Malliavin Calculus. Matheus Grasselli Tom Hurd Department of Mathematics & Statistics, McMaster University Hamilton, Ontario, Canada L8S 4K1

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1 Malliavin Calculus Maheus Grasselli Tom Hurd Deparmen of Mahemaics & Saisics, McMaser Universiy Hamilon, Onario, Canada L8S 4K1 April, 25 1 Inroducion 2 Malliavin Calculus 2.1 The Derivaive Operaor Consider a probabiliy space (Ω, F, P ) equipped wih a filraion (F ) generaed by a onedimensional Brownian moion W and le L 2 (Ω) be he space of square inegrable random variables. Le L 2 [, T ] denoe he Hilber space of deerminisic square inegrable funcions h : [, T ] R. For each h L 2 [, T ], define he random variable W (h) = h()dw using he usual Io inegral wih respec o a Brownian moion. Observe ha his is a Gaussian random variable wih E[W (h)] = and ha, due o he Io isomery, E[W (h)w (g)] = for all h, g L 2 [, T ]. h()g()d =< h, g > L 2 [,T ], The closed subspace H 1 L 2 (Ω) of such random variables is hen isomeric o L 2 [, T ] and is called he space of zero-mean Gaussian random variables. 1

2 Now le C p (R n ) be he se of smooh funcions f : R n R wih parial derivaives of polynomial growh. Denoe by S he class of random variables of he form F = f(w (h 1 ),..., W (h n )) where f C p and h 1,..., h n L 2 ([, T ]). Noe ha S is hen a dense subspace of L 2 (Ω). Definiion 2.1 : For F S, we define he sochasic process D F = n i=1 f x i (W (h 1 ),..., W (h n ))h i () One can hen show ha DF L 2 (Ω T ). So for we have obained a linear operaor D : S L 2 (Ω) L 2 (Ω T ). We can now exend he domain of D by considering he norm F 1,2 = F L 2 (Ω) + D F L 2 (Ω T ) We hen define D 1,2 as he closure of S in he norm 1,2. Then D : D 1,2 L 2 (Ω) L 2 (Ω T ) is a closed unbounded operaor wih a dense domain D 1,2. Properies: 1. Lineariy: D (af + G) = ad F + D G, F, G D 1,2 2. Chain Rule: D (f(f )) = f i (F )D F, f Cp (R), F i D 1,2 3. Produc Rule:D (F G) = F (D G) + G(D F ), F, G D 1,2 s.. F and DF L 2 (Ω T ) are bounded. Simple Examples: 1. D ( h()dw ) = h() 2. D W s = 1 { s} 2

3 3. D f(w s ) = f (W s )1 { s} Exercises: Try Oksendal (4.1) (a), (b), (c), (d). Fréche Derivaive Inerpreaion Recall ha he sample space Ω can be idenified wih he space of coninuous funcion C([, T ]). Then we can look a he subspace H 1 = {γ Ω : γ = h(s)ds, h L 2 [, T ]}, ha is, he space of coninuous funcions wih square inegrable derivaives. This space is isomorphic o L 2 [, T ] and is called he Cameron-Marin space. Then we can prove ha (D )F h()d is he Fréche derivaive of F in he direcion of he curve γ = h(s)ds, ha is D F h()d = d dε F (w + εγ) ε= 2.2 The Skorohod Inegral Riemann Sums Recall ha for elemenary adaped process u = n F i 1 (i, i+1 ](), i= F i F i he Io inegral is iniially defined as u dw = n F i (W i+1 W i ) i= Since ε is dense in L 2 (Ω T ), we obain he usual Io inegral in L 2 (Ω T ) by aking limis. Alernaively, insead of approximaing a general process u L 2 (Ω T ) by a sep process of he form above, we could approximae i by he sep process û() = n i= 1 i+1 i ( i+1 ) ( E[u s F [i, i+1 ] c]ds 1 (i, i+1 ]() i 3

