 + dx 1 + d = r Y d + A dw 1, B dw 2,, A : v v X 1 C = X 2 ( X ) 1 B := 1 2 X 2 C, X 2 X 2 C X 2, ( ) dy = r Y + X 2 X 1 d A dw 1, B dw 2,, C C X 1 + p X 2 + p = + 1 2, v A, [, T ] dy = (r Y + v B ) d B dw 2,, Duffie, Ma and Yong Hogan Y Y = E e s ru du ds F,

20 dr = (r, Y ) d + (r, Y ). dw. r, Y BSDE sa(.) dy = (r Y 1)d + (r, Y ). dw. r SDEX r = l ( X ) dx = b(x, Y )d + (X, Y ). dw, X = x. FBSDE s dx = b (X, Y )d + (X, Y ). dw, dx = (l (X )Y 1)d + Z. dw, Y a.s. X, Y, Z FBSDE s Oaka and Yoshida FBSDE s{w 1 ( ), }{W 2 ( ), } X =((X 1, X 2 ))FBSDE s dx = b(, X, P (, T )) d + (, X ). dw, dp (, T ) = h (, X, P(, T ))d + Z. dw, X 1 = x 1, X 2 = x 2, P (T, T ) = g (X(T )) = 1, h(, X(), P(, T )) = l(x 1 ) P (, T ) = r P (, T ), 11(, X ) 12 (, X ) (, X ) = (, 22 (, X )) 11 (),. 12 (),. 22 ():posiive. funcions, Z Z 1, Z 2 = ( ), W = ( W 1, W 2 ).

21 Forward Backward Sochasic Differenial Equaions 1(, X )(X 2 X 1 ) b(, X, P(, X()) = 2(, X )( (, X, P(, T )) X 2 ), Vasicek X 1 = r 2 (.) = l (X 1 ) = X 1 11 Cox, Ingersoll and Ross Model Vasicek 11 (, X ) = X 1 Hull-Whie Vasicek 1 (, X ) = 1 ( ) 2 (, X )(, X, P(,T )) = ( ) 22 (.) = Black-Karasinski Hull-Whie l (X 1 ) = exp (X 1 ) Duffie-Kan h (, X, P(,T )) P(,T ) X 1 X 2 (.) Moneary Policy Rules SvenssonX insrumensi X +1 = AX + Bi + +1, i = fx. Σ iid X E (1 δ ) δ = L + F, L =(Y Y ) K(Y Y ), Y = CX + Di. Oaka and Yoshida

22 ( < < 1) CDK Y Svensson y i +1= + α y y + +1, ( +1 = E [ +1 F ] = + α y ), y +1 = y y + z z r (i r )+ +1, z z z = +1 y y > y r > z < 1r 2 iid 2 iid 2 L = 1 2 [ ] ( ) 2 + y 2. i = r c ( ) y r ( ) + y 1+ 1 c ( ) y r + y r y + z r z, c ( ) = 2 y k ( ), + δ k( ) = ( 1 δ ) δ 2 y ( 1 δ ) δ 2 y y 1. m log moneary aggregae money growh = m m 1 L = 1 2 ( ) 2, money demand

23 Forward Backward Sochasic Differenial Equaions +1 = m +1 m = +1 + y ( y y 1 ) i ( i i 1 ) ( +1 ), y > i > 2 iid i i = 1 1 ( +1 )+ i y i 1 i (y y ) = +1 + y (y y 1) r c ( ) ( )+ i +1 y r y y + r z r z i 1 +. Taylor rule i = i ( )+.5 y. i Henderson-McKibbin rule i = i + 2 ( + y ( + y)). QPMBank of Canada, Reserve Bank of New Zealand i = r L + (E Q [ +T F ] ), r L () Oaka and Yoshida r r ins (X 1 X 2 )=(ln r, ln r ins )

24 ins r arge = i = ln P(, T )/( T ) T = Y (, T ) T, Y (, T ) (T ) T Oaka and Yoshida T = Y(, T )(1 e 1(T ) ), 1 (, X(), P (, T )) = ins ln r arge = ln i = ln Y (, T ) 1 (T ). Y (, T ) sliding bond yield lnp (, +T )/T JGB Heah-Jarrow-Moron D (, T ) = P(, +T ) dd(, T ) = D(, T )((r () f (, +T ))d + (, +T ). dw()), f (, T )Rukowski Four Sep Scheme Jensen Y (, T )= ln P (, T ) T = 1 T T ln E Q [ e r (u) du F ] E Q [r ( ω ) F ]=C (r,, T). r () = sup u [, ] r (u,), Ω lim C (r,, T ) = r C (r,, T) = r 1 [ ] (, T )= ln e r (, T )(T ) T 1 = ln E Q e r ( ω) ( T ) [ F ] Y (, T ). T r () = inf u [, ] r (u,), Ω lim C (r,, T ) = r

