Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics
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186 7 Stochastic Analysis on Manifolds
From the proofs of Theorems 7.95 and 7.96 it is not hard to see that for
processes with unit diffusion coefficients their developments (as well as line
integrals with parallel translation) are well-defined on stochastically com-
plete Riemannian manifolds, i.e., on a broader class than the uniformly com-
plete Riemannian manifolds and manifolds satisfying the hypothesis of The-
orem 7.76. In addition, for such processes it is possible to weaken the require-
ment of boundedness and yet still obtain the existence of an Itˆo development
on every apriorigiven non-random time interval.
Consider a stochastic process β(t)onT
m
0
M non-anticipative with respect
to B
t
and such that
P
∞
0
β(τ )
2
dτ<∞
=1. (7.54)
Define an Itˆo process z(t)onT
m
0
M by the formula
z(t)=
t
0
β(τ )dτ + w(t). (7.55)
Theorem 7.98 Let M be stochastically complete. Then for any l>0 the
development R
I
z(t) exists on [0,l] and is weakly unique.
Proof. Let (
˜
Ω,
˜
F,ν)andB
t
be as in the proof of Theorem 7.95. Denote
by μ
z
the probability measure on (
˜
Ω,
˜
F) which corresponds to z.Thenit
follows from (7.54)thatμ
z
is absolutely continuous with respect to ν (see
[175, Chapter 7]). Since M is stochastically complete, the development of a
standard Wiener process exists. In other words, the development can be a.s.
extended from the space of smooth curves to
˜
Ω.Sinceμ
z
ν,thesame
holds true when μ
z
is replaced by ν.
One can easily see that z(t) coincides with the coordinate process z
t, x(·)
= x(t)on(
˜
Ω,
˜
F,μ
z
). The theorem follows.
In the Euclidean space R
n
consider a process β(t)thatsatisfies(7.54)and
is non-anticipative with respect to B
t
and a Wiener process w(t) adapted
to B
t
. Construct the Itˆo process ϑ(t)=
t
0
β(τ )dτ + w(t). Let x
0
: Ω → M
be a random element independent of B
t
,andletβ
0
be a Borel measurable
cross-section of OM.
Consider the development R
I
(β
0
(x
0
))ϑ(t) from Definition 7.68.
Theorem 7.99 If a Riemannian manifold M is stochastically complete, the
development R
I
(β
0
(x
0
))ϑ(t) exists, is well-defined for t ∈ [0,l] and is weakly
unique.
The proof of Theorem 7.99 is analogous to that of Theorem 7.98.
Chapter 8
Mean Derivatives in Linear Spaces
8.1 General Definitions and Results
In this section we briefly describe some preliminary facts about mean deriva-
tives. For details, see [7, 106, 107, 115, 188, 190]. This notion was first in-
troduced by E. Nelson [187, 188, 190] for the needs of so-called stochastic
mechanics (see Chapter 15) but it turns out to be useful in some other prob-
lems of mathematical physics, economics, and elsewhere.
Consider a stochastic process ξ(t), t ∈ [0,T], given on a probability space
(Ω,F, P), taking values in a separable Hilbert space and such that ξ(t)isan
L
1
random element for all t. For the sake of convenience in this section we
work mainly with the Euclidean space R
n
. The general case of Hilbert space
is quite analogous and we shall describe its features only when necessary.
In Section 6.1.1 for a stochastic process ξ(t) three families of σ-subalgebras
of the σ-algebra F were introduced: “the past” P
ξ
t
of ξ(t), “the future” F
ξ
t
of
ξ(t) and “the present” (“now”) N
ξ
t
of ξ(t). All the above families we suppose
to be complete, i.e., contain all sets of measure zero.
For the sake of convenience we denote by E
ξ
t
the conditional expectation
E(·|N
ξ
t
) with respect to the “present” N
ξ
t
for ξ(t).