4 and consider he Riemann sum n i= 1 i+1 i ( i+1 i ) E[u s F [i, i+1 ] c]ds )(W i+1 W i ). If hese sums converge o a limi in L 2 (Ω) as he size of he pariion goes o zero, hen he limi is called he Skorohod inegral of u and is denoed by δ(u). This is clearly no a friendly definiion! Skorohod Inegral via Malliavin Calculus An alernaive definiion for he Skorohod inegral of a process u L 2 [T Ω] is as he adjoin o he Malliavin derivae operaor. Define he operaor δ : Dom(δ) L 2 [T Ω] L 2 (Ω) wih domain Dom(δ) = {u L 2 [T Ω] : E( D F u d) c(u) F L 2 (Ω), F D 1,2 } characerized by E[F δ(u)] = E[in T D F u d] ha is, < F, δ(u) > L 2=< D F, u > L 2 [T Ω] Tha is, δ is a closed, unbounded operaor aking square inegrable process o square inegrable random variables. Observe ha E[δ(u)] = for all u Domδ. Lemma 2..1 :Le u Dom(δ), F D 1,2 s.. F u L 2 [T Ω]. Then δ(f u) = F δ(u) D F u d (1) in he sense ha (F u) is Skorohod inegrals iff he righ hand side belongs o L 2 (Ω). Proof:Le G = g(w (G 1,..., G n )) be smooh wih g of compac suppor. Then from he produc rule 4

5 E[GF δ(u)] = E[ = E[G = E[G D (GF )u d] D F u d] + E[F D F u d] + E[Gδ(F u)] D Gu d] For he nex resul, le us denoe by L 2 a[t Ω] he subse of L 2 [T Ω] consising of adaped processes. Proposiion 2..1 : L 2 a[t Ω] Dom(δ) and δ resriced o L 2 a coincides wih he Io inegral. Exercises: Now do Oksendal 2.6, 2.1 (a),(b),(c),(d), 4.1 (f), as well as examples (2.3) and (2.4). 3 Wiener Chaos Consider he Hermie polynomials H n (x) = ( 1) n e x2 /2 dn dx n e x2 /2, H (x) = 1 which are he coefficiens for he power expansion of F (x, ) = e x 2 2 = e x2 2 e 1 2 (x )2 = e x2 2 = n= n= n n! ( dn n n! H n(x) 1 d n e 2 (x )2 ) = I hen follows ha we have he recurrence relaion (x d dx )H n(x) = H n+1 (x). 5

6 The firs polynomials are H = 1 H 1 = x H 2 (x) = x 2 1 H 3 (x) = x 3 3x. Lemma 3..2 : Le x, y be wo random variables wih join Gaussian disribuion s.. E(x) = E(y) = and E(x 2 ) = E(y 2 ) = 1. Then for all n, m,, if n m E[H n (x)h m (y)] = n!e(xy) n if n = m Proof: From he characerisic funcion of a join Gaussian random variable we obain Taking he parial derivaive Le us now define he spaces E[e sx+vy ] = e s2 2 +sve[xy ]+ v2 2 E[e sx s2 2 e vy v2 2 ] = e sve[xy ] n+m s n v m on boh sides gives, if n m E[H n (x)h m (y)] = n!e(xy) n if n = m H n = span{h n (W (h)), h L 2 [, T ]} These are called he Wiener chaos of order n. We see ha H corresponds o consans while H 1 are he random variables in he closed linear space generaed by {W (h) : h L 2 [, T ]} (as before). Theorem 3.1 : L 2 (Ω, F T, P ) = n= H n Proof: The previous lemma shows ha he spaces H n, H m are orhogonal for n m. 6

