Malliavin's calculus and applications in stochastic control and ...

Malliavin’s calculus and applications in stochastic control and finance Warsaw, March, April 2008 Peter Imkeller version of 5. April 2008 Malliavin’s calculus has been developed for the study of the smoothness of measures on infinite dimensional spaces. It provides a stochastic access to the analytic problem of smoothness of solutions of parabolic partial differential equations. The mathematical framework for this access is given by measures on spaces of trajectories. In the one-dimensional framework it is clear what is meant by smoothness of measures. We look for a direct analogy to the smoothness problem in infinite-dimensional spaces. For this purpose we start interpreting the Wiener space as a sequence space, to which the theory of differentiation and integration in Euclidean spaces is generalized by extension to infinite families of real numbers instead of finite ones. The calculus possesses applications to many areas of stochastics, in particular finance stochastics, as is underpinned by the recently published book by Malliavin and Thalmayer. In this course I will report on recent applications to the theory of backward stochastic differential equations (BSDE), and their application to problems of the fine structure of option pricing and hedging in incomplete finance or insurance markets. At first we want to present an access to the Wiener space as sequence space.

1

The Wiener space as sequence space

Definition 1.1 A probability space (Ω, F, P ) is called Gaussian if there is a family (Xk )1≤k≤n or a sequence (Xk )k∈N of independent Gaussian unit random variables such that F = σ(Xk : 1 ≤ k ≤ n) (completed by sets of P -measure 0).

1

resp.

σ(Xk : k ∈ N)

Example 1: Let Ω = C(R+ , Rm ), F the Borel sets on Ω generated by the topology of uniform convergence on compact sets of R+ , P the m-dimensional canonical Wiener measure on F. Let further W = (W 1 , · · · , W m ) be the canonical m-dimensional Wiener process defined by the projections on the coordinates. Claim: (Ω, F, P ) is Gaussian. Proof: Let (gi )i∈N be an orthonormal basis of L2 (R+ ), j

Z

W (gi ) =

gi (s)dWsj ,

i ∈ N, 1 ≤ j ≤ m,

in the sense of L2 -limits of Itô integrals. Then (modulo completion) we have F = σ(Wt : t ≥ 0). Let t ≥ 0, (ai )i∈N a sequence in l2 such that 1[0,t] =

X

ai gi .

i∈N

Then we have for 1 ≤ j ≤ m Wtj = n→∞ lim

n X

∞ X

ai W j (gi ) =

i=1

ai W j (gi ),

i=1

hence Wtj is (modulo completion) measurable with respect to σ(W j (gi ) : i ∈ N). Therefore (modulo completion) F = σ(W j (gi ) : i ∈ N, 1 ≤ j ≤ m). Moreover, due to E(W j (gi )W k (gl )) = δjk hgi , gl i = δjk δil ,

i, l ∈ N, 1 ≤ j, k ≤ m,

hence the W j (gi ) are independent Gaussian unit variables. • In the following we shall construct an abstract isomorphism between the canonical Wiener space and a sequence space. Since we are finally interested in infinite dimensional spaces, we assume from now on Assumption: the Gaussian space considered is generated by infinitely many independent Gaussian unit variables. 2

Let RN = {(xi )i∈N : xi ∈ R, i ∈ N} be the set of all real-valued sequences, and for n ∈ N denote by πn : RN → Rn ,

(xi )i∈N 7→ (xi )1≤i≤n ,

the projection on the first n coordinates. Let Bn be the σ-algebra of Borel sets in Rn , −1 BN = σ(∪n∈N πn [Bn ]). Let for n ∈ N 1 x2 ν1 (dx) = √ exp(− ) dx, 2 2π

ν = P(Xn )n∈N = ⊗i∈N ν1 ,

νn = ν ◦ πn−1 .

This notation is consistent for n = 1. We want to construct an isomorphism between the spaces of integrable functions on (Ω, F, P ) and (RN , BN , ν). For this purpose, it is necessary to know how functions on the two spaces are mapped to each other. It is clear that for BN -measurable f on RN we have F = f ◦ ((Xn )n∈N ) is F-measurable on Ω. Lemma 1.1 Let F be F-measurable on Ω. Then there exists a BN measurable function f on RN such that F = f ◦ ((Xn )n∈N ). Proof 1. Let F = 1A with A = ((Xi )1≤i≤n )−1 [B], B ∈ Bn . Then set f = 1πn−1 [B] . f is by definition BN -measurable and we have f ((Xn )n∈N ) = 1B ((Xi )1≤i≤n ) = 1A = F. Hence the asserted equation is verified by indicators of a generating set of F which is stable for intersections. Hence by Dynkin’s theorem it is valid for all indicators of sets in F. 2. By part 1. and by linearity the claim is then verified for linear combinations of indicator functions of F-measurable sets. The assertion is stable for monotone limits in the set of functions for which it is verified. Hence it is valid for all F-measurable functions by the monotone class theorem. • 3

Theorem 1.1 Let p ≥ 1. Then the mapping Lp (RN , BN , ν)) 3 f 7→ F = f ◦ ((Xn )n∈N ) ∈ Lp (Ω, F, P )) defines a linear isomorphism. Proof The mapping is well defined due to ||F ||pp = E(|f ((Xn )n∈N )|p ) Z

=

|f (x)|p ν(dx)

(transformation theorem)

= ||f ||pp , and bijective due to Lemma 1.1. Linearity is trivial. • Theorem 1.1 allows us to develop a differential calculus on the sequence space (RN , BN , ν), and then to transfer it to the canonical space (Ω, F, P ). For this purpose we are stimulated by the treatment of the one-dimensional situation. Questions of smoothness of probability measures are prevalent. We start considering them in the setting of R.

2

Absolute continuity of measures on R

Our aim is to study laws of random variables defined on (Ω, F, P ), i.e. the probability measures PX for random variables X. By means of Theorem 1.1 these measures correspond to the measures ν ◦f −1 for BN -measurable functions f on RN . The one-dimensional version of these measures is given by ν1 ◦ f −1 for B1 -measurable functions f defined on R. We first discuss a simple analytic criterion for absolute continuity of measures of this type. Lemma 2.1 Let µ be a finite measure on B1 . Suppose there exists c ∈ R such that for all φ ∈ C 1 (R) we have Z

|

0

φ (x)µ(dx)| ≤ c||φ||∞ .

Then µ 0 1Zε 1Zε || f (· + ξa)dξ − g(· + ξa)dξ||p ε 0 ε 0 1Zε ||f (· + ξa) − g(· + ξa)||p dξ ≤ ε 0 = ||g − f ||p . 13

By means of this observation we can transfer the desired result from C0∞ (Rk ) to Lp (Rk ), since C0∞ (Rk ) is dense in Lp (Rk ). • Corollary 4.1 Let e1 , · · · , ek denote the canonical basis of Rk , let (fn )n∈N be a sequence in W1p such that (i) ||fn − f ||p → 0 as n → ∞, (ii) for any 1 ≤ i ≤ k the sequence (di fn )n∈N converges in Lp (Rk ). Then f ∈ W1p and ||fn − f ||1,p → 0 as n → ∞. Proof We have to show that f is weakly differentiable in direction ei for 1 ≤ i ≤ k, and di f = limn→∞ di fn ∈ Lp (Rk ). For this purpose let ui = n→∞ lim di fn , which exists due to assumption (ii). Then by (i) for any φ ∈ C0∞ (Rk ), 1 ≤ i≤k Z

Z

f (x)di φ(x)dx = n→∞ lim = − n→∞ lim

fn (x)di φ(x)dx Z

di fn (x)φ(x)dx = −

Z

ui (x)φ(x)dx.

This means that f possesses weak directional derivatives in direction ei and di f = ui ∈ Lp (Rk ). Now Theorem 4.1 is applicable and finishes the proof. • Corollary 4.2 Let p ≥ 1. Then W1p is a Banach space with respect to the norm || · ||1,p , and for any a ∈ Rk the mapping da : W1p → Lp (Rk ) is continuous. Proof We have to prove that W1p is complete with respect to ||·||1,p . Let therefore (fn )n∈N be a Cauchy sequence in W1p . Then setting f = limn→∞ fn in Lp (Rk ), we see that the hypotheses (i) and (ii) of Corollary 4.1 are satisfied, and it suffices to apply this Corollary. • We finally need a local version of Sobolev spaces. Definition 4.4 For p ≥ 1, s ∈ N let p Ws,loc = {f : f : Rk → R measurable f φ ∈ Wsp for φ ∈ C0∞ (Rk ).}

( local Sobolev space of order (s, p)). 14

p Theorem 4.2 Let p ≥ 1, s ∈ N. Then f ∈ Ws,loc iff for any x0 ∈ Rk there exists an open neighborhood Vx0 of x0 such that for any φ ∈ C0∞ (Rk ) with support in Vx0 we have φf ∈ Wsp .