Generally speaking, the sample trajectories of ξ(·) a.s. are not differentiable
(see, e.g., Theorem 6.10 for Wiener processes) and so we cannot determine
the derivative of ξ(·) in the ordinary way. Following Nelson (see e.g. [187,
188, 190]) we give the following:
Definition 8.1.
(i) The forward mean derivative Dξ(t) of the process ξ(t)attimet is the
L
1
-random variable of the form
Dξ(t) = lim
Δt→+0
E
ξ
t
ξ(t + Δt) − ξ(t)
Δt
(8.1)
Y.E. Gliklikh, Global and Stochastic Analysis with Applications
to Mathematical Physics, Theoretical and Mathematical Physics,
DOI 10.1007/978-0-85729-163-9
8, © Springer-Verlag London Limited 2011
187
188 8 Mean Derivatives in Linear Spaces
where the limit is assumed to exist in L
1
(Ω,F, P)andΔt → +0 means
that Δt → 0andΔt > 0.
(ii) The backward mean derivative D
∗
ξ(t)ofξ(t)att is the L
1
-random
variable
D
∗
ξ(t) = lim
Δt→+0
E
ξ
t
ξ(t) − ξ(t −Δt)
Δt
(8.2)
where (as in (i)) the limit is assumed to exist in L
1
(Ω,F, P)andΔt →
+0 means that Δt → 0andΔt > 0.
If ξ(t) is a Markov process (see Section 6.1.3)thenE
ξ
t
can be replaced
by E(·|P
ξ
t
)in(8.1)andbyE(·|F
ξ
t
)in(8.2). In order to distinguish these
constructions for non-Markovian processes, we introduce the following defi-
nition:
Definition 8.2. The forward mean derivative relative to the past (P-mean
derivative) D
P
ξ(t) of a process ξ(t)attimet is the L
1
-random element of
the form
D
P
ξ(t) = lim
t→+0
E
ξ(t + t) − ξ(t)
t
P
ξ
t
, (8.3)
where the limit is assumed to exist in L
1
and t → +0 means that t tends
to 0 and t>0.
The backward mean derivative relative to the future (F-mean derivative)
D
F
∗
ξ(t) of a process ξ(t)attimet is the L
1
-random element of the form
D
F
∗
ξ(t) = lim
t→+0
E
ξ(t) − ξ(t −t)
t
F
ξ
t
, (8.4)
where the limit is assumed to exist in L
1
and t → +0 means that t tends
to 0 and t>0.
Remark 8.3. In fact Nelson considered at most the case of Markov processes,
and so he gave in different works two equivalent definitions of mean deriva-
tives: with E
ξ
t
and with E(·|P
ξ
t
)orE(·|F
ξ
t
), respectively. We mainly consider
Itˆo diffusion type processes which are, generally speaking, non-Markovian,
and so these definitions become non-equivalent. Definition 8.1 is compati-
ble with the principle of locality in physics: the derivative depends on the
present but neither on the entire past nor on the entire future. Nevertheless
the P-mean and F-mean derivatives as in Definition 8.2 also arise in many
problems.
It should be noted that in general Dξ(t) = D
∗
ξ(t) (but if ξ(t) a.s. has
smooth sample trajectories, these derivatives coincide).
From the properties of conditional expectation it follows that Dξ(t)and
D
∗
ξ(t) are expressed as compositions of ξ(t) and the Borel measurable vector
fields, namely the regressions (see Section 6.1.2)
8.1 General Definitions and Results 189
Y
0
(t, x) = lim
Δt→+0
E
ξ(t + Δt) − ξ(t)
Δt
ξ(t)=x
(8.5)
Y
0
∗
(t, x) = lim
Δt→+0
E
ξ(t) − ξ(t − Δt)
Δt
ξ(t)=x
(8.6)
on R
n
, i.e., Dξ(t)=Y
0
(t, ξ(t)) and D
∗
ξ(t)=Y
0
∗
(t, ξ(t)).