7 Now suppose X L 2 (Ω, F T, P ) is such ha E[XH n (W (h))] =, n, h L 2 [, T ]. Then E[XW (h) n ] =, E[Xe W (h) ] =, n, h h E[Xe P m i=1 iw (h i ) ] =, 1,..., m R h 1,..., h m R X =. Now le O n = {( 1,..., n ) [, T ] n, n T }. We define he ieraed inegral for a deerminisic funcion f L 2 [O n ] as J n (f) = n Observe ha, due o Io isomery, 2 f( 1,..., n )dw 1 dw n 1 dw n J n (f) L 2 (Ω) = f L 2 [O n] So he image of L 2 [S n ] under J n is closed in L 2 (Ω, F T, P ). Moreover, in he special case where f( 1,..., n ) = h( 1 )... h( n ), for some h L 2 [, T ], we have n!j n (h( 1 )h( 2 )... h( n )) = h n H( W (h) h ), as can be seen by inducion. Therefore, H n J n (L 2 [O n ]). Finally, a furher applicaion of Io isomery shows ha, if n m E[J m (g)j n (h)] =. < g, h > L 2 [O n] if n = m Therefore J m (L 2 [O n ]) is orhogonal o H n for all n m. Bu his implies ha H n = J n (L 2 [O n ]). We have jus given an absrac proof for he following heorem. Theorem 3.2 (ime-ordered Wiener Chaos): Le F L 2 (Ω, F T, P ). Then F = J m (f m ) m= for (unique) deerminisic funcion f m L 2 [O m ]. Moreover, F L 2 [Ω] = f m L 2 [O m] m= 7

8 Example: Find he Wiener chaos expansion of F = W 2 T. Soluion: We use he fac ha H 2(x) = x 2 1. So wriing W T = 1 { T }dw = h()dw T we obain h = ( 12 { T } d)1/2 = T 1/2, so From We find H 2 ( W T h ) = W 2 T T { T } 1 { T } dw 1 dw 2 = T ( W T 2 T 1) W 2 T = T + 2J 2 (1). The connecion beween Wiener chaos and Malliavian calculus is bes explained hrough he symmeric expansion, as opposed o he ime- ordered expansion jus presened. We say ha a funcion g : [, T ] n R is symmeric if g(x σ1,..., X σn ) = g(x 1,..., X n ) for all permuaions σ of he se {1,..., n}. The closed subspace of square inegrable symmeric funcions is denoed L 2 s([, T ] n ). Now observe ha O n occupies only he fracion 1 n! of he box [, T ] n. Therefore, for a symmeric funcion we have g 2 L 2 [,T ] n = = n! n = n! g 2 L 2 [O n] 2 g 2 ( 1,..., n )d 1 d n g 2 ( 1,..., n )d 1 d n Try o do a wo dimensional example o convince yourself of his. We can now exend he definiion of muliple sochasic inegrals o funcion g L 2 s[, T ] n by seing I n (g) := n!j n (g) = n! n g( 1,..., n )dw 1 dw n 2 g( 1,..., n )dw 1 dw n 8

9 I hen follows ha E[I 2 n(g)] = E[n! 2 J 2 n(g)] = n! 2 g L 2 [O n] = n! g L 2 [,T ] n so he image of L 2 s[, T ] n under I n is closed. Also, for he paricular case g( 1,..., n ) = h( 1 )... h( n ), he previous resul for he ime-ordered inegral gives I n (h( 1 )... h( n )) = h n H n ( W (h) h ) Therefore we again have ha H n I n (L 2 s). Moreover, he orhogonaliy relaion beween inegrals of differen orders sill holds, ha is, if n m E[J n (g)j m (h)] = n! < g, h > if n = m Therefore I n (L 2 s) is orhogonal o all H m, n m, which implies I n (L 2 s) = H n. Theorem 3.3 (symmeric Wiener chaos) Le L 2 (Ω, F T, P ). Then F = I m (g m ) m= for (unique) deerminisic funcions g m L 2 s[, T ] n. Moreover, F L 2 (s) = m= m! g m L 2 [,T ] n To go from he wo ordered expansion o he symmeric one, we have o proceed as following. Suppose we found F = J m (f m ), f m L 2 [O n ] m= Firs exend f m ( 1,..., m ) from O m o [, T ] m by seing f m ( 1,..., m ) = if ( 1,..., m ) [, T ] m \O m Then define a symmeric funcion g m ( 1,..., m ) = 1 f m ( σ1,..., σm ) m! σ Then I m (g m ) = m!j m (g m ) = J m (f m ) 9