25 Forward Backward Sochasic Differenial Equaions [L T 1, U T 1 ] [L T 2, U T 2 ]T X 1 ( )X 2 ( ) L T 1 U T 1 L T 2 U T 2 T Problem B s dx () = b(, X(), P(, T ))d +. dw(), dp(, T) = h(x(), P(, T )) d + Z(). dw(), h(x(), P(, T )) = e X1( ) P (, T ) = r() P(, T ), X 1 () = ln r() = lnr, X 2 () lnr ins () = lnr, P(T, T) = g(x(t )) = 1, = ins X() D = { X() L T 1 X 1 () U1 T, L T 2 X 2 () U2 T }, 1(X 2 () X 1 ()) b (, X(), P(, X())) = (, 2(ln Y(, T) 1(T ) X 2 ()) ) 1 >, = ( ) , 11, 12, 22 >, Z() (Z 1 (), Z 2 ()),W() W 1 (), W 2 = = ( ()). Four Sep Scheme P(, T ) = (X(), ) P(, T ) dp(, T ) = d (X(), ) = (X(), ) +1(X 2 () X 1 ()) x 1 (X(), ) + 2(ln( ln (X(), )/ (T )) 1(T ) X 2 ()) x 2 (X(), ) 1 2 ( ) x1 x 1 (X(), ) 1222 x 1x 2 (X(), ) x d 2 2 x 2 (X(), ) + 11 x 1 (X(), ) dw 1 () + ( 12 x1 (X(), ) +22 x 2 (X(), )) dw 2 ().

26 B s (X(), ) ( ) x 1 x x 1 x x 2 2 x (X 2 () X 1 ()) x (ln ( ln / (T )) 1 (T ) X 2 ()) x 2 e X 1 ( ) =, (X (T ), T) =1, Z = x (X ( ), ).. B s PDE ( ) x 1 x x 1 x x 2 2 x (x 2 () x 1 ()) x1 + 2 (ln ( ln / (T )) 1(T ) x 2 ()) x 2 e x 1 ( ) =, (x, T )= 1. SDE dx 1 () = 1 (X 2 () X 1 ()) d + 11 dw 1 ()+ 12 dw 2 (), dx 2 () = 2 (ln( ln (X (), ) /(T )) 1(T ) X 2 ()) d + 22 dw 2 (), X 1 () =ln r () =ln r, X 2 () =l n r ins () =lnr ins. P(, T ) = (X(), ), Z = x (X(), ).. BSDE s FBSDE s BSDE s BSDE s

27 Forward Backward Sochasic Differenial Equaions BSDE s FBSDE s EFBSDE s

28 FBSDE s Schwarz Disribuion locally convex linear opological spaced(ω) T Ω f (x) Ωa.e.dx = dx 1 dx 2,,dx n locally inegrable T f ( ) = Ω f (x ) (x) dx, D ( Ω), T f generalized funcion m (B) R n ΩBaireB complex-valued measure, D ( Ω), T m ( ) = (x )m (dx ) Ω T m T p () = (p) p Ω D(Ω) T p Dirac disribuion Weak Derivaivef (x) C k T D p f () = ( 1) p T f (D p ) p k (D p T f ) () = ( 1) p T f (D p )(), D(Ω), R n Ω(x){x (x) } Ω(x) supp() p = (p 1,,p n ) p =p p n p k C k (D p )(x) = p (x)/ x p1 1,, x pn n (D (,, ) )(x) = (x ) D(Ω)ΩC (x) m T( m ). D(Ω)YosidaProposiion

29 Forward Backward Sochasic Differenial Equaions D p T f f derivaive in he sense of disribuion Schwarz Disribuion wih Parameers T D(Ω) T () T () T D p x T D p x T / = D p x ( T / ) T [a, b]d(ω) D (Ω) T = b a T s ds D (Ω) T ( ) = b a T s ( )ds, Weak Soluion {(X, W ), (Ω, F, P), F } dx = b(, X )d + (, X ). dw. Ω, F, P F filraion X = {X ; < }F R n W = {W ; < } d P[ { b i (s, X s ) + 2 ij (s, X s )} ds < ] = 1 i n, 1 j d X = X + b (s, X s ) ds + (s, X s ). dw s, < almos surely (Γ)= P [ X Γ], Γ B n iniial disribuion D p = (p) dx + /dx = 1x + x xx < d1(x) / dx = 1(x)Heavisie

