We will suppose in this subsection that the separable Hilbert space H is an L2 space of the form H = L2(T,B, à), where à is aσ-finite atomless measure on a measurable space (T,B).
The derivative of a random variableF∈D1,2will be a stochastic process denoted by {DtF, t ∈ T} due to the identification between the Hilbert spacesL2(Ω;H) andL2(T×Ω). Notice thatDtF is defined almost every- where (a.e.) with respect to the measure àìP. More generally, if k ≥2 andF ∈Dk,2, the derivative
DkF ={Dtk1,...,tkF, ti∈T},
is a measurable function on the product spaceTk×Ω, which is defined a.e.
with respect to the measure àkìP.
Example 1.2.1 Consider the example of a d-dimensional Brownian mo- tion on the interval[0,1], defined on the canonical spaceΩ =C0([0,1];Rd).
In this case DF, hH can be interpreted as a directional Fr´echet deriva- tive. In fact, let us introduce the subspace H1 of Ω which consists of all absolutely continuous functions x : [0,1] → Rd with a square integrable derivative, i.e., x(t) = t
0x(s)ds,˙ x˙ ∈ H = L2([0,1];Rd). The space H1 is usually called the Cameron-Martin space. We can transport the Hilbert space structure of H to the spaceH1 by putting
x, yH1 =x,˙ y˙H = d i=1
1 0
˙
xi(s) ˙yi(s)ds.
In this way H1 becomes a Hilbert space isomorphic toH. The injection of H1 intoΩis continuous because we have
sup
0≤t≤1|x(t)| ≤ 1
0 |x(s)˙ |ds≤ x˙H=xH1.
Assume d = 1 and consider a smooth functional of the particular form F = f(W(t1), . . . , W(tn)), f ∈ Cp∞(Rn), 0 ≤ t1 < ã ã ã < tn ≤ 1, where W(ti) =ti
0 dWt=W(1[0,ti]). Notice that such a functional is continuous inΩ. Then, for any functionhinH, the scalar productDF, hH coincides with the directional derivative ofF in the direction of the elementã
0h(s)ds, which belongs toH1. In fact,
DF, hH = n i=1
∂if(W(t1), . . . , W(tn))1[0,ti], hH
= n i=1
∂if(W(t1), . . . , W(tn)) ti
0
h(s)ds
= d
dǫF(ω+ǫ ã
0
h(s)ds)|ǫ=0.
On the other hand, ifF is Fr´echet differentiable andλF denotes the signed measure associated with the Fr´echet derivative ofF, thenDtF =λF((t,1]).
In fact, for anyh∈H we have DF, hH=
1 0
λF(dt)(
t 0
h(s)ds)dt= 1
0
λF((t,1])h(t)dt.
Suppose thatF is a square integrable random variable having an orthog- onal Wiener series of the form
F = ∞ n=0
In(fn), (1.38)
where the kernels fn are symmetric functions of L2(Tn). The derivative DtF can be easily computed using this expression.
Proposition 1.2.7 Let F ∈ D1,2 be a square integrable random variable with a development of the form (1.38). Then we have
DtF = ∞ n=1
nIn−1(fn(ã, t)). (1.39)
Proof: Suppose first that F = Im(fm), where fm is a symmetric and elementary function of the form (1.10). Then
DtF = m j=1
m i1,...,im=1
ai1ãããimW(Ai1)ã ã ã1Aij(t)ã ã ãW(Aim) =mIm−1(fm(ã, t)).
Then the result follows easily.
The heuristic meaning of the preceding proposition is clear. Suppose that F is a multiple stochastic integral of the formIn(fn), which has also been denoted by
F =
Tã ã ã
T
fn(t1, . . . , tn)W(dt1)ã ã ãW(dtn).
Then, F belongs to the domain of the derivation operator and DtF is obtained simply by removing one of the stochastic integrals, letting the variablet be free, and multiplying by the factorn.
Now we will compute the derivative of a conditional expectation with respect to aσ-field generated by Gaussian stochastic integrals. LetA∈ B. We will denote byFAtheσ-field (completed with respect to the probability P) generated by the random variables{W(B), B ⊂A, B ∈ B0}. We need the following technical result:
Lemma 1.2.5 Suppose thatF is a square integrable random variable with the representation (1.38). Let A∈ B. Then
E(F|FA) = ∞ n=0
In(fn1⊗An). (1.40) Proof: It suffices to assume that F = In(fn), where fn is a function in En. Also, by linearity we can assume that the kernel fn is of the form 1B1ìãããìBn, whereB1, . . . , Bn are mutually disjoint sets of finite measure.
In this case we have
E(F|FA) = E(W(B1)ã ã ãW(Bm)|FA)
= En
i=1
(W(Bi∩A) +W(Bi∩Ac))| FA
= In(1(B1∩A)ìãããì(Bn∩A)).
Proposition 1.2.8 Suppose that F belongs to D1,2, and let A∈ B. Then the conditional expectationE(F|FA)also belongs to the spaceD1,2, and we have:
Dt(E(F|FA)) =E(DtF|FA)1A(t) a.e. in T×Ω.
Proof: By Lemma 1.2.5 and Proposition 1.2.7 we obtain Dt(E(F|FA)) =
∞ n=1
nIn−1(fn(ã, t)1⊗A(n−1))1A(t) =E(DtF|FA)1A(t).