10 Example: Wha is he symmeric Wiener chaos expansion of F = W (W T W )? Soluion: Observe ha W (W (T ) W ) = W = = dw 2 2 dw 1 dw 2 1 {1 << 2 }dw 1 dw 2 Therefore F = J 2 (1 {1 << 2 }). To find he symmeric expansion, we have o find he symmerizaion of f( 1, 2 ) = 1 {1 << 2 }. This is g( 1, 2 ) = 1 2 [f( 1, 2 ) + f( 2, 1 )] = 1 2 [1 { 1 << 2 } + 1 {2 << 1 }] Then F = I 2 [ 1 2 (1 { 1 << 2 } + 1 {2 << 1 })] Exercises: Oksendal 1.2(a),(b),(c),(d). 3.1 Wiener Chaos and Malliavin Derivaive Suppose noe ha F L 2 (Ω, F, P ) wih expansion F = I m (g m ), m= g m L 2 s[, T ] m Proposiion F D 1,2 if and only if and in his case mm! g m 2 L 2 (T m ) < m=1 D F = mi m 1 (g m (, )) (2) m=1 Moreover D F 2 L 2 (Ω T ) = m=1 mm! g m 2 L 2. 1

11 Remarks: 1. If F D 1,2 wih D F = hen he series expansion above shows ha F mus be consan, ha is F = E(F ). 2. Le A F. Then Therefore D (1 A ) = D (1 A ) 2 = 21 A D (1 A ). D (1 A ) =. Bu since 1 A = E(1 A ) = P (A), he previous iem implies ha P (A) = or 1 Examples:(1) (Oksendal Exercises 4.1(d)) Le F = 2 cos( )dw 1 dw 2 = 1 2 cos( )dw 1 dw 2 Using he (2) we find ha D F = (2) Try doing D WT 2 using Wiener chaos. cos( 1 + ) = cos( 1 + ). Exercise: 4.2(a),(b). 3.2 Wiener Chaos and he Skorohod Inegral Now le u L 2 (Ω T ). Then i follows from he Wiener-Io expansion ha u = I m (g m (, )), g m (, ) L 2 s[, T ] m m= Moreover u 2 L 2 (Ω T ) = m! g m L 2 [,T ] m+1 m= 11

12 Proposiion : u Domδ if and only if (m + 1)! g m 2 L 2 (T m+1 ) < m= in which case δ(u) = I m+1 ( g m ) m= Examples: (1) Find δ(w T ) using Wiener Chaos. Soluion: Therefore W T = 1dW 1 = I 1 (1), f =, f 1 = 1, f n = n 2 δ(w T ) = I 2 (1) = 2J 2 (1) = W 2 T T (2) Find δ(wt 2 ) using Wiener Chaos. Soluion: W 2 T = T + I 2 (1), f = T, f 1 =, f 2 = 1, f n = n 3 δ(w 2 T ) = I 1 (T ) + I 2 () + I 3 (1) = T W T + T 3/2 H 3 ( W (T ) T 1/2 ) = T W T + T 3/2 W (T )3 ( T 3/2 = T W T + W (T ) 3 3W (T )T = W (T ) 3 2T W (T ) 3W (T ) T 1/2 ) 3.3 Furher Properies of he Skorohod Inegral The Wiener chaos decomposiion allows us o prove wo ineresing properies of Skorohod inegrals. Proposiion : Suppose ha u L 2 (Ω T ). Then u = DF for some F D 1/2 if and only if he Kernels for u = I m (f m (, )) is symmeric funcions of all variables. m= 12

