Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics
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196 8 Mean Derivatives in Linear Spaces
E
f(ξ(t))E
ξ
t
(w(t) − w(t − Δt))
= E
t
t−Δt
(gradf(ξ(s)) · a(s))ds dw(s)+
σ
2
2
t
t−Δt
trf
(ξ(s))ds dw
+
t
t−Δt
σgradf(ξ(s))ds
= E
t
t−Δt
σgradf(ξ(s))ds
,
and so
E[f ((ξ(t))u
ξ
(t, ξ(t))] = −
1
2
E
σf(ξ(t)) lim
Δt→+0
1
Δt
E
ξ
t
(w(t) −w(t − Δt)
= −
1
2
E[σ
2
grad f(ξ(t))]
= −
1
2
[0,l]×R
n
σ
2
grad f(x)ρ
ξ
(t, x)dλ
=
1
2
[0,l]×R
n
σ
2
f(x)grad ρ
ξ
(t, x)dλ
=
1
2
[0,l]×R
n
σ
2
f(x)
grad ρ
ξ
(t, x)
ρ
ξ
(t, x)
ρ
ξ
(t, x)dλ
=
1
2
E[σ
2
f(ξ(t))grad log ρ
ξ
(t, ξ(t))].
Since this is valid for every smooth function f with compact support, (8.18)
holds.
From (8.18) it follows that the osmotic velocity does indeed characterize
the ‘variation of randomness’ of the process.
Lemma 8.18 For the process (8.15) in R
n
, the vector field v
ξ
(t, x) and the
density ρ
ξ
(t, x) satisfy the continuity equation
∂ρ
ξ
(t, x)
∂t
= −div(ρ
ξ
v
ξ
). (8.19)
Proof. Let f(t, x) be a smooth real-valued function on [0,l]×R
n
with compact
support. Note that for such f and a vector Y the equality f
(Y )=(gradf ·Y )
holds where the dot denotes the inner product in R
n
. Recall that E(E
ξ
t
(·)) =
E(·). Then by formula (8.16)fort>sthe equality
E[f (t, ξ(t)) − f(s, ξ(s))]
= E
t
s
∂f
∂t
dτ
t
s
(gradf ·Y
0
(t, ξ(t)))dτ +
t
s
σ
2
2
∇
2
fdτ
holds. On the other hand, by formula (8.17) we get the equality
8.1 General Definitions and Results 197
E[f (t, ξ(t)) − f(s, ξ(s))]
= E
t
s
∂f
∂t
dτ +
t
s
(gradf ·Y
0
∗
(t, ξ(t)))dτ −
t
s
σ
2
2
∇
2
fdτ
.
Thus
E[f (t, ξ(t)) − f(s, ξ(s))] = E
t
s
∂f
∂t
dτ +
t
s
(gradf ·v
ξ
(t, ξ(t)))dτ
,
where v
ξ
(t, x)=
1
2
(Y
0
(t, x)+Y
0
∗
(t, x)) (see above). But
E
t
s
∂f
∂t
dτ +
t
s
(gradf ·v
ξ
(t, ξ(t)))dτ
=
[s,t]×R
n
∂f(τ, x)
∂τ
ρ(τ,x)+
gradf ·v
ξ
(τ,x)ρ(τ, x)
dλ
=
[s,t]×R
n
∂
∂t
f(τ, x)ρ
ξ
(τ,x)
dλ −
[s,t]×R
n
f(τ, x)
∂ρ
ξ
(τ,x)
∂τ
dλ
−
[s,t]×R
n
f(τ, x)div
v
ξ
(τ,x)ρ
ξ
(τ,x)
dλ
= E[f (t, ξ(t)) − f(s, ξ(s))] −
[s,t]×R
n
f(τ, x)
∂ρ
ξ
(τ,x)
∂τ
dλ
−
[s,t]×R
n
f(τ, x)div
v
ξ
(τ,x)ρ
ξ
(τ,x)
dλ.
Hence
[s,t]×R
n
f(τ, x)
∂ρ
ξ
(τ,x)
∂τ
dλ = −
[s,t]×R
n
f(τ, x)div
v
ξ
(τ,x)ρ
ξ
(τ,x)
dλ.
Since the last equality is valid for any f, the Lemma follows. An alternative
proof for a Markovian diffusion ξ(t) can be found, e.g., in [187, 188, 190].