Corollary 1.2.1 Let A∈ B and suppose thatF ∈D1,2 isFA-measurable.
ThenDtF is zero almost everywhere inAc×Ω.
Given a measurable set A ∈ B, we can introduce the space DA,2 of random variables which are differentiable on A as the closure of S with respect to the seminorm
F2A,2=E(F2) +E
A
(DtF)2à(dt)
.
Exercises
1.2.1 LetW ={W(t),0≤t≤1}be a one-dimensional Brownian motion.
Leth∈L2([0,1]), and consider the stochastic integralF =1
0 htdWt. Show thatFhas a continuous modification onC0([0,1]) if and only if there exists a signed measure à on (0,1] such that h(t) = à((t,1]), for all t ∈ [0,1], almost everywhere with respect to the Lebesgue measure.
Hint: Ifh is given by a signed measure, the result is achieved through integrating by parts. For the converse implication, show first that the con- tinuous modification of F must be linear, and then use the Riesz repre- sentation theorem of linear continuous functionals onC([0,1]). For a more general treatment of this problem, refer to Nualart and Zakai [268].
1.2.2 Show that the expression of the derivative given in Definition 1.2.1 does not depend on the particular representation ofF as a smooth func- tional.
1.2.3 Show that the operatorDk is closable from S intoLp(Ω;H⊗k).
Hint: Let {FN, N ≥ 1} be a sequence of smooth functionals that con- verges to zero inLpand such thatDkFN converges to someηinLp(Ω;H⊗k).
Iterating the integration-by-parts formula (1.31), show thatE(η, h1⊗ã ã ã⊗
hkF ξ) = 0 for allh1, . . . , hk∈H,F ∈ Sb, and ξ= exp
−ǫ k i=1
W(hi)2 .
1.2.4 Letfn be a symmetric function inL2([0,1]n). Deduce the following expression for the derivative ofF =In(fn):
DtF = n!
n i=1
{t1<ããã<ti−1<t<tiããã<tn−1}
fn(t1, . . . , tn−1, t)
ìdWt1ã ã ãdWtn−1, with the conventiontn= 1.
1.2.5 Let F ∈ Dk,2 be given by the expansion F = ∞
n=0In(fn). Show that
Dtk1,...,tkF = ∞ n=k
n(n−1)ã ã ã(n−k+ 1)In−k(fn(ã, t1, . . . , tk)), and
E(DkF2L2(Tk)) = ∞ n=k
n!2
(n−k)!fn2L2(Tn). 1.2.6 Suppose that F = ∞
n=0In(fn) is a random variable belonging to the space D∞,2=∩kDk,2. Show thatfn =n!1E(DnF) for everyn≥0 (cf.
Stroock [321]).
1.2.7LetF = exp(W(h)−12
Th2sà(ds)),h∈L2(T). Compute the iterated derivatives of F and the kernels of its expansion into the Wiener chaos.
1.2.8 Lete1, . . . , en be orthonormal elements in the Hilbert spaceH. De- note byFntheσ-field generated by the random variablesW(e1), . . . , W(en).
Show that anFn-measurable random variableF belongs toD1,2if and only if there exists a functionf in the weighted Sobolev spaceW1,2(Rn, N(0, In)) such that
F =f(W(e1), . . . , W(en)).
Moreover, it holds thatDF =n
i=1∂if(W(e1), . . . , W(en))ei.
1.2.9 Let (Ω,F, P) be the canonical probability space of the standard Brownian motion on the time interval [0,1]. Let F be a random variable that satisfies the following Lipschitz property:
|F(ω+ ã
0
hsds)−F(ω)| ≤chH a.s., h∈H =L2([0,1]).
Show thatF ∈D1,2andDFH ≤ca.s. In [92] Enchev and Stroock proved the reciprocal implication.
Hint:Suppose thatF ∈L2(Ω) (the general case is treated by a truncation argument). Consider a complete orthonormal system {ei, i ≥ 1} in H.
Define Fn = E(F|Fn), where Fn is the σ-field generated by the random variablesW(e1), . . . , W(en). Show thatFn =fn(W(e1), . . . , W(en)), where
fn is a Lipschitz function with a Lipschitz constant bounded by c. Use Exercise 1.2.8 to prove that Fn belongs to D1,2 and DFnH ≤ c, a.s.
Conclude using Lemma 1.2.3.
1.2.10 Show that the operator defined in (1.33) is closable in Lp(Ω), for allp≥1.
1.2.11Suppose thatW ={W(t),0≤t≤1}is a standard one-dimensional Brownian motion. Show that the random variable M = sup0≤t≤1W(t) belongs to the spaceD1,2, andDtM =1[0,T](t), whereT is the a.s. unique point whereW attains its maximum.
Hint:Approximate the supremum ofW by the maximum on a finite set (see Section 2.1.4).
1.2.12 LetF1 andF2 be two elements ofD1,2 such thatF1 and DF1H are bounded. Show thatF1F2∈D1,2 andD(F1F2) =F1DF2+F2DF1. 1.2.13Show the following Leibnitz rule for the operatorDk:
Dkt1,...,tk(F G) =
I⊂{t1,...,tk}
D|II|(F)DkIc−|I|(G), F, G∈ S,
where for any subsetI of{t1, . . . , tk},|I|denotes the cardinality ofI.
1.2.14Show that the setS0 is dense inDk,p for anyk≥1,p≥1.