Proof We only need to prove the only if part of the claim. For any x0 ∈ Rk let therefore Vx0 be given according to the statement of the assertion. Then (Vx0 )x0 ∈Rk is an open covering of Rk . Then there exists a locally finite partition of the unit (φk )k∈N ⊂ C0∞ (Rk ) which is subordinate to the covering, i.e. such that (i) 0 ≤ φn ≤ 1, for any n ∈ N, (ii) for any n ∈ N there exists x0 (n) such that supp(φn ) ⊂ Vx0 (n) , P (iii) n∈N φn = 1, (iv) for any compact set K ⊂ Rk the intersection of K and supp(φn ) is non-empty for at most finitely many n. Now let φ ∈ C0∞ (Rk ). Then for any k ∈ N (ii) gives supp(φφk ) ⊂ Vx0 (k) and thus by assumption φk φf ∈ Wsp ,

k ∈ N.

Since by (iv) the support of φk φ is non-trivial for at most finitely many k, (iii) and linearity yield the desired φf ∈ Wsp . • We now turn to Gaussian Sobolev spaces. Our analysis will again be based on the differential operator we know from the above sketched classical calculus. Only the measure with respect to which we consider duality changes from the Lebesgue to the Gaussian measure. Since we thereby pass from an infinite to a finite measure, integrability properties for functions and therefore the domains of the dual operators change. This is why the notion of local Sobolev spaces is important. On these spaces, we can define our operators locally, without reference to integrability first. In fact, using Theorem 4.2, and for p s ∈ N, p ≥ 1, 1 ≤ j1 , · · · , js ≤ k, f ∈ Ws,loc we can define dj1 dj2 · · · djs f 15

locally on an open neighborhood Vx0 of an arbitrary point x0 ∈ Rk by the corresponding generalized derivative of φf with φ ∈ C0∞ (Rk ) such that φ = 1 on an open neighborhood Ux0 ⊂ Vx0 of x0 . This gives a globally unique notion, since x0 is arbitrary. p Definition 4.5 Let s ∈ N, p ≥ 1, 1 ≤ j ≤ k, f ∈ Ws,loc , and denote by dj the directional derivative in direction of the jth unit vector in Rk according to the preceding remark. Let then

∇f = (d1 f, · · · , dk f ), δj f = −dj f + xj f, Lf =

k X

δj d j f =

j=1

k X

[−dj dj f + xj dj f ].

j=1

For any 1 ≤ r ≤ s we define more generally ∇r f = (dj1 dj2 · · · djr f : 1 ≤ j1 , j2 , · · · , jr ≤ k). This definition gives rise to the following notion of Gaussian Sobolev spaces. Definition 4.6 Let p ≥ 1, s ∈ N. Then let p Dsp (Rk ) = {f ∈ Ws,loc :

||f ||s,p =

s X r=0

s X r=0

|| |∇r f | ||p < ∞},

|| |∇r f | ||p

( k-dimensional Gaussian Sobolev space of order (s, p)). Remark is a Banach space. This is seen by arguments as for the proof of Corollary 4.2.

Dsp (Rk )

Since our calculus will be based mostly on the Hilbert case p = 2, we shall restrict our attention to this case whenever convenient. In this case, our ONB composed of k-dimensional Hermite polynomials as investigated in the previous chapter will play a central role, and adds structure to the setting. To get acquaintance with Gaussian Sobolev spaces, let us 16

compute the operators defined on the series expansions with respect to this ONB. For f ∈ L2 (Rk , νk ) we can write f=

cp (f ) Hp p! p∈Ek X

with coefficients cp (f ) ∈ R, p ∈ Ek . Due to orthogonality, the Gaussian norm is given by X cp (f )2 cp (f )2 ||f ||2 = . hHp |Hp i = 2 p! p∈Ek p! p∈Ek X

We also write f ∼ (cp (f )) to denote this series expansion. Denote by P the linear hull of the k-dimensional Hermite polynomials. Plainly, p P ⊂ Ws,loc for any s ∈ N, p ≥ 1. According to chapter 3, P is dense 2 k in L (R , νk ). And for functions in P, the generalized derivatives dj are just identical to the usual partial derivatives in direction j, 1 ≤ j ≤ k. We first calculate the operators on Hermite polynomials. In fact, for p ∈ Ek , 1 ≤ j ≤ k we have in the non-trivial cases dj Hp = pj

Y i6=j

Hpi Hpj −1 ,

Y

δj Hp =

i6=j

Hpi Hpj +1 ,

LHp = |p|Hp .

Hence for f ∼ (cp (f )) ⊂ P, 1 ≤ j ≤ k we may write cp (f ) Y pj Hpi Hpj −1 , p∈Ek p! i6=j X cp (f ) Y δj f = Hpi Hpj +1 , p∈Ek p! i6=j X cp (f ) |p|Hp . Lf = p∈Ek p!

dj f =

X

According to Corollary 4.2 and the calculations just sketched, the natural domains of the operators extending ∇, δj and L beyond P must be those distributions in Rk for which the formulas just given generate convergent series in the L2 -norm with respect to νk . The most important domain is the one of ∇, the Sobolev space D12 (Rk ). For f ∼ (cp (f )) ∈ P 17

we have || |∇f | ||22

Z

= = = = =

|∇f |2 (x)νk (dx)

Rk k Z X

k j=1 R k X X

|dj f |2 (x)νk (dx)

2 2 cp (f ) Y pj pi !(pj p!2 i6=j j=1 p∈Ek k X X cp (f )2 pj p! j=1 p∈Ek X cp (f )2

|p|

p∈Ek

p!

− 1)!

.

If in addition f ∈ L2 (Rk , νk ), we may write f ∼ (cp (f )) and approximate P it by fn = p∈Ek ,|p|≤n cpp!(f ) ∈ P, n ∈ N. Hence, according to corollary 4.2, f belongs to D12 (Rk ) if the following series converges || |∇f | ||22 = n→∞ lim || |∇fn | ||22 cp (f )2 = n→∞ lim |p| p! p∈Ek ,|p|≤n X

cp (f )2 < ∞. = |p| p! p∈Ek X

Along these lines, we now turn to describing Gaussian Sobolev spaces and the domains of our principal operators for p = 2 by means of Hermite expansions. We start with the case k = 1. 2 Theorem 4.3 Let r ∈ N, f ∼ (cp (f )) ∈ L2 (R, ν1 ) ∩ Wr,loc . Denote cp (f ) fp = p! Hp , p ≥ 0. Then the following are equivalent: (i) ∇r f ∈ L2 (R, ν1 ), P (ii) p≥0 pr ||fp ||22 < ∞, (iii) f ∈ Dr2 (R), (iv) δ r f ∈ L2 (R, ν1 ). In particular, Dr2 (R) is the domain of ∇r , δ r in L2 (R, ν1 ). For f, g ∈ D12 (R) we have h∇f |gi = hf |δgi.

18

Proof 1. We prove equivalence of (i) and (ii). We have ∇f =

X cp+1 (f ) p cp (f ) Hp−1 = Hp , p! p! p≥1 p≥0 X

and therefore by iteration ∇r f = Therefore ||∇

r

f ||22

cp+r (f )2 = , p! p≥0 X

and hence ||∇r f ||22 = if and only if

cp+r (f ) Hp−1 . p! p≥0 X

X

||fp ||22

cp (f )2 = , p!

(p + r)! ||fp+r ||22 < ∞ p! p≥0 X

(p + r)r ||fp+r ||22 < ∞,

p≥0

and this is the case if and only if X p≥0

pr ||fp ||22 < ∞.