13 Proof: Pu F = 1 m= I m+1 m+1(f m (, )). Proposiion : Every process u L 2 (Ω T ) can be uniquely decomposed as u DF +u, where F D 1/2 and E( D Gu d) = for all G D 1/2. Furhermore, u Dom(δ) and δ(u ) =. Proof: I follows from he previous proposiion ha an elemen of he form DF, F D 1/2 form a closed subspace of L 2 (Ω T ). 4 The Clark-Ocone formula (Opional) Recall from Io calculus ha any F L 2 (ω) can be wrien as F = E(F ) + φ dw for a unique process φ L 2 (Ω T ). If F D 1/2, his can be made more explici, since in his case F = E(F ) + E[D F F ]dw We can also obain a generalized Clark-Ocone formula by considering and he measure dq dp condiions we obain d W Q = Z T = e R T λdw 1 R T 2 F = E Q [F ] + = dw + λ d E Q [(D F λ2 d. Then for F D 1/2 wih addiional echnical D λ s dw Q s ) F ]dw Q Exercises: 5.2 (a),(b),(c),(d),(e),(f) 5.3 (a),(b),(c) 5 The Malliavin Derivaive of a Diffusion Le us begin wih a general resul concerning he commuaor of he Malliavin derivaive and he Skorohod inegral. 13

14 Theorem 5.1 : Le u L 2 (Ω T ) be a process such ha u s D 1/2 for each s [, T ]. Assume furher ha, for each fixed, he process D u s is Skorohod Inegrable (D u s Dom(δ)). Furhermore, suppose ha δ(d u) L 2 (Ω T ). Then δ(u) D 1/2 and D (δ(u)) = u + δ(d u) (3) Proof: Le u s = m= I m(f m (, s)). Then δ(u) = I m+1 ( f m ) m= where f m is he symmerizaion of f m (, s). Then Now noe ha D (δ(u)) = (m + 1)I m ( f m (, )) m= f m ( 1,..., m, ) = 1 m+1 [f m( 1,..., m, ) + f m (, 2,..., m, 1 ) f m ( 1,..., m 1,, m )] Therefore On he oher hand = 1 m+1 [f m( 1,..., m, ) + f m ( 1,..., m 1,, m ) + f m ( m, 2,..., m 1,, 1 ) + D (δ(u)) = I m (f m (, )) + m= δ(d u) = δ[d ( = δ[ =... + f m ( 1,..., m,, n 1 )] mi m (symmf m (,, ). m= I m (f m (, s)))] m= mi m 1 (f m (,, s))] m= mi m (symmf m (,, )) m= Comparing he wo expressions now gives he resul. Corollary : If, in addiion o he condiions of he previous heorem, u s is F s adaped, we obain D ( u s dw s ) = u + D u s dw s 14

15 Now suppose ha dx = b(, X )d + σ(, X )dw for funcions b and σ saisfying he usual Lipschiz and growh condiions o enssure exisence and uniqueness of he soluion in he form X = X + b(u, X u )du + σ(u, X u )dw u I is hen possible o prove ha X D 1,2 for each [, T ]. Moreover, is Malliavian derivaive saisfies he linear equaion D (X ) = D s ( = b(u, X u )du) + D s ( b (u, X u )D s X u du + σ(s, X s ) + σ(u, X u )dw u ) s s σ (u, X u )D s X u dw u Tha is, In oher words, where Y is he soluion o D s X = σ(s, X s )exp[ s (b 1 2 (σ ) 2 )du + D s X = Y Y s σ(s, X s )1 {s } s σ dw u ] dy = b (, X )Y d + σ (, X )Y dw, Y = 1. This is called he firs variaion process and plays a cenral role in wha follows. Examples: (1) dx = r()x d + σ()x dw, X = x = dy = r()y d + σ()y dw = Y = X x (2) dx = (θ() kx )d + σdw, X = x = D s X = X X s σ(s)x s = σ(s)x = dy = ky d = Y = e k 15