Lemmas 8.17 and 8.18 can be generalized for processes with more compli-
cated diffusion terms in the following way.
Consider an autonomous smooth field of non-degenerate linear operators
A(x):R
n
→ R
n
, x ∈ R
n
. Suppose that ξ(t) is a diffusion type process
whose diffusion integrand is A(ξ(t)). Then its diffusion coefficient A(x)A
∗
(x)
is a smooth field of symmetric positive definite matrices α(x)=(α
ij
(x)).
Since all such matrices are invertible, the field of inverse matrices (α
ij
)exists
and is smooth and at any x the matrix (α
ij
)(x) is symmetric and positive
definite. Thus it defines a new Riemannian metric α(·, ·)=α
ij
dx
i
⊗ dx
j
on R
n
. Consider the Riemannian volume form of this Riemannian metric
Λ
α
=
det(α
ij
)dx
1
∧ dx
2
∧···∧dx
n
(see (1.31)).
198 8 Mean Derivatives in Linear Spaces
Denote by ρ
ξ
(t, x) the probability density of ξ(t) with respect to the volume
form dt ∧ Λ
α
=
det(α
ij
)dt ∧ dx
1
∧ dx
2
∧···∧dx
n
on [0,T] × R
n
, i.e., for
any continuous bounded function f :[0,T] × R
n
→ R the relation
T
0
E(f (t, ξ(t)))dt =
T
0
⎛
⎝
Ω
f(t, ξ(t))dP
⎞
⎠
dt =
[0,T ]×R
n
f(t, x)ρ
ξ
(t, x)dt ∧ Λ
α
holds. We have the following generalization of formula (8.18):
u
ξ
(t, x)=
1
2
Grad log ρ
ξ
(t, x) = Grad log
ρ
ξ
(t, x), (8.20)
where Grad denotes the gradient with respect to the Riemannian metric
α(·, ·). Analogously for v
ξ
(t, x)andρ
ξ
(t, x) we have the following generaliza-
tion of formula (8.19) (equation of continuity)
∂ρ
ξ
(t, x)
∂t
= −Div(v
ξ
(t, x)ρ
ξ
(t, x)) (8.21)
where Div denotes divergence with respect to the Riemannian metric α(·, ·).
The arguments for the derivation of (8.20)and(8.21) are analogous to those
in the proofs of Lemmas 8.17 and 8.18 with a natural modification for using
Grad and Div. For an alternative proof for Markovian diffusion processes, see
[190].
Let Z(t, x)beaC
2
-smooth vector field, and ξ(t) be a stochastic process.
Definition 8.19. The forward and backward mean derivatives of Z along ξ(·)
at t (denoted by DZ(t, ξ(t)) and D
∗
Z(t, ξ(t)), respectively) are the L
1
-limits
of the form
DZ(t, ξ(t)) = lim
Δt→+0
E
ξ
t
Z(t + Δt, ξ(t + Δt)) −Z(t, ξ(t))
Δt
(8.22)
D
∗
Z(t, ξ(t)) = lim
Δt→+0
E
ξ
t
Z(t, ξ(t)) − Z(t −Δt, ξ(t − Δt))
Δt
(8.23)
As in Definition 8.1,ifξ(t) is a Markov process, E
ξ
t
can be replaced by
E(·|P
ξ
t
)in(8.22)andbyE(·|F
ξ
t
)in(8.23), see Remark 8.3.
Of course DZ(t, ξ(t)) and D
∗
Z(t, ξ(t)) can be presented as compositions
of ξ(t) with certain Borel vector fields on R
n
. These vector fields (regressions)
will also be denoted by DZ and D
∗
Z, respectively.
Lemma 8.20 For the process (8.15) in R
n
the following formulae hold
DZ =
∂
∂t
Z +(Y
0
·∇)Z +
σ
2
2
∇
2
Z, (8.24)
D
∗
Z =
∂
∂t
Z +(Y
0
∗
·∇)Z −
σ
2
2
∇
2
Z, (8.25)
8.2 Calculation of Mean Derivatives for a Wiener Process and for Diffusion Processes 199
where ∇ =(
∂
∂x
1
, ...,
∂
∂x
n
), ∇
2
is the Laplacian, the dot denotes the inner
product in R
n
and Y
0
(t, x) and Y
0
∗
(t, x) are as introduced in formulae (8.5)
and (8.6).