2. We next prove that (ii) and (iv) are equivalent. Note that δf =

cp (f ) Hp+1 , p≥0 p! X

and therefore δ r f =

cp (f ) Hp+r . p≥0 p! X

This implies that ||δ

r

f ||22

X cp (f )2 cp (f )2 = (p + r)! = (p + r) · · · (p + 1) 1, f ∈ Lp (RN ), (fn )n∈N the corresponding sequence according to the above remarks. Suppose that supn∈N ||fn ||1,p < ∞. Then for any j ∈ N the sequence (dj fn ◦ π n )n∈N converges in Lp (RN ), to a limit that we denote by dj f . Corresponding statements hold true for higher order derivatives. Proof Let n ∈ N, j ∈ N. Then for n ≥ j we have E(dj fn+1 ◦ π n+1 |Cn ) = dj fn ◦ π n . This means that (dj fn ◦ π n )n≥j is a martingale with respect to (Cn )n≥j which, due to sup ||dj fn ◦ π n ||p ≤ sup ||fn ||1,p < ∞, n≥j

n∈N

21

is bounded in Lp (RN ) and hence converges in Lp (RN ), due to p > 1. • The preceding Lemmas give rise to the following definition of Sobolev spaces. Definition 5.1 Let p ≥ 1, s ∈ N. Then Dsp (RN ) = {f ∈ Lp (RN , ν) : fn ∈ Dsp (Rn ), n ∈ N, sup ||fn ||s,p < ∞}, n∈N

( infinite dimensional Sobolev space of order (s, p)), endowed with the norm ||f ||s,p = sup ||fn ||s,p , f ∈ Dsp (RN ). n∈N

This definition makes sense, for the following reasons. Theorem 5.1 Let p > 1, s ∈ N. Then Dsp (RN ) is a Banach space with the norm || · ||s,p . Proof We prove the claim for s = 1. Let (f m )m∈N be a Cauchy sequence in D1p (RN ), and (fnm )n,m∈N the corresponding finite dimensional functions according to the remarks above. Then for m, l ∈ N, n ∈ N Jensen’s inequality and the martingale statement in the preceding proof give the following estimate lim sup ||fnm − fnl ||1,p ≤ lim ||f m − f l ||1,p = 0. m,l→∞

m,l→∞

D1p (Rn ) being a Banach space for n ∈ N, we know that fn = m→∞ lim fnm ∈ D1p (Rn ) exists. Let fˆn = fn ◦ π n . Now let f = limm→∞ f m in Lp (RN ). Then by uniform integrability E(f |Cn ) = E(m→∞ lim f m |Cn ) = m→∞ lim E(f m |Cn ) = m→∞ lim fˆnm = fˆn . Moreover sup ||fn ||1,p ≤ sup ||fnm ||1,p ≤ sup ||f m ||1,p < ∞.

n∈N

m,n∈N

m∈N

22

Hence by definition f ∈ D1p (RN ), and by Fatou’s lemma ||f − f m ||1,p ≤ lim inf ||f m − f l ||1,p → 0 l→∞

as m → ∞. • According to Lemma 5.2, the gradient on the infinite dimensional Gaussian Sobolev spaces is defined as follows. Definition 5.2 Let p > 1, f ∈ D1p (RN ). Then let ∇f = (dj f )j∈N ( Malliavin gradient or Malliavin derivative), where for any j ∈ N according to Lemma 5.2 lim dj fn ◦ π n . dj f = n→∞ Accordingly, for s ∈ N we define ∇r f, 1 ≤ r ≤ s, for f ∈ Dsp (RN ). Remark The gradient ∇ being a continuous mapping from D1p (Rn ) to Lp (Rn , νn ) for any finite dimension n, Lemma 5.2 and the definition of the Malliavin gradient imply, that ∇ is a continuous mapping from D1p (RN ) to Lp (RN , ν). Let us now again restrict our attention to p = 2 and describe Gaussian Sobolev spaces by means of the generalized Hermite polynomials. First P of all, suppose f = p∈E cpp!(f ) ∈ L2 (RN , ν). We shall continue to use the P

c

(f )

notation f ∼ (cp (f )). Then for n ∈ N, we have fn = p∈En (p,0) p! Hp , where we put (p, 0) = (p1 , · · · , pn , 0, 0, · · ·) for p = (p1 , · · · , pn ) ∈ En . P c (f ) Therefore, we also have fˆn = p∈En (p,0) p! H(p,0) . Let again P be the linear hull generated by all generalized Hermite polynomials. As in the preceding chapter, we may calculate the gradient norms for f ∼ (cp (f )) ∈ D12 (RN ). In fact, we have for j ∈ N c(p,0) (f ) Y Hpi Hpj −1 pj p! p∈En i6=j X cp (f ) Y = Hpi Hpj −1 . pj p! p∈E i6=j

dj f = n→∞ lim dj fn ◦ π n = n→∞ lim

23

X

Furthermore, for f ∈ D12 (RN ) let us compute the norm of |∇f | = 1 P [ j∈N (dj f )2 ] 2 in L2 (RN , ν). In fact, we have, using the calculation of gradient norms in the preceding chapter, ∞ > sup |||∇fn ◦ π n |||22 = |||∇f |||22 n∈N

X c(p,0) (f )2 cp (f )2 = sup |p| = |p| . p! p! n∈N p∈En p∈E X

We therefore obtain the following main result about the description of the infinite dimensional Gaussian Sobolev spaces or order (1, 2). Theorem 5.2 For f ∈ L2 (RN , ν) the following are equivalent: (i) f ∈ D12 (RN ), P )2 (ii) p∈E |p| cp (f p! < ∞, 1 P (iii) |∇fn | ◦ π n = [ j∈N (dj fn )2 ◦ π n ] 2 converges in L2 (RN , ν) to |∇f |. Moreover, D12 (RN ) is a Hilbert space with respect to the scalar product (f, g)1,2 = hf |gi +

X

hdj f |dj gi,

j∈N

f, g ∈ D12 (RN ).

For p ≥ 2 P is dense in D1p (RN ). Analogous results hold for Sobolev spaces of order (s, 2) with s ∈ N.

6

Absolute continuity in infinite dimensional Gaussian space

We are now in a position to discuss the main result of Malliavin’s calculus in the framework of infinite dimensional Gaussian sequence spaces. The result is about the smoothness of laws of random variables defined on the Gaussian space. We start with a generalization of Lemma 1.1 to finite measures on Bd for d ∈ N. Lemma 6.1 Let µ|Bd be a finite measure. Assume there exists c ∈ R such that for all φ ∈ C 1 (Rd ) with bounded partial derivatives, and any 1 ≤ j ≤ d we have Z ∂ | φ(x) µ(dx)| ≤ c ||φ||∞ . ∂xj Then µ 0 uε = ε12 u( 1ε ·). Moreover, let Z

ψε =

uε (· − y)µ(dy)

be a smoothed version of µ. Then we obtain for h continuous with compact support, using Fubini’s theorem, Z

Z Z

ψε (x)h(x)dx =

[ uε (x − y)µ(dy)]h(x)dx

Z Z

=

[ uε (x − y)h(x)dx]µ(dy)

Z Z

=

[ u(x)h(εx + y)dx]µ(dy) Z

→

h(y)µ(dy).

3. We show: L2 (R2 ) 3 g 7→

Z

gdµ ∈ R

is a continuous linear functional. In fact, let φ ∈ C 1 (R2 ) have compact support, and let ε > 0. Then by hypothesis and smoothness of ψε with a calculation as in 2. Z

|

Z ∂ ∂ ψε (x)φ(x)dx| = | ψε (x) φ(x)dx| ∂xi ∂xi Z Z ∂ = | [ uε (x − y) φ(x)dx]µ(dy)| ∂xi

25

Z Z

∂ uε (x − y)φ(x)dx]µ(dy)| ∂xi Z Z ∂ uε (x − y)φ(x)dx]µ(dy)| = | [ ∂y i Z ≤ c|| uε (x − ·)φ(x)dx||∞ = | [

≤ c||φ||∞ . Generalizing this inequality to bounded measurable φ, and then taking φ = sgn(ψε ) yields the inequality Z

|

∂ ψε |dλ ≤ c ∂xi

for any ε > 0. Now let ε > 0, g ∈ L2 (R2 ) be given. Then, using 1. and the estimate above Z

|

Z

2

Z

1

ψε (x)g(x)dx| ≤ [ |ψε (x)| dx |g(x)|2 dx] 2 Z 1 ∂ 1Z ∂ ψε |dλ + | ψε |dλ] 2 ||g||2 ≤ [ | 2 ∂x1 ∂x2 ≤ c||g||2 .

Applying this inequality in the special case, in which g is continuous with compact support, and using 2. we get Z

|

g(x)µ(dx)| ≤ c||g||2 .

Finally extend this inequality to g ∈ L2 (R2 ) by approximating it with continuous functions of compact support. This yields the desired continuity of the linear functional. 4. It remains to apply Riesz’ representation theorem to find a square integrable density for µ. • We now consider a vector f = (f 1 , · · · , f d ) with components in L2 (RN , BN , ν). Our aim is to study the absolute continuity with respect to λd of the law of f under ν, i.e. of the probability measure ν ◦ f −1 . For this purpose we plan to apply the criterion of Lemma 6.1. Let φ ∈ C 1 (Rd ) possess bounded partial derivatives. Then, the integral transformation theorem gives Z ∂ Z ∂ −1 φdν ◦ f = φ ◦ f dν. ∂xi ∂xi 26

In case d = 1 at this place we use integration by parts hidden in the representations d(φ ◦ f ) = φ0 (f )df, 1 φ0 (f ) = d(φ ◦ f ) . df Our infinite dimensional analogue of d is the Malliavin gradient ∇. Hence, we need a chain rule for ∇. Theorem 6.1 Let p ≥ 2, f ∈ D1p (RN )d , φ ∈ C 1 (Rd ) with bounded partial derivatives. Then φ ◦ f ∈ D1p (RN ) and ∇[φ ◦ f ] =

d X

∂ φ(f ) · ∇f i . i=1 ∂xi

Proof Use Theorem 5.2 to choose a sequence (fn )n∈N ⊂ P d such that for any 1≤i≤d ||fni − f i ||1,p → 0. For each n ∈ N we have ∇[φ ◦ fn ] =

d X

∂ φ(fn ) · ∇fni . i=1 ∂xi

Since ∇ is continuous on D1p (RN ), and since the partial derivatives of φ are bounded, we furthermore obtain that ∇[φ ◦ f ] = n→∞ lim ∇[φ ◦ fn ] =

d X

∂ φ(f ) · ∇f i i=1 ∂xi

in L2 (RN , ν). This completes the proof.• We next present a calculation leading to the verification of the absolute continuity criterion of Lemma 6.1. We concentrate on the algebraic steps, and remark that their analytic background can be easily provided with the theory of chapter 5. The first aim of the calculations must be to isolate, for a given test function φ ∈ C 1 (Rd ) with bounded partial derivatives, the expression ∂x∂ i φ(f ), 1 ≤ i ≤ d. Recall the notation (x, y) =

∞ X

xi y i ,

i=1

27

x, y ∈ l2 .