16 D s X = e k( s) σ (3) dx = k(θ X )d + σ X dw, X = x D s X = σ X s exp[ dy = ky d + 1 σ Y dw 2 X 6 Malliavin Weighed Scheme s σ 2 (k + 1 )du X u 2 s σ Xu dw u ] Suppose now ha we wan o calculae E[φ (X)G] where X is an underlying asse, φ is a pay-off funcion and X λ X λ G as λ, corresponding o a perurbaion of he process X. Then using he chain rule for he Malliavin derivaive we have ha, for an arbirary process h s, Inegraing boh sides of he equaion gives Tha is Define u s = Gh s D s (φ(x)) = Gh s φ (X)D s.x Gh s D s (φ(x))ds = Gφ (X) Gφ (X) = Gh R s T hsdsxds. Then from dualiy E[φ G] = E[ Therefore E[φ (X)G] = E[φ(X)π], where Gh sd s (φ(x))ds h sd s Xds h s D s Xds. D s (φ(x))u s ds] = E[φ(X)δ(u)] Gh s π = ( h ). (4) sd s Xds σ2 (r Example: Le S T = S e 2 )T + σw T. Then is dela is given by = E[e rt φ(s T )] = E[e rt φ (S T ) S T S ] = e rt S E[φ (S T )S T ] 16

17 Using (4) wih h s = 1 we ge π = δ( S T D ss T ds ) S T = δ( σ T in T 1 {s T } ds ) = δ( 1 σt ) = W T σt Therefore For he vega we have = e rt S E[φ(S T ) W T σt ]. V = σ E[e rt φ(s T )] Again, applying (4) wih h s = 1 we obain = e rt E[φ (S T ) S T σ ] = e rt E[φ (S T )(W T σt )S T ] π = δ( (W T σt )S T D ss T ds ) = δ( (W T σt ) ) σt = δ( W T σt 1) = 1 σt δ(w T ) W T = W 2 T T σt W T = W 2 T σt W T 1 σ, ha is V = e rt E[φ (S T )( W 2 T σt W T 1 σ )] 17

18 Finally, for he gamma we have o evaluae a second derivaive. Tha is Γ = 2 E[e rt φ(s S 2 T )] = e rt S 2 E[φ (S T )S 2 T ]. We begin by applying (4) o his las exapression. This gives Tha is π 1 = δ( We now use he formula again for such ha γ = v S 2 σt E[φ (S T )S 2 T ] = E[φ (S T )π 1 ] S2 T ) = δ( S T σt S T σt ) = S T W T σt S T E[φ (S T )S 2 T ] = E[φ (S T )( S T W T σt S T )]. W T σt π 2 = δ( 1 σt ) = δ( W T σ 2 T 2 ) δ( 1 σt ) = W 2 T T σ 2 T 2 W T σt Γ = e rt S 2 σt E[φ(S T )( W 2 T σt W T 1 σ )] To formalize our discussion, suppose f D 1/2 and denoe by W he se of random variables π such ha E[φ (F )G] = E[φ(F )φ], φ C p Proposiion 6..1 : A necessary and sufficien condiion for a weigh o be of he form π = δ(u) where u Dom(δ), is ha E[ D F u d F(F )] = E[G F(F )] ( ) Moreover, π = E[π F(F )] is he minimum over all weighs of he correc funcional V ar = E[(φ(F )π E[φ (F )G]) 2 ] 18