Proof. The vector field Z(t, x) can be considered as a map Z :[0,l]×R
n
→ R
n
and so one may apply formulae (8.16)and(8.17). Note that Z
(Y )=(Y ·∇)Z
for a vector Y . Formula (8.24) follows immediately from (8.16)and(8.22)
while (8.25) follows from (8.17)and(8.23).
In fact the forward mean derivative and the quadratic mean derivative
together make it possible in principle to recover a stochastic process from
its mean derivatives: the forward mean derivative gives information about
the drift while the quadratic mean derivative gives information about the
diffusion term. It turns out that such recovery is also possible for more com-
plicated relations with mean derivatives that we call equations with mean
derivatives (EMDs).
Specify a homogeneous polynomial f(x, y)oforderk of two variables,
analogous polynomials g
i
(x, y), i =1, ..., k −1oforderi and two mappings:
F : R ×R
n
× ... × R
n
→ R
n
and g : R × R
n
→ R
n
.
Definition 8.21. A k-th-order stochastic equation with mean derivatives
(EMD) is a system
f(D, D
∗
)ξ(t)=F (t, ξ(t),g
1
(D, D
∗
)ξ(t), ..., g
k−1
(D, D
∗
)ξ(t)), (8.26)
D
2
ξ(t)=g(t, ξ(t)).
The definition of a differential inclusion with mean derivatives is anal-
ogous. In Section 8.4 below we prove some existence theorems for several
types of first order equations and inclusion with mean derivatives. Various
second order equations and inclusions with mean derivatives are considered
in Chapters 14 and 15 and in Section 16.4. For the construction of their so-
lutions we need to have formulae of mean derivatives for processes from a
sufficiently broad class. The next two sections are devoted to the derivation
of such formulae.
8.2 Calculation of Mean Derivatives for a Wiener
Process and for Diffusion Processes
For a Wiener process w(t)inR
n
Dw(t)=0,t ∈ [0,l), by Lemma 8.5(i) since
w(t) is a martingale.
Lemma 8.22 For t ∈ (0,l] the equality D
∗
w(t)=
w(t)
t
holds.
200 8 Mean Derivatives in Linear Spaces
Proof. In this case, from the definition of the osmotic velocity u
w
(t, w(t)) it
follows that D
∗
w(t)=−2u
w
(t, w(t)). Recall that the density ρ
w
(t, x) is given
in formula (6.1). Thus according to formula (8.18)wehave
u
w
(t, x)=−
1
2
·
x
t
i.e. D
∗
w(t)=
w(t)
t
.
Obviously the process
w(t)
t
does not exist at t = 0. Nevertheless the fol-
lowing statement holds:
Lemma 8.23 The integral
t
0
w(s)
s
ds exists a.s. for all t ∈ [0,l].
Proof. By standard calculations, using the density ρ
w
(t, x) one can easily
obtain the estimate E
t
0
w(s)
s
ds<C·
√
t,whereE denotes the expectation
and the constant C>0 depends only on the dimension n. Then the result
follows from the classical Tchebyshev inequality.
Remark 8.24. To emphasize that mean derivatives essentially depend on
the σ-algebras with respect to which they are calculated, we consider the
following example. By Lemma 8.23 the process η(t)=−
t
0
w(s)
s
ds + w(t)is
well-defined. From Lemma 8.22 one can easily derive that D
w
∗
η(t)=0and
D
w
η(t)=−D
∗
w(t). But it is shown in [153]thatη(t) is a Wiener process
with respect to its own ‘past’ family of σ-algebras and so Dη(t)=0.
Lemma 8.25
(i) D
w
w(t)
t
= −
w(t)
t
2
for t ∈ (0,l).
(ii) D
w
∗
w(t)
t
=0for t ∈ (0,l].
Proof. It is easy to see that D
w
w(t)
t
=(
d
dt
1
t
)w(t)+
1
t
Dw(t)=−
w(t)
t
2
and
D
w
∗
w(t)
t
=(
d
dt
1
t
)w(t)+
1
t
D
∗
w(t)=0.
Lemma 8.26 Let a Markovian diffusion process ξ(t) be a solution of the Itˆo
equation (6.16).Then:
(i) Dξ(t)=a(t, ξ(t)) for t ∈ (0,l];
(ii) D
∗
ξ(t)=a(t, ξ(t)) −trA
(A(t, ξ(t))) + A(t, ξ(t))D
ξ
∗
w(t) for t ∈ (0,l],
where D
ξ
∗
w(t) is the backward mean derivative introduced in (8.8).