For 1 ≤ i, k ≤ d let σik = (∇f i , ∇f k ). Then we have for 1 ≤ k ≤ d k

(∇(φ ◦ f ), ∇f ) =

∞ X

dj (φ ◦ f )dj f k

j=1 ∞ X X

∂ φ(f )dj f i dj f k j=1 1≤i≤d ∂xi X ∂ = φ(f )σik . ∂x i 1≤i≤d

=

We now assume that the matrix σ is (almost everywhere) invertible. Then, denoting its inverse by σ −1 we may write ∂ φ(f ) = ∂xi =

X

−1 (∇(φ ◦ f ), ∇f k σki )

1≤k≤d ∞ X X

1≤k≤d j=1

−1 dj (φ ◦ f )σki dj f k .

We next assume, that the dual operator δj of dj , which is defined in the usual way on P, is well defined and the series appearing is summable. Then we have Z ∂ Z ∞ X X −1 φ(f )dν = dj (φ ◦ f )σki dj f k dν ∂xi 1≤k≤d j=1 Z

=

φ ◦ f[

X

∞ X

1≤k≤d j=1

−1 )]dν. δj (dj f k σki

The right hand side can be estimated by c||φ||∞ with c = ||

X

∞ X

1≤k≤d j=1

−1 δj (dj f k σki )||2

in L2 (RN , ν). It can be seen (in analogy to Theorem 4.3) that this series makes sense under the hypotheses of the following main theorem. Theorem 6.2 Suppose that f = (f 1 , · · · , f d ) ∈ L2 (RN , ν) satisfies (i) f i ∈ D24 (RN ) for 1 ≤ i ≤ d, −1 (ii) σik = (∇f i , ∇f k ), 1 ≤ i, k ≤ d, is ν-a.s. invertible and σki ∈ 4 N D1 (R ) for 1 ≤ i, k ≤ d. Then we have ν ◦ f −1 0, and a dimension m ∈ N. We start by explaining some notation. Let (Ω, F, P ) be the canonical n-dimensional Wiener space, with canonical Wiener process W = (W 1 , · · · , W n ). Denote by (Ft )t≥0 the filtration of the canonical space, i.e. the natural filtration completed by sets of P -measure 0. 51

Let L2 (Rm ) be the linear space of Rm -valued FT -measurable random 1 variables, endowed with norm E(|X|2 ) 2 . Let H 2 (Rm ) denote the linear space of (Ft )0≤t≤T -adapted measurable processes X : Ω × [0, T ] → Rm 1 R endowed with the norm ||X||2 = E( 0T |Xt |2 dt) 2 . Further let H 1 (Rm ) denote the space of (Ft )0≤t≤T -adapted measurable processes X : Ω × R 1 [0, T ] → Rm with the norm ||X||1 = E([ 0T |Xt |2 dt] 2 ). Finally, for β > 0 and X ∈ H 2 (Rm ) let ||X||22,β

Z T

= E(

0

eβt |Xt |2 dt),

and H 2,β (Rm ) the space H 2 (Rm ) endowed with the norm || · ||2,β . We next describe the general hypotheses we want to require for the parameters of our BSDE. The terminal condition ξ will be supposed to belong to L2 (Rm ). The generator will be a function f : Ω × R+ × Rm × Rn×m → Rm , which is product measurable, adapted in the time parameter, and which fulfills (H1) f (·, 0, 0) ∈ H 2 (Rm ), f is uniformly Lipschitz, i.e. there exists C ∈ R such that for any (y1 , z1 ), (y2 , z2 ) ∈ Rm × Rn×m , P ⊗ λ-a.e. (ω, t) ∈ Ω × R+ (H2) |f (ω, t, y1 , z1 ) − f (ω, t, y2 , z2 )| ≤ C[|y1 − y2 | + |z1 − z2 |]. 1

Here for z ∈ Rn×m we denote |z| = (tr(zz ∗ )) 2 . Definition 10.1 A pair of functions (f, ξ) fulfilling, besides the mentioned measurement requirements, hypotheses (H1), (H2), is said to be a standard parameter. Given standard parameters, we shall solve the problem of finding a pair of (Ft )0≤t≤T -adapted processes (Yt , Zt )0≤t≤T such that the backward stochastic differential equation (BSDE) (∗) dYt = Zt∗ dWt − f (·, t, Yt , Zt )dt,

YT = ξ,

is satisfied. In order to construct a solution, a contraction argument on suitable Banach spaces will be used. For its derivation we shall need the following a priori inequalities. 52

Lemma 10.1 For i = 1, 2 let (f i , ξ i ) be standard parameters, (Y i , Z i ) ∈ H 2 (Rm ) × H 2 (Rn×m ) solutions of (*) with corresponding standard parameters. Let C be a Lipschitz constant for f 1 . Define for 0 ≤ t ≤ T δYt = Yt1 − Yt2 , δ2 ft = f 1 (·, t, Yt2 , Zt2 ) − f 2 (·, t, Yt2 , Zt2 ). Then for any triple (λ, µ, β) with λ > 0, λ2 > C, β ≥ C(2 + λ2 ) + µ2 we have 1 ||δY ||22,β ≤ T [eβT E(|δYT |2 ) + 2 ||δ2 f ||22,β ], µ 2 λ 1 ||δZ||22,β ≤ 2 [eβT E(|δYT |2 ) + 2 ||δ2 f ||22,β ]. λ −C µ Proof 1. Let (Y, Z) ∈ H 2 (Rm ) × H 2 (Rn×m ) be a solution of (*) with standard parameters (f, ξ). This means that we may write for 0 ≤ t ≤ T Z T

(∗) Yt = ξ −

t

Zs∗ dWs

+

Z T t

f (·, s, Ys , Zs )ds.

We show: sup |Yt | ∈ L2 (Rm ).

0≤t≤T

In fact, due to (*) we have sup |Yt | ≤ |ξ| +

Z T

0≤t≤T

0

|f [·, s, Ys , Zs )|ds + sup | 0≤t≤T

Z T t

Zs∗ dWs |,

and, with the help of Doob’s inequality E( sup | 0≤t≤T

Z T t

Zs∗ dWs |2 )

≤ 4E( sup | 0≤t≤T

Z t

Zs∗ dWs |2 ) 0

Since in addition (H1) and (H2) guarantee that |ξ|+ L2 (R), we obtain the desired

Z T

≤ 8E( RT 0

0

|Zs |2 ds).

|f (·, s, Ys , Zs )|ds ∈

E( sup |Yt |2 ) < ∞. 0≤t≤T

2. Now we derive a preliminary bound. Apply Itô’s formula to the semimartingale (eβs |δYs |2 )0≤s≤T to obtain for 0 ≤ t ≤ T eβT |δYT |2 − eβt |δYt |2 =β

Z T t

βs

2

e |δYs | ds + 2

Z T t

−f 2 (·, s, Ys2 , Zs2 )ids−2

eβs hδYs , f 1 (·, s, Ys1 , Zs1 ) Z T t

53

eβs hδYs , δZs∗ dWs i+

Z T t

eβs |δZs |2 ds.

By reordering the terms in the equation we obtain βt

2

e |δYt |

+ β

Z T t

βs

2

e |δYs | ds +

= eβT |δYT |2 + 2 Z T

−2

t

Z T t

Z T t

eβs |δZs |2 ds

eβs hδYs , δZs∗ dWs i

eβs hδYs , f 1 (·, s, Ys1 , Zs1 ) − f 2 (·, s, Ys2 , Zs2 )ids.