30 Pahwise Uniqueness{(X, W ), (Ω, F, P), F } {( X, W ), (Ω, F, P), F }Ω, F, P P[X = X ] = 1P[X = X, < ] = 1 Uniqueness in Disribuion {(X, W ), (Ω, F, P), F } {( X, W ), ( Ω, F, P ), F } P[X Γ] = P [ X Γ] ; Γ B (R d ), XX WeakUniquness Space-Time-Harmonic FuncionL diffusion operaor C 1, 2 (R +,R n )space-imeharmonic ( + L ) (, x ) =. Parabolic Operaor - Semimaringale n X P[X Γ ] = 1 C 1, 2 (R +, R n ) n d (, X ) = ( + L ) (, X ) d + [ i (, X ij ] dw j ) (, X ).. i =1 j=1 d E [ ( i (s, X s ) ij (s, X s )) 2 ds ] < X = (, X ) (, X ) ( s + L s ) (s, X s ) ds maringale von Weizsäcker and WinklerTheorem Maringale Problem X diffusion process C (R n ) Q X =(X ) (X ) L s (X s ) ds. Maringale Problem-Weak Soluion R n QC (R +, R n )

31 Forward Backward Sochasic Differenial Equaions iniial disribuionvq v = Z T dp x (dx) Z T = dq x / dp x vq v C 1, 2 (R +, R n ) (, X ) (, X ) ( s + L s ) (s, X s ) ds. (Ω, F, P), F W SDE dx = b(, X )d + (, X ). dw, (X, W)(Ω, F, Q v ), F von Weizsäcker and WinklerTheorem P x { X, F ; T }, (Ω, F ) n SDE dx = b(, X )d + dw, P x [X = x] = 1, b (, x ) K(1 + x ) Q x W Q x [W = ] = 1 Z T = dq x / dp x T 1 T Z T = exp b (s, X dx s b 2 s ). ds, 2 (s, X s ) a = Ω = C (R +, R n ), P = QW = a 1 (s, X s ). dy s,y = X b(s, X s ) ds Karazas and ShreveDefiniion Karazas and ShreveProposiion

32 Fundamenal SoluionG(, x ;, ), < T, x R n, R n ( + L )V (, x ) = kv (, x ). G(, x ;, )f C (R n ) V(, x) = R lim V(, x) = f (x ), G(, x ;, )f ( )d, n G(, x ;, )ransiion probabiliy densiy Cauchy Problem f (x) : R n R, g (, x) : [, T ] R n R f (x ) L(1 + x 2 ), 1, g (, x) ; g (, x) L(1 + x 2 ), 1, g (, x) ; ( + L )V(, x) = kv (, x) + g, V(T, x) = f (x ), max V(, x) M (1 + x 2 ), M>, 1, T V(, x) = R ng(, x ; T, ) f ( )d + T G(, x ;, ) g (, )d d R n T = E, x f (X T )exp k(, X ) d T + g (s, X s ) exp k(, X ) d ds s. Karazas and ShreveTheorem xx, x P[X, x A]= A G(, x ;, )d,

33 Forward Backward Sochasic Differenial Equaions M ([, T ] U) (, A) = s (A) ds L(Ω) L(Ω )L ([, T ] U ) M (Ω) M (Ω ) = (, A, ) (, A, ) P a.e. Ω s (A, )ds, a. s. P, = s (A, ) ds, a.s.p, A B(U ) (, T ]. (A, ) A B(U) lim (A, )= (A, ) modificaion (A, )P a.e. Ω. M(Ω). M(Ω ) M ([, T ] U ) Ma and YongLemma (, A) (, A) = A B(U) (, U) = (,.) B(U) (, A)A B(U ) sup A B(U) (s, A) (, A) = s

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CEV(Constant Elasticity of Variance) FE 1

CEV(Constant Elasticity of Variance) FE 1 23 6 1 CEV(Consan Elasiciy of Variance) FE 1 1 (Karazas and Shreve[1998] Carr, Jarrow and Myneni[1992] J. Huang, Subrahmanyam and Yu[1996] ) S 1 ds =(r δ)s d + σ(s )dw,s = x(> ) r δ K T sup E[e rτ (K S