Proof. Assertion (i) is a corollary of Theorem 8.7.Toprove(ii),represent
ξ(t) by formula (6.26). Then using the fact that the first two summands on
the right hand side of (6.26) are processes with a.s. smooth trajectories, as
well as the properties of the conditional expectation and formula (8.8), we
obtain
8.2 Calculation of Mean Derivatives for a Wiener Process and for Diffusion Processes 201
D
∗
ξ(t)=D
ξ
∗
t
0
a(s, ξ(s))ds −
t
0
trA
(A(s, ξ(s))
ds
+
t
0
A(s, ξ(s))d
∗
w(s))
= a(t, ξ(t)) − trA
(A(t, ξ(t)))
+ lim
Δt→+0
E
ξ
t
A(t, ξ(t))
w(t) − w(t − Δt)
Δt
= a(t, ξ(t)) − trA
(A(t, ξ(t)))
+ A(t, ξ(t)) lim
Δt→+0
E
ξ
t
w(t) − w(t − Δt)
Δt
= a(t, ξ(t)) − trA
(A(t, ξ(t)) + A(t, ξ(t))D
ξ
∗
w(t).
Theorem 8.27 Let ξ(t) be a solution of the Itˆo equation
ξ(t)=ξ
0
+
t
0
a(s, ξ(s))ds +
t
0
trA
(A(s, ξ(s)))ds
−
t
0
A(s, ξ(s))D
ξ
∗
w(s)ds +
t
0
A(s, ξ(s))dw(s). (8.27)
Then for t ∈ (0,l] we have D
∗
ξ(t)=a(t, ξ(t)).
In fact Theorem 8.27 is a corollary of Lemma 8.26 and the proof is abso-
lutely analogous to the proof of the latter.
Lemma 8.28 For a diffusion type process ξ(t) the relation ED
ξ
∗
w(t)=0
holds.
Proof. By the definition of D
ξ
∗
and the properties of conditional expectation
ED
ξ
∗
w(t)=E lim
Δt→+0
E
ξ
t
w(t) −w(t − Δt)
Δt
= lim
Δt→+0
E
w(t) −w(t − Δt)
Δt
=0. 2
Let ξ(t) be a Markovian diffusion process given on a finite interval t ∈
[0,T].
Definition 8.29. The process w
ξ
∗
(t)=
T
t
D
ξ
∗
w(s)ds + w(t) − w(T ) is called
the backward Wiener process with respect to ξ(t).
Lemma 8.30 The process w
ξ
∗
(t) is a backward martingale with zero mean
relative to F
ξ
t
.
202 8 Mean Derivatives in Linear Spaces
Proof. Since ξ(t) is Markovian, D
ξ
∗
w
x
i
∗
(t)=D
F
ξ
∗
w
ξ
∗
(t). Note that D
ξ
∗
w
ξ
∗
(t)=
−D
ξ
∗
w(t)+D
ξ
∗
w(t) = 0. Hence, D
F
ξ
∗
w
ξ
∗
(t) = 0 and the assertion of the
Theorem follows from Lemma 8.5(ii) and Lemma 8.28.
We should emphasize that w
ξ
∗
(t) depends on the given process ξ(t); from
Lemmas 8.22 and 8.23 it follows that w
w
∗
(t)=
T
t
w(s)
s
ds + w(t) − w(T )and
it is well-defined. Below in formula (8.37) we find D
ξ
∗
w(s) for an Itˆo diffusion
type process ξ(t) with unit diffusion coefficient and so it is possible to find
w
ξ
∗
(t) in this case. The same can also be done for more general processes ξ(t).
As mentioned above (after Theorem 8.7), it is convenient to use w
ξ
∗
(t)for
representing ξ(t) via this backward martingale and hence for calculating the
backward derivatives (see below).