3. We prove for 0 ≤ t ≤ T : 1 Z T βs E(e |δYt | ) ≤ E(e |δYT | ) + 2 E( e |δ2 fs |2 ds). t µ βt

2

βT

2

To prove this, first take expectations on both sides of the inequality obtained in 2., with the result βt

Z T

2

E(e |δYt | ) + βE(

t βT

βs

2

e |δYs | ds) + E

Z T t

eβs |δZs |2 ds)

2

≤ E(e |δYT | ) Z T

+2E(

t

eβs hδYs , f 1 (·, s, Ys1 , Zs1 ) − f 2 (·, s, Ys2 , Zs2 )ids).

Now by our assumptions for 0 ≤ s ≤ T |f 1 (·, s, Ys1 , Zs1 ) − f 2 (·, s, Ys2 , Zs2 )| ≤ |f 1 (·, s, Ys1 , Zs1 ) − f 1 (·, s, Ys2 , Zs2 )| +|δ2 fs | ≤ C[|δs Y | + |δs Z|] + |δ2 fs |. The latter implies Z T t

E(2eβs |hδYs , f 1 (·, s, Ys1 , Zs1 ) − f 2 (·, s, Ys2 , Zs2 )i|ds

≤ =

Z T t

Z T t

2eβs E(|δYs |[C(|δs Y | + |δs Z|) + |δ2 fs |]ds 2eβs [CE(|δYs |2 ) + E(|δs Y |(C|δs Z|) + |δ2 fs |)]ds.

Now for C, y, z, t > 0 with µ, λ > 0 2y(Cz + t) = 2Cyz + 2yt t z ≤ C[(yλ)2 + ( )2 ] + (yµ)2 + ( )2 λ µ z t = C( )2 + ( )2 + y 2 (µ2 + Cλ2 ). λ µ 54

With this we can estimate the last term in our inequality further: Z T t

2eβs [CE(|δYs |2 ) + E(|δs Y |(C|δs Z| + |δ2 fs |))]ds ≤

Z T

eβs [2CE(|δYs |2 ) +

t

+ =

Z T

C E(|δs Z|2 ) 2 λ

1 E(|δ2 fs |2 ) + (µ2 + Cλ2 )E(|δs Y |2 ]ds 2 µ

eβs [(µ2 + C(2 + λ2 ))E(|δYs |2 )

t

+

C 1 2 E(|δ Z| ) + E(|δ2 fs |2 )]ds. s 2 2 λ µ

Summarizing, we obtain, using our assumptions on the parameters βt

Z T

2

(∗∗) E(e |δYt | ) ≤ E(

t

eβs |δYs |2 ds)[−β + C(2 + λ2 ) + µ2 ]

Z T

+E(

t

eβs |δZs |2 ds)[

C − 1] + E(eβT |δYT |2 ) 2 λ

1 Z T βs + 2 E( e |δ2 fs |2 ds) + E(eβT |δYT |2 ) t µ 1 Z T βs βT 2 ≤ E(e |δYT | ) + 2 E( e |δ2 fs |2 ds). t µ

This is the claimed inequality. 4. In order to obtain the first inequality in the assertion, it remains to integrate the inequality resulting from 3. in t ∈ [0, T ]. 5. The second inequality in the assertion follows from (**) by taking the second term from the right hand side to the left. This completes the proof. • We are in a position to state existence and uniqueness results for our BSDE (*). Theorem 10.1 Let (ξ, f ) be standard parameters. Then there exists a uniquely determined pair (Y, Z) ∈ H 2 (Rm )×H 2 (Rn×m ) with the property (BSDE)

Yt = ξ −

Z T t

Zs∗ dWs

+

Z T t

f (·, s, Ys , Zs )ds,

0 ≤ t ≤ T.

Proof Consider Γ : H 2,β (Rm ) × H 2,β (Rn×m ) → H 2,β (Rm ) × H 2,β (Rn×m ), (y, z) 7→ (Y, Z), 55

where (Y, Z) is a solution of the BSDE (∗) Yt = ξ −

Z T

Zs∗ dWs +

t

Z T t

f (·, s, ys , zs )ds,

0 ≤ t ≤ T.

1. We prove: (Y, Z) is well defined. First of all, our assumptions yield ξ+

Z T t

f (·, s, ys , zs )ds ∈ L2 (Ω),

0 ≤ t ≤ T.

Therefore Mt = E(ξ +

Z T 0

f (·, s, ys , zs )ds|Ft ),

0 ≤ t ≤ T,

is a well defined martingale. M possesses a continuous version, due to the fact that we are working in a Wiener filtration. M is square integrable. Hence we may apply the martingale representation theorem, which provides (a unique) Z ∈ H 2 (Rn×m ) such that Mt = M0 +

Z t 0

Let now Yt = M t −

Zs∗ dWs , Z t 0

0 ≤ t ≤ T.

f (·, s, ys , zs )ds.

Then Y is square integrable, and we have Yt = E(ξ +

Z T t

Hence YT = ξ = M 0 +

f (·, s, ys , zs )ds|Ft ), 0 ≤ t ≤ T. Z T

Zs∗ dWs 0

−

Z T 0

f (·, s, ys , zs )ds,

and thus for 0 ≤ t ≤ T Yt = ξ − M 0 − + M0 + = ξ−

Z t

Z T t

Z T 0

Zs∗ dWs +

Zs∗ dWs 0

−

Zs∗ dWs +

Z t

0 Z T t

Z T 0

f (·, s, ys , zs )ds

f (·, s, ys , zs )ds

f (·, s, ys , zs )ds.

2. We prove: For β > 2(1 + T )C the mapping Γ is a contraction. For this purpose, let (y 1 , z 1 ), (y 2 , z 2 ) ∈ H 2,β (Rm ) × H 2,β (Rn×m ), (Y 1 , Z 1 ), (Y 2 , Z 2 ) corresponding solutions of (*) according to 1. We apply Lemma 10.1 with C = 0, β = µ2 , and f i = f (·, y i , z i ). With this 56

choice we obtain ||δY ||2,β ||δZ||2,β

Z T 1 T ≤ [E( eβs |f (·, s, ys1 , zs1 ) − f (·, s, ys2 , zs2 )|2 ds)] 2 , 0 β Z T 1 1 ≤ [E( eβs |f (·, s, ys1 , zs1 ) − f (·, s, ys2 , zs2 )|2 ds)] 2 . 0 β

Since f is Lipschitz continuous, we further obtain 2T C [||δy||2,β + ||δz||2,β ], β 2C ≤ [||δy||2,β + ||δz||2,β ]. β

||δY ||2,β ≤ ||δZ||2,β We summarize to obtain

(∗∗) ||δY ||2,β + ||δZ||2,β ≤

2C(T + 1) [||δy||2,β + ||δz||2,β ]. β

By choice of β, Γ is a contraction. 3. Now let (Y , Z) be the fixed point of Γ, which exists due to 2. Let Yt = E(ξ +

Z T t

f (·, s, Y s , Z s )ds|Ft ),

0 ≤ t ≤ T.

Then Y is continuous and P -a.s. identical to Y . Then (Y, Z) is a solution of our BSDE. 4. Uniqueness follows from the contraction property of Γ and the uniqueness of the fixed point. • The construction of solutions in the preceding proof rests upon a recursive algorithm. The algorithm converges, as we shall note in the following Corollary. Corollary 10.1 Let β > 2(1 + T )C, ((Y k , Z k ))k≥0 the sequence of processes, given by Y 0 = Z 0 = 0, Ytk+1

=ξ−

Z T

(Zsk+1 )∗ dWs t

+

Z T t

f (·, s, Ysk , Zsk )ds

according to the proof of the preceding Theorem. Then ((Y k , Z k ))k≥0 converges in H 2,β (Rm ) × H 2,β (Rn×m ) to the uniquely determined solution (Y, Z) of (BSDE). 57

Proof The inequality (**) in the proof of Theorem 10.1 recursively yields ||Y k+1 − Y k ||2,β + ||Z k+1 − Z k ||2,β ≤ εk [||Y 1 − Y 0 ||2,β + ||Z 1 − Z 0 ||2,β ], with ε =

2C(T +1) β X

< 1. This implies [||Y k+1 − Y k ||2,β + ||Z k+1 − Z k ||2,β ] < ∞.

k∈N

Now a standard argument applies. •

11

Interpretation of backward stochastic differential equations in Malliavin’s calculus

In this chapter we shall establish the vital connection between Malliavin’s calculus and the structure of solutions of BSDE. We shall see that, provided the standard parameters are sufficiently smooth, the process Z can be interpreted as the Malliavin trace of the process Y . For doing this, we first have to introduce the version of the space L21 which corresponds to integrability in some arbitrary power p ≥ 2. For simplicity, we let the dimension of our underlying Wiener process be one, i.e. for this chapter we set n = 1. Definition 11.1 Let p ≥ 2, and Z T

Lp1 (Rm ) = {u|u adapted, [

(D1p )m

0

1

|ut |2 dt] 2 ∈ Lp (Ω),

ut ∈ for λ − a.a. t ≥ 0, for some measurable version

Z T Z T

of (s, t) 7→ Ds ut we have E([

0

0

p

|Ds ut |2 dsdt] 2 ) < ∞}.