19 Proof: Suppose ha u Dom(δ) saisfies Then E[ D F u d σ(f )] = E[G σ(f )] E[φ (F )G] = E[E[φ (F )G σ(f )]] = E[φ (F )E[G σ(f )]] = E[φ (F )E[ D F u d σ(f )]] = E[ D φ(f )u d] = E[φ(F )δ(u)d] so π = δ(u) is a weigh. Conversely, if π = δ(u) for some u Dom(δ) is a weigh, hen E[φ (F )G] = E[φδ(u)] = E[ = E[φ (F ) D φ(f )u d] D F u d] Therefore, E[ D F u d σ(f )] = E[G σ(f )]. To prove he minimal variance claim, observe firs ha for any wo weighs π 1, π 2 we mus have E[π 1 σ(f )] = E[π 2 σ(f )]. Therefore, seing π = E[π σ(f )] for a generic weigh π we obain var π = E[(φ(F )π E[φ (F )G]) 2 ] = E[(φ(F )(π π ) + φ(f )π E[φ (F )G]) 2 ] = E[(φ(F )(π π )) 2 ] + E[(φ(F )π E[φ (F )G]) 2 ] + 2E[φ(F )(π π )(φ(f )π E[φ (F )G])] 19

20 Bu E[φ(F )(π π )(φ(f )π E[φ (F )G])] = E[E[φ(F )(π π )(φ(f )π E[φ (F )G])] σ(f )] =. Therefore he minimum mus be achieved for π = π. 7 Generalized Greeks Consider now dx = r()x d + σ(, X )dw, X = x where r() is a deerminisic funcion and σ(, ) saisfies he Lipschiz, boundedness and uniform ellipiciy condiion. Le r(), σ(, ) be wo direcions such ha (r + ɛ r) and (σ + ɛ σ) saisfy he same condiions for any ɛ [ 1, 1]. Define dx ɛ 1 = (r() + ɛ 1 r)x ɛ 1 + σ(, X ɛ 1 )dw dx ɛ 2 = r()x ɛ 2 + [σ(, X ɛ 2 ) + ɛ 2 σ(, X ɛ 2 )]dw Consider also he price funcionals, for a square inegral pay-off funcion of he form φ : R m R. Then he generalized Greeks are defined as P (x) = E Q x [e R T r()d φ(x 1,..., X m )] P ɛ 1 (x) = E Q x [e R T (r()+ɛ 1 r())d φ(x ɛ 1 1,..., X ɛ 1 m )] P ɛ 2 (x) = E Q x [e R T r()d φ(x ɛ 2 1,..., X ɛ 2 m )] = P (x) x, Γ = 2 P x 2 ρ = P ɛ 1 ɛ 1 ɛ1 =, r, V = P ɛ2 ɛ 2 ɛ2 =, σ The nex proposiion shows ha he variaions wih respec o boh he drif and he diffusion coefficiens for he process X can be expressed in erms of he firs variaion process Y defined previously. Proposiion 7..2 : The following limis hold in L 2 convergence 2

21 1. lim ɛ1 Xɛ 1 2. lim ɛ2 Xɛ 2 X ɛ 1 X ɛ 2 = = Y r(s)x s Y s ds Y σ(s,xs) Y s dw s Y σ (s,x s) σ(s,x s) Y s ds 8 General One Dimensional Diffusions Now we reurn o he general diffusion SDE dx = b(, X )d + σ(, X )dw where he deerminisic funcions b, σ saisfy he usual condiions for exisence and uniqueness. We will find several ways o specify Malliavin weighs for dela: similar formulas can be found for he oher greeks. An imporan poin no addressed in hese noes is o undersand he compuaional issues when using Mone Carlo simulaion. This will involve he pracical problem of knowing which processes in addiion o X iself need o be sampled o compue he weigh: he answer in general is o compue he variaion processes Y, Y (2), Y (3),.... Recall he condiions for π = δ(w ) o be a weigh for dela: E[ σ Y w d σ(x i )] = E[Y T σ(x i )] Our soluions will solve he sronger, sufficien condiions: σ Y w d = Y T We invesigae in more deail wo special forms for w. We will see he need from his o look a higher order Malliavin derivaives: he calculus for his is given in he final secion. 1. We obain a independen weigh analogous o he consrucion in Ben-hamou (21) by leing [ w = w T = ] 1 σ d Y To compue δ(w) use (1) o give δ(w) = ww T D wd. From he quoien rule D (A 1 ) = A 1 D A A 1 and he commuaion relaion (3) we obain [ D w = w 2 Y 1 D σ s Y s σ s D Y s = w 2 Y 1 21 Ys 2 [ D σ s Y s σ s D Y s Y 2 s ] ds (5) ] ds (6) (7)