Let ξ(t) be a diffusion process, i.e., a solution of the Itˆo equation
ξ(t)=ξ
0
+
t
0
a(τ,ξ(τ))dτ +
t
0
A(τ,ξ(τ))dw(τ ) (8.28)
for t ∈ [0,T]. Specify a time t. From the above formulae it follows that the
process η(t) satisfying for s<tthe relation
η(t) −η(s)=
t
s
a
∗
(τ,η(τ))dτ +
t
s
A(τ,η(τ))d
∗
w
η
∗
(τ) (8.29)
with a
∗
(t, x)=a(t, x) − trA
(A(t, x)) + A(t, x)D
η
∗
w(t)andsuchthatη(t)=
ξ(t), has the same backward mean derivative at t as ξ(t).
Lemma 8.31
t
s
A(τ,η(τ))d
∗
w
η
∗
(τ) is a backward martingale with respect to
F
η
t
.
Proof. By Lemma 8.30 and Lemma 8.5(ii), D
F
η
∗
w
η
∗
(t) = 0. Approximate the
backward increment of
t
0
A(τ,η(τ))d
∗
w
η
∗
(s) by a summand of the backward
integral sum (6.4), A(t, η(t))(w
η
∗
(t) −w
η
∗
(t −t)). Since A(t, η(t)) is measur-
able with respect to N
ξ
t
and hence with respect to F
η
t
,
D
F
η
∗
t
s
A(τ,η(τ))d
∗
w
η
∗
(τ)
= lim
t→+0
E
A(t, η(t))
(w
η
∗
(t) −w
η
∗
(t −t))
t
F
η
t
= A(t, η(t))D
F
η
∗
w
η
∗
(t)
=0.
Thus the assertion of the Theorem follows from Lemma 8.5(ii).
Lemma 8.31 is ‘symmetric’ to Theorem 6.11(2).
8.3 Calculation of Mean Derivatives for Itˆo Processes 203
Definition 8.32. The equality (8.29) is called an equation in backward dif-
ferentials and is denoted as follows
d
∗
ξ(t)=a(t, ξ(t))d
∗
t + A(t, ξ(t))d
∗
w
ξ
∗
(t). (8.30)
Here d
∗
t means the increment in the negative time direction. We use this
notation only for convenience. From the above arguments it follows that for
s<tclose enough to t, a solution of (8.30) approximates a solution of (8.29).
Lemma 8.33 Let f(t, x) be a function that is C
1
in t ∈ R and C
2
in x ∈ R
n
.
Then for the process ξ(t) satisfying (8.30), the backward Itˆoformula
d
∗
f(ξ(t)) =
∂f
∂t
d
∗
t + f
(a
∗
(t, ξ(t))d
∗
t
+
1
2
trf
(A(t, ξ(t)),A(t, ξ(t)))d
∗
t
+ f
(A(t, ξ(t))d
∗
w
ξ
∗
(t)
is valid.
The proof of Lemma 8.33 is analogous to that of formula (10.11). Here we
have to use the Taylor expansion at the right point and so the summands with
even numbers change sign. Taking into account the properties of integrals of
higher order, we obtain the assertion of the Lemma.
Remark 8.34. We refer the reader to Nelson’s books [188, 190]wherean-
other approach to backward processes and equations is developed.
8.3 Calculation of Mean Derivatives for Itˆo Processes
This section is devoted to the calculation of mean derivatives for processes of
diffusion type of the form (8.15). To do this, we first describe a method for
calculating conditional expectations under a change of probability measure.
On a probability space (Ω,F, P) consider a new probability measure
μ.Letμ be absolutely continuous with respect to P with a certain den-
sity θ,letB be a σ-subalgebra of F and ψ be a measurable map from
(Ω,F)intoR
n
equipped with the Borel σ-algebra. Denote by E
0
(ψ|B)
the conditional expectation of ψ with respect to B on the probability
space (Ω,F, P)andbyE
(ψ|B) the same expectation on the probability
space (Ω,F,μ). Using E
0
(ψ|B) we can calculate E
(ψ|B) as follows (cf.,
e.g., [175]). For any function λ, measurable with respect to B,wehave
E
(λψ)=E
(λE
(ψ|B)) = E
0
(λE
(ψ|B)θ)=E
0
(λE
(ψ|B)E
0
(θ|B)), and
on the other hand E
(λψ)=E
0
(λψθ)=E
0
(λE
0
(ψθ|B)). Thus
204 8 Mean Derivatives in Linear Spaces
E
(ψ|B)=E
0
(θ|B)
−1
E
0
(ψθ|B). (8.31)
Consider a process of diffusion-type (8.15) and for the sake of simplicity
suppose that σ = 1. For the space of continuous curves
˜
Ω = C
0
([0,l], R
n
)
and for the σ-algebra
˜
F of cylinder sets on
˜
Ω, consider two probability spaces
(
˜
Ω,
˜
F,ν)and(
˜
Ω,
˜
F,μ)whereν is the Wiener measure and the measure μ
corresponds to the process ξ(t). Denote the coordinate process on (
˜
Ω,
˜
F)
by ζ(t). Recall that ζ(t), considered as a process on (
˜
Ω,
˜
F,ν), is a Wiener
process. Let us denote this process by W (t). The process ζ(t), considered on
(
˜
Ω,
˜
F,μ), is ξ(t).