For u ∈ Lp1 (Rm ) define Z T

||u||1,p = E([

0

2

Z T Z T

p 2

|ut | dt] ) + E([

0

0

To abbreviate, for k ∈ N, v ∈ L2 (Ω × [0, T ]k ) denote Z

||v|| = [

1

|vt |2 dt] 2 . k

[0,T ]

58

p

|Ds ut |2 dsdt] 2 ).

In these terms, Jensen’s inequality gives p

E(||Du|| ) ≤ T

p 2 −1

Z T 0

||Ds u||pp ds.

We next prove that solutions of BSDE for regular standard parameters are Malliavin differentiable, and that Z allows an interpretation as a Malliavin trace of Y . We need some versions of the process spaces considered in the previous chapter that correspond to p-integrable random variables. For p ≥ 2 denote by S p (Rm ) the linear space of all measurable (Ft )adapted continuous processes X : Ω × [0, T ] → Rm , endowed with the 1 norm ||X||S p = E(sup0≤t≤T |Xt |p ) p . Let further H p (Rm ) denote the linear space of measurable (Ft )-adapted processes X : Ω × [0, T ] → Rm endo1 wed with the norm ||X||p = E(||X||p ) p . To abbreviate, let B p (Rm ) = 1 S p (Rm ) × H p (Rm ), with the norm ||(Y, Z)||p = [||Y ||pS p + ||Z||pp ] p . We are ready to state our main result. Theorem 11.1 Let (f, ξ) be standard parameters such that ξ ∈ D12 ∩ L4 (Rm ), f : Ω × [0, T ] × Rm × Rm → Rm continuously differentiable in (y, z), with uniformly bounded and continuous partial derivatives, and such that for (y, z) ∈ Rm × Rm we have (H3)

f (·, y, z) ∈ L21 ,

f (·, 0, 0) ∈ H 4 (Rm ),

for t ∈ [0, T ] and (y 1 , z 1 , y 2 , z 2 ) ∈ (Rm × Rm )2 we have (H4)

|Ds f (·, t, y 1 , z 1 ) − Ds f (·, t, y 2 , z 2 )| ≤ Ks (t)[|y 1 − y 2 | + |z 1 − z 2 |]

(in L2 (Ω × [0, t]2 )), with a real-valued measurable process (Ks (t))0≤s≤t which is (Ft )-adapted in t, and satisfies Z T 0

||Ks ||44 ds < ∞.

For the unique solution (Y, Z) of the BSDE (*) we moreover suppose Z T 0

||Ds f (·, Y, Z)||2 ds < ∞

P -a.s.. Then we have: (Y, Z) ∈ L21 (Rm ) × L21 (Rm ), 59

and a (measurable) version of (Ds Yt , Ds Zt )0≤s,t≤T ) possesses the properties Ds Yt = Ds Zt = 0, Ds Yt = Ds ξ − Z T

(Ds Ys )0≤s≤T

Z T t

0 ≤ t < s ≤ T, Ds Zu∗ dWu

∂ ∂ [ f (·, u, Yu , Zu ) Ds Yu + f (·, u, Yu , Zu ) Ds Zu + t ∂y ∂z +Ds f (·, u, Yu , Zu )]du, 0 ≤ s ≤ t ≤ T, is a version of (Zs )0≤s≤T .

Proof 1. For further simplifying notation, we assume m = 1. As in chapter 10, our arguments are mainly based upon several a priori estimates. The first one is an analogue of Lemma 10.1 and investigates the properties of the contraction map on B 2 (R). Lemma 11.1 Let p ≥ 2, assume f (·, 0, 0) ∈ H p (R), and define Γ : B p (R) → B p (R), (y, z) 7→ (Y, Z), where (Y, Z) is the solution of the BSDE (+)

Yt = ξ −

Z T t

Zs dWs +

Z T

f (·, s, ys , zs )ds.

t

Let further for i = 1, 2 (Y i , Z i ) be the solutions corresponding to (y i , z i ) in (+), and let δY = Y 1 − Y 2 ,

δZ = Z 1 − Z 2 ,

δy = y 1 − y 2 ,

δz = z 1 − z 2 .

Then there exists a constant Cp not depending on (y, z, Y, Z) such that (i) (ii) (iii)

p

Z T

||Y ||pS p ≤ Cp E([|ξ|p ) + T 2 ( p 2

0 Z T

||Z||pp ≤ Cp E([|ξ|p ) + T (

||(δY, δZ)||pp

≤ Cp T

p 2

0

p

|f (·, s, ys , zs )|2 ds) 2 ]), p

|f (·, s, ys , zs )|2 ds) 2 ]),

||(δy, δz)||pp .

Proof a) Taking up the notation of the proof of Lemma 10.1, we show: Γ is well defined. 60

To do this, recall for 0 ≤ t ≤ T Yt = E(ξ +

Z T

f (·, s, ys , zs )ds|Ft ).

t

We have |Yt | ≤ E(|ξ| +

Z T t

|f (·, s, ys , zs )|ds|Ft ),

hence Doob’s inequality provides a universal constant Cp1 such that ||Y

||pS p

≤

Cp1 E([|ξ|

+

Z T

|f (·, s, ys , zs )|ds]p ).

0

Moreover, by Cauchy-Schwarz’ inequality Z T 0

1

Z T

|f (·, s, ys , zs )|ds ≤ T 2 [

1

|f (·, s, ys , zs )|2 ds] 2 ,

0

hence with another universal constant Cp2 we have (∗) ||Y

||pS p

≤

Cp2 E([|ξ|p

+[

Z T 0

p

|f (·, s, ys , zs )|2 ds] 2 ]).

p

Invoke f (·, 0, 0) ∈ H (R), that f is Lipschitz continuous, and that (y, z) ∈ B p (R), to see that the right hand side of the preceding inequality is finite. We next prove that Z ∈ H p (R). For this purpose we shall use the inequality of Burkholder. It yields further universal constants Cp3 , ·, Cp5 such that Z T 3 Cp E(| 0

p

(∗∗) E(||Z|| ) ≤

Zs dWs |p )

≤ Cp4 E(|ξ + Cp5 E(|ξ|p

≤

Z T 0

f (·, s, ys , zs )ds − Y0 |p ) p 2

Z T

+T [

0

p

|f (·, s, ys , zs )|2 ds] 2 ).

p

Hence we obtain Z ∈ H (R), and therefore (Y, Z) ∈ B p (R). The inequalities (i) and (ii) have also been established. b) We prove: (iii). The solution (δY, δZ) belongs to the generator f (·, t, yt1 , zt1 ) − f (·, t, yt2 , zt2 ), and ξ = 0. Therefore (i) and (ii), as well as an appeal to the Lipschitz condition, give, with universal constants Cp6 , Cp7 ||(δY, δZ)||pp

Z T

≤

Cp6 T

p 2

≤

Cp7 T Cp7 T

p 2

[||δy||Sp p + ||δz||pp ]

p 2

||(δy, δz)||pp .

=

E([

0

p

|f (·, t, yt1 , zt1 ) − f (·, t, yt2 , zt2 )|2 dt] 2 )

61

This completes the proof. • Let us return to the proof of Theorem 11.1. We define approximations of the solution of the BSDE recursively. Let for k ≥ 0, 0 ≤ t ≤ T Y 0 = Z 0 = 0, Ytk+1 = ξ −

Z T t

Zsk+1 dWs +

Z T t

f (·, s, Ysk , Zsk )ds.