22 where we use he general formula for D Y s derived in he nex secion. This yields he final formula π = w 2 Y 1 [ σ s σ Y s σ s D Y s Y 2 s ] ds d Compuing his weigh will be compuaionally inensive: in addiion o sampling X, one needs Y, Y (2). 2. We obain a dependen weigh analogous o he consrucion in Ben-hamou (21) by leing w = Y T σ. This yields he weigh as an ordinary Io inegral: π = Y T σ dw Numerical simulaion of X, Y will be sufficien o compue his weigh.. 9 Higher Order Malliavin Derivaives In his secion, we skech ou he properies of he higher order variaion processes Y (k) s = k X s, < s X k and use hem o compue muliple Malliavin derivaives D 1... D k X s. These formulas are usually necessary for higher order greeks like γ. Bu as seen in he previous secion i may also ener formulas for dela when cerain weighs are chosen. Here are he SDEs saisfied by he firs hree variaion processes: dy = b Y d + σ Y dw [ ] [ ] (8) dy (2) = b Y (2) + b Y 2 d + σ Y (2) + σ Y 2 dw [ ] [ ] (9) dy (3) = b Y (3) + 3b Y Y (2) + b Y 3 d + σ Y (3) + 3σ Y Y (2) + σ Y 3 dw (1) 22 (11)

23 One can see ha he paern is where F (k), G (k) dy (k) = b Y (k) d + σ Y (k) dw + F (k) d + G (k) dw are explici funcions of, X, (Y (j) ) j<k. This paricular form of SDE can be inegraed by use of he semigroup defined by Y s. Proposiion 9..3 Proof: Y k) = Y [ Yu 1 (F (k) u ] G (k) u σ u )du + G (k) u dw u From he produc rule dxy = dx(y + dy ) + XdY for sochasic processes [ d(rhs) = [b Y d + σ Y dw ] [ +Y Y 1 = b Y (k) = d(lhs) (F (k) + σ Y (k) Y 1 Y (k) + Y 1 (F (k) ] G (k) σ )d + G (k) dw dw + G (k) σ d + (F (k) G (k) ] G (k) σ )d + G (k) dw σ )d + G (k) dw Since he higher variaion processes have he inerpreaion of higher derivaives of X wih respec o he iniial value x, we can exend he noion o derivaives wih respec o X, any. For ha we define and hen noe If we define Ỹ (k) s Ỹ (k) = s Ỹ (k) s Ỹ (k) [ Yu 1 (F (k) u = Y (k) Y := Y [Ỹ (k) s ] G (k) u σ u )du + G (k) u dw u Ỹ (k) ] hen Ỹ (k) s solves (9..3) for s >, subjec o he iniial condiion Ỹ (k) =. Therefore Ỹ (k) s he inerpreaion of k X s. X k The following rules exend he Malliavin calculus o higher order derivaives: has 1. Chain rule: D F (X s ) = F (X s )D X s, < s; 23

24 X s := Y s I( < s) := X Ỹ (1) s I( < s); Ỹ (k) s X = Ỹ (k+1) s Ỹ (k) s := k+1 X s, < s; X k+1 = Ys 1 Y (k) Y (k) s ; D X = σ. Examples: 1. D X T = X T X D X = Y T σ by (2), (5) by chain rule 2. D Y s = (D Y s )Y = Ỹ (2) s (D X )Y 3. For < s < T : D [D s X T ] = = [Y T s σ s I(s < T )] D X s X s ] (2) [Ỹ T s σ + Y T s σ s Y s σ by chain rule 4. For < s < T : D s [D X T ] = D s [Y T s Y s σ ] = Ỹ (2) T s σ sy s σ (12) 24