It is well-known that if ξ(t) satisfies the condition
P
l
0
a(s)
2
ds<∞
=1, (8.32)
then μ is absolutely continuous with respect to ν. Under some additional
assumptions (see, e.g., [175]) one can show that the density of μ with respect
to ν has the form
θ(l) = exp
−
1
2
l
0
a(s)
2
ds +
l
0
(a(s) ·dW (s))
(8.33)
(the above assumptions mean that θ(l) is a probability density) and so μ and
ν are equivalent. For the remainder of this Section we suppose that (8.32)
and the assumptions are satisfied. Clearly
θ(l)
−1
=exp
1
2
l
0
a
2
ds −
l
0
(a(s) ·dW (s))
.
Determine θ(t) by analogy with formula (8.33)wherel is replaced by t.Then
using the Itˆo formula one can easily show that
θ(l)=1+
l
0
θ(s)(a(s) · dW (s)) (8.34)
(for details see, e.g., [175]).
Denote by E
0
the (conditional) expectation on (
˜
Ω,
˜
F,ν)andbyE
the
(conditional) expectation on (
˜
Ω,
˜
F,μ). Then using formulae (8.31)and(8.34)
we can calculate
8.3 Calculation of Mean Derivatives for Itˆo Processes 205
Dξ(t) = lim
Δt→+0
E
ζ
t
ζ(t + Δt) −ζ(t)
Δt
= lim
Δt→+0
E
0ζ
t
(θ(l))
−1
E
0ζ
t
ζ(t + Δt) −ζ(t)
Δt
θ(l)
= E
0ζ
t
(θ(l))
−1
lim
Δt→+0
E
0ζ
t
ζ(t + Δt) −ζ(t)
Δt
+E
0ζ
t
(θ(l))
−1
lim
Δt→+0
E
0ζ
t
ζ(t + Δt) −ζ(t)
Δt
l
0
θ(t)(a(t) ·dW (t)
.
Let f : M → R be a smooth function with compact support. Then since
here ζ(t)=W (t)andsof (ζ(t))(ζ(t + Δt) − ζ(t))) =
t+Δt
t
f(W (t))dW (s),
we can apply the usual properties of the multiplication of Itˆo integrals (see
Theorem 6.12) to obtain
E
0
(f(ζ(t)) lim
Δt→+0
E
0ζ
t
ζ(t + Δt) −ζ(t)
Δt
= lim
Δt→+0
E
0
f(ζ(t))
ζ(t + Δt) −ζ(t)
Δt
= lim
Δt→+0
E
0
t+Δt
t
f(W (t))θ(s)a(s)ds
Δt
= E
0
(f(ζ(t))θ(t)a(t)).
Thus, since f is an arbitrary function of the above-mentioned type,
lim
Δt→+0
E
0ζ
t
ζ(t + Δt) −ζ(t)
Δt
t
0
θ(s)(a(s) ·dW (s)
= E
0ζ
t
(θ(t)a(t))
On the other hand, by Theorem 8.7 Dξ(t)=E
ξ
t
(a(t)), thus
E
0ζ
t
(θ(l))
−1
E
0ζ
t
(θ(t)a(t)) = E
ξ
t
(a(t)) (8.35)
(note that formula (8.35) can also easily be obtained by direct calculation).
Then
D
∗
ξ(t) = lim
Δt→+0
E
ζ
t
ζ(t) − ζ(t − Δt)
Δt
= E
0
ζ
t
(θ(l))
−1
lim
Δt→+0
E
0
ζ
t
ζ(t) − ζ(t − Δt)
Δt
θ(l)
.
Using the same arguments as above we easily obtain