We show: ||(Y k , Z k ) − (Y, Z)||4 → 0 (k → ∞). Recall the universal constant C4 from Lemma 11.1, (iii). We may (modulo repeating the argument finitely often on successive subintervals of 1 [0, T ]) assume that [C4 T 2 ] 4 < 1. With this condition, Lemma 11.1 implies that Γ is a contraction, and the solution (Y, Z) of the BSDE its unique fixed point in B 4 (R). From this observation, we obtain our assertion via the Cauchy sequence property of the approximate solutions which follows from l X

||(Y k , Z k )−(Y l , Z l )||4 ≤

||(Y r , Z r )−(Y r−1 , Z r−1 )||4 → 0 (k, l → ∞).

r=k+1

2. We prove by recursion on k: (Y k , Z k ) ∈ L21 (R) × L21 (R). This is trivial for k = 0. Let it be guaranteed for k. According to the chain rule for the Malliavin derivative and our hypotheses concerning the standard parameters we know for 0 ≤ t ≤ T ξ+

Z T t

f (·, s, Ysk , Zsk )ds ∈ D12 ,

with Malliavin derivative Z T

∂ ∂ f (·, u, Yuk , Zuk ) Ds Yuk + f (·, u, Yuk , Zuk ) Ds Zuk t ∂y ∂z k k + Ds f (·, u, Yu , Zu )]du. Ds ξ +

[

This is seen by discretizing the Lebesgue integral, using the chain rule, and then approximating by means of the boundedness properties of 62

the partial derivatives, the Lipschitz continuity properties of f and closedness of the operator D. Consequently, Lemma 9.2 yields for fixed 0≤t≤T Z T k+1 = E(ξ + f (·, s, Ysk , Zsk )ds|Ft ) ∈ D12 Yt t

as well. Consequently, also Z T t

Zsk+1 dWs = ξ +

Z T

f (·, s, Ysk , Zsk )ds − Ytk+1 ∈ D12 .

t

Now an appeal to Theorem 9.4 implies that Z k+1 ∈ L21 (R) and in L2 (Ω× [0, T ]2 ) we have the equation Ds Ds

Z T t Z T t

Zuk+1 dWu

Z T

=

t

Ds Zuk+1 dWu ,

Zuk+1 dWu = Zsk+1 +

Z T s

s ≤ t,

Ds Zuk+1 dWu ,

s > t.

All stated differentiabilities go along with square integrability of the Malliavin derivatives in all variables. This means that (Y k+1 , Z k+1 ) ∈ L21 (R) × L21 (R), and the recursion step is completed. We also can identify the Malliavin derivative by the formula valid for 0 ≤ s ≤ t ≤ T in the usual sense (∗ ∗ ∗)

Ds Xtk+1

= Ds ξ − +

Z T t

[

Z T t

Ds Zuk+1 dWu

∂ ∂ f (·, u, Yuk , Zuk ) Ds Yuk + f (·, u, Yuk , Zuk ) Ds Zuk ∂y ∂z +Ds f (·, u, Yuk , Zuk )]du.

3. In this step we show: (DY k , DZ k ) → (Y · , Z · ) in L2 (Ω × [0, T ]2 ), where for 0 ≤ s ≤ T (Y s , Z s ) is the solution of the BSDE (\)Yts Yts

Z T

Z T

∂ ∂ f (·, u, Yu , Zu ) Yus + f (·, u, Yu , Zu ) Zus t t ∂y ∂z +Ds f (·, u, Yu , Zu )]du, 0 ≤ s ≤ t ≤ T, s = Zt = 0, 0 ≤ t < s ≤ T.

= Ds ξ −

Zus dWu

+

[

63

We first consult our hypotheses to verify that, at least for λ-a.e. 0 ≤ s ≤ T the parameters (F s , Ds ξ) with F s (·, t, y, z) =

∂ ∂ f (·, t, Yt , Zt ) y + f (·, t, Yt , Zt ) z + Ds f (·, t, Yt , Zt ), ∂y ∂z

0 ≤ t ≤ T, y, z ∈ R, are standard. Hence (Y s , Z s ) is well defined (and set trivial on the set of s where the parameters eventually fail to be standard). Also in this case our arguments will be based on a priori inequalities. Lemma 11.2 Let (f i , ξ i ), i = 1, 2, be standard parameters of a BSDE, p ≥ 2. Suppose ξ i ∈ Lp (Ω),

f i (·, 0, 0) ∈ H p (R),

i = 1, 2.

Let (Y i , Z i ) ∈ B p (R) be the corresponding solutions, C a Lipschitz constant for f 1 . Put δY = Y 1 − Y 2 ,

δZ = Z 1 − Z 2 ,

δ2 ft = f 1 (·, t, Yt2 , Zt2 ) − f 2 (·, t, Yt2 , Zt2 ),

0 ≤ t ≤ T. Then for T small enough there exists a constant Cp,T such that ||(δY, δZ)||pp

Z T

p

≤ Cp,T [E(|δYT | ) + E(( p

0

|δ2 fs |ds)p )]

≤ Cp,T [E(|δYT |p ) + T 2 ||δ2 fs ||p ]. Proof With a calculation analogous to the one used to prove (i) and (ii) in Lemma 11.1 we arrive at the following inequality which is valid with universal constants Cp1 , ·, Cp3 , and for which we also use Doob’s and CauchySchwarz’ inequalities, ||δY ||pS p + ||δZ||pp ≤ Cp1 E(|δYT |p Z T

+(

0

|f 1 (·, t, Yt1 , Zt1 ) − f 2 (·, t, Yt2 , Zt2 )|dt)p )

≤ Cp2 E(|δYT |p + ( ≤ Cp3 [E(|δYT |p Z T

+(

0

Z T 0

[|δYs | + |δZs | + |δ2 fs |]ds)p ) p

|δ2 fs |ds)p ) + (T p ||δY ||pS p + T 2 ||δZ||pp )]. 64

p

Now choose T small enough to ensure Cp3 (T p +T 2 ) < 1. This being done, we may take the last two expressions in the previous inequality from the right to the left hand side, to obtain the desired estimate. • Let us now apply Lemma 11.2 to prove that for λ-a.a. 0 ≤ s ≤ T we have (Y s , Z s ) ∈ B 2 (R). For this purpose, we apply the Lemma with Y 1 = Y s , Y 2 = 0, ξ 1 = Ds ξ, ξ 2 = 0, ∂ ∂ f 1 (·, t, y, z) = ( f (·, t, Yt , Zt ) y + f (·, t, Yt , Zt ) z + Ds f (·, t, Yt , Zt ), ∂y ∂z 2 f = 0, 0 ≤ t ≤ T, y, z ∈ R. Then we have δ2 ft = Ds f (·, t, Yt , Zt ),

0 ≤ t ≤ T.

We obtain with some universal constant C the inequality ||(Y s , Z s )||22 ≤ CE(|Ds ξ|2 + ||Ds f (·, Y, Z)||2 ), and therefore

Z T 0

||(Y s , Z s )||22 ds < ∞.

This implies the desired integrability. To obtain estimates for differences of (DY k , DZ k ) and (Y · , Z · ), let us next, fixing k ∈ N, apply Lemma 11.2 to the following parameters ξ 1 = ξ 2 = Ds ξ, ∂ ∂ f 1 (t) = f (·, t, Ytk , Ztk ) Ds Ytk + f (·, t, Ytk , Ztk ) Ds Ztk +Ds f (·, t, Ytk , Ztk ), ∂y ∂z ∂ ∂ f (·, t, Yt , Zt ) Yts + f (·, t, Yt , Zt ) Zts + Ds f (·, t, Yt , Zt ), f 2 (t) = ∂y ∂z s ≤ t ≤ T. Set for abbreviation δtk = f 1 (t) − f 2 (t),

s ≤ t ≤ T.

The Lemma yields the inequality ||(Ds Y k+1 − Y s , Ds Z k+1 − Z s )||22 ≤ C1 E(( 65

Z T s

|δtk |dt)2 )

with a universal constant C1 . Let us now further estimate the right hand side of this inequality. We have, fixing 0 ≤ s ≤ T E((

Z T

|δtk |dt)2 ) ≤ C2 [Ask (T ) + Bks (T ) + Cks (T )],

s

where Ask (T )

Z T

= E([

s Z T

|Ds f (·, t, Yt , Zt ) − Ds f (·, t, Ytk , Ztk )|dt]2 ),

∂ f (·, t, Ytk , Ztk )(Yts − Ds Ytk )|dt]2 ) s ∂y Z T ∂ + E([ | f (·, t, Ytk , Ztk )(Zts − Ds Ztk )|dt]2 ), s ∂z Z T ∂ ∂ Cks (T ) = E([ | f (·, t, Yt , Zt ) − f (·, t, Ytk , Ztk )| |Yts |dt]2 ) s ∂y ∂y Z T ∂ ∂ + E([ | f (·, t, Yt , Zt ) − f (·, t, Ytk , Ztk )| |Zts |dt]2 ). s ∂z ∂z

Bks (T ) = E([

|

With further universal constants C3 , C4 we deduce, using (H4) Ask (T )

Z T

≤ E([

s

Ks (t)[|Yt − Ytk | + |Zt − Ztk |]dt]2 )

Z T

≤ C3 E(

≤ C4 [E(

s Z T s

2

Z T

Ks (t) dt[

s

|Yt − Z T

1 2

Ks (t)4 dt] [E(

s

Ytk |2 dt

+

Z T

|Zt − Ztk |2 dt])

s 1

|Yt −Ytk |4 dt) 2 + E(

Z T s

1

|Zt −Ztk |4 dt) 2 ].

Hence by part 1. of the proof lim

Z T

k→∞ 0

Ask (T )ds = 0.

Furthermore, since the partial derivatives of f with respect to y, z are R bounded and continuous, and since E( 0T ||(Y s , Z s )||2 ds < ∞, dominated convergence allows to conclude lim

Z T

k→∞ 0

Cks (T )ds = 0.

Let us finally discuss the convergence of the Bks (T ) as k → ∞. Again by boundedness of the partial derivatives of f we obtain with a universal constant C5 Bks (T ) ≤ C5 T 2 ||Ds Y k − Y s , Ds Z k − Z s )||22 . 66

Now choose T small enough to ensure α = C5 T 2 < 1. Let ε > 0. Then by what has been shown there exists N ∈ N large enough so that for k ≥ N we have Z T

E(

0

||(Ds Y k+1 − Y s , Ds Z k+1 − Z s )||2 ds) Z T

≤ ε + αE(

0

||(Ds Y k − Y s , Ds Z k − Z s )||2 ds).

By recursion we obtain for k ≥ N Z T

E(

0

||(Ds Y k − Y s , Ds Z k − Z s )||2 ds) ≤ ε(1 + α + α2 + · · · + αk−N −1 ) Z T

+εαk−N E( ||(Ds Y N − Y s , Ds Z N − Z s )||2 ds) 0 Z T ε k−N ≤ +α E( ||(Ds Y N − Y s , Ds Z N − Z s )||2 ds). 0 1−α Now let k → ∞. Since ε is arbitrary, we conclude lim

Z T

k→∞ 0

Bks (T )ds = 0.

3. Since L21 (R) is a Hilbert space, and D is a closed operator, we obtain that (Y, Z) ∈ L21 (R) × L21 (R), and that (Y s , Z s )0≤s≤T is a version of (Ds Y, Ds Z)0≤s≤T in the usual sense. 4. We show: (Dt Yt )0≤t≤T is a version of (Zt )0≤t≤T . For t ≤ s we have Ys = Yt +

Z s t

Zr dWr −

Z s t

f (·, r, Yr , Zr )dr.

Hence by Theorem 9.4 for t < u ≤ s Du Ys = Zu + −

Z s

Z s u

Du Zr dWr

∂ ∂ f (·, r, Yr , Zr ) Du Yr + f (·, r, Yr , Zr ) Du Zr u ∂y ∂z +Du f (·, r, Yr , Zr )]dr. [

By continuity in t of (Y s , Z s ) we may choose u = s, to obtain the desired identity. • 67

Literatur [1] Amendinger, J., Imkeller, P., Schweizer, M. Additional logarithmic utility of an insider. Stoch. Proc. Appl. 75 (1998), 263-286. [2] S. Ankirchner, P. Imkeller, A. Popier On measure solutions of backward stochastic differential equations. Preprint, HU Berlin (2008). [3] S. Ankirchner, P. Imkeller, G. Reis Classical and Variational Differentiability of BSDEs with Quadratic Growth. Electron. J. Probab. 12 (2007), 1418–1453 (electronic). [4] S. Ankirchner, P. Imkeller, G. Reis Pricing and hedging of derivatives based on non-tradable underlyings. Preprint, HU Berlin (2007). [5] S. Ankirchner, P. Imkeller Quadratic hedging of weather and catastrophe risk by using short term climate predictions. Preprint, HU Berlin (2008). [6] Bell, D. The Malliavin calculus. Pitman Monographs and Surveys in Pure and Applied Math. 34. Longman and Wiley: 1987. [7] Bell, D. Degenerate stochastic differential equations and hypoellipticity. Pitman Monographs and Surveys in Pure and Applied Mathematics. 79. Harlow, Essex: Longman (1995). [8] Bismut, J. M. Martingales, the Malliavin Calculus and Hypoellipticity under General Hörmander’s Conditions. Z. Wahrscheinlichkeitstheorie verw. Geb. 56 (1981), 469-505. [9] Bismut, J. M. Large deviations and the Malliavin calculus. Progress in Math. 45, Birkhäuser, Basel 1984. [10] Bouleau, N., Hirsch, F. Dirichlet forms and analysis on Wiener space. W. de Gruyter, Berlin 1991. [11] El Karoui, N.; Peng, S.; Quenez, M.C. Backward stochastic differential equations in finance. Math. Finance 7, No.1, 171 (1997).

68

[12] Fournié, E., Lasry, J.-M., Lebouchoux, J., Lions, P.-L., Touzi, N. Applications of Malliavin’s calculus to Monte Carlo methods in finance. Finance Stochast. 3 (1999), 391-412. [13] Huang, Z-Y.; Yan, J-A. Introduction to infinite dimensional stochastic analysis. Mathematics and its Applications (Dordrecht). 502. Dordrecht: Kluwer Academic Publishers (2000). [14] Ikeda, N., Watanabe, S. Stochastic differential equations and diffusion processes. North Holland (2 nd edition): 1989. [15] Imkeller, P. Malliavin’s calculus in insider models: additional utility and free lunches. Preprint, HU Berlin (2002). [16] Imkeller, P., Pontier, M., Weisz, F. Free lunch and arbitrage possibilities in a financial market model with an insider. Stochastic Proc. Appl. 92 (2001), 103-130. [17] Imkeller, P. Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus. PTRF 106(1996), 105-135. [18] Imkeller, P., Perez-Abreu, V., Vives, J. Chaos expansions of double intersection local time of Brownian motion in Rd and renormalization. Stoch. Proc. Appl. 56(1994), 1-34. [19] Jacod, J. Grossissement initial, hypothèse (H’), et théorème de Girsanov. in: Grossissements de filtrations: exemples et applications. T. Jeulin, M.Yor (eds.). LNM 1118. Springer: Berlin 1985. [20] Janson, Svante Gaussian Hilbert spaces. Cambridge Tracts in Mathematics. 129. Cambridge: Cambridge University Press (1997). [21] Kuo, H. H. Donsker’s delta function as a generalized Brownian functional and its application. Lecture Notes in Control and Information Sciences, vol. 49 (1983), 167-178. Springer, Berlin. [22] Kusuoka, S. , Stroock, D.W. The partial Malliavin calculus and its applications to nonlinear filtering. Stochastics 12 (1984), 83-142.

69

[23] Ma, J., and Yong, J. Forward-Backward Stochastic Differential Equations and Their Applications. LNM 1702. Springer: Berlin (1999). [24] Malliavin, P. Stochastic analysis. Grundlehren der Mathematischen Wissenschaften. 313. Berlin: Springer (1997). [25] Malliavin, P. Integration and probability. Graduate Texts in Mathematics. 157. New York, NY: Springer (1995). [26] Malliavin, P. Stochastic calculus of variation and hypoelliptic operators. Proc. Intern. Symp. SDE Kyoto, 1976, 195-263. Kinokynia: Tokyo 1978. [27] Malliavin, P. C k −hypoellipticity with degeneracy. Stochastic Analysis. ed: A. Friedman and M. Pinsky, 199-214, 327-340. Acad. Press: New York 1978. [28] Malliavin, P., Thalmaier, A. Stochastic calculus of variations in mathematical finance. Springer: Berlin 2006. [29] Norris, J. Simplified Malliavin calculus. Séminaire de Probabilités XX. LNM 1204, 101-130. Springer: Berlin 1986. [30] Nualart, D., Pardoux, E. Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields 78 (1988), 535-581. [31] Nualart, D. The Malliavin calculus and related topics. Springer: Berlin 1995. [32] Nualart, D. Malliavin’s calculus and anticipative calculus. St Flour Lecture Notes (1996). [33] Nualart, D., Vives, J. Chaos expansion and local times. Publicacions Matematiques 36(2), 827-836. [34] Revuz, D., Yor, M. Continuous martingales and Brownian motion. Springer, Berlin 1991. [35] Stroock, D. W. The Malliavin Calculus. Functional Analytic Approach. J. Funct. Anal. 44 (1981), 212-257. 70

[36] Stroock, D. W. The Malliavin calculus and its applications to second order parabolic differential equations. Math. Systems Theory, Part I, 14 (1981), 25-65, Part II, 14 (1981), 141-171. [37] Stroock, D. W. Some applications of stochastic calculus to partial differential equations. In: Ecole d’Eté de Probabilité de St Flour. LNM 976 (1983), 267-382. [38] Watanabe, H. The local time of self-intersections of Brownian motions as generalized Brownian functionals. Letters in Math. Physics 23 (1991), 1-9. [39] Watanabe, S. Stochastic differential equations and Malliavin calculus. Tata Institut of Fundamental Research. Springer: Berlin 1984. [40] Yor, M. Grossissement de filtrations et absolue continuité de noyaux. in: Grossissements de filtrations: exemples et applications. T. Jeulin, M.Yor (eds.). LNM 1118. Springer: Berlin 1985. [41] Yor, M. Entropie d’une partition, et grossissement initial d’une filtration. in: Grossissements de filtrations: exemples et applications. T. Jeulin, M.Yor (eds.). LNM 1118. Springer: Berlin 1985.

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