The mean derivatives of Definition 8.1 are particular cases of the follow-
ing notions. Let x(t)andy(t)beL
1
-stochastic processes given on (Ω,F, P).
Introduce the y-forward derivative of x(t) by the formula
D
y
x(t) = lim
Δt→+0
E
y
t
x(t + Δt) − x(t)
Δt
(8.7)
and the y-backward derivative of x(t) by the formula
D
y
∗
x(t) = lim
Δt→+0
E
y
t
x(t) −x(t − Δt)
Δt
(8.8)
where, of course, the limits are assumed to exist in L
1
(Ω,F, P). If by anal-
ogy with (8.3)and(8.4) we replace E
y
t
by E(·|P
y
t
)in(8.7)andby
E(·|F
y
t
)in(8.8), we obtain the y-forward P-derivative D
P
y
x(t)ofx(t)
and the y-backward F-derivative D
F
y
∗
x(t)ofx(t), respectively. As above, if
y(t) is Markovian, D
P
y
x(t) coincides with D
y
x(t)andD
F
y
∗
x(t) coincides
with D
y
∗
x(t).
Lemma 8.4 For s<t
E (x(t) − x(s) |P
y
s
)=E
t
s
D
P
y
x(τ)
dτ
P
y
s
, (8.9)
E (x(t) − x(s) |F
y
t
)=E
t
s
D
F
y
∗
x(τ)
dτ
F
y
t
. (8.10)
Proof. Take a partition q =(s = t
0
<t
1
< ···<t
N
= t) of the interval [s, t]
and consider the following integral sum
N−1
i=0
E
x(t
i+1
) −x(t
i
)
t
i+1
− t
i
P
y
t
i
(t
i+1
− t
i
)
=
N−1
i=0
E
(x(t
i+1
) −x(t
i
))|P
y
t
i
whose limit as diam q → 0is
t
s
(D
P
y
x(τ))dτ .Sincet
i
≥ s, by the properties
of conditional expectation E(E(·|P
y
t
i
) |P
y
s
)=E(·|P
y
s
). Thus
190 8 Mean Derivatives in Linear Spaces
E
N−1
i=0
E (x (t
i+1
) −x (t
i
)|P
y
t
i
P
y
s
= E
N−1
i=0
(x (t
i+1
) −x (t
i
))|P
y
s
= E ( x(t) −x(s)|P
y
s
) .
This proves (8.9). The proof of (8.10) is similar, replacing the “past” by the
“future”.
Lemma 8.5
(i) x(t) is a martingale with respect to P
y
t
if and only if D
P
y
x(t)=0.
(ii) x(t) is a backward martingale with respect to F
y
t
if and only if
D
F
y
∗
ξ(t)=0.
Proof. Let x(t) be a martingale with respect to P
y
t
. By the martingale prop-
erty (see Section 6.1.4) E(x(t + Δt)|P
y
t
)=x(t)andsoE(x(t + Δt) − x(t) |
P
y
t
)=0.HenceD
P
y
x(t)=0.
Let D
P
y
x(t) = 0. Then by Lemma 8.4 for t>swe have E(x(t) − x(s) |
P
y
s
) = 0. Thus E(x(t) |P
y
s
)=E(x(s) |P
y
s
). But E(x(s) |P
y
t
)=x(s)andso
x(t) is a martingale with respect to P
y
t
.
This proves (i). The proof of (ii) is similar.
Corollary 8.6
(i) If x(t) is a martingale with respect to P
y
t
, D
y
x(t)=0.
(ii) If x(t) is a backward martingale with respect to F
y
t
, D
∗
ξ(t)=0.
In particular, if a process ξ(t) is a martingale, Dξ(t)=0and if it is a
backward martingale, D
∗
ξ(t)=0.
This follows from the fact that Dξ(t)=E
ξ
t
(D
P
ξ(t)) and D
∗
ξ(t)=
E
ξ
t
(D
F
∗
ξ(t)).
Of course, when we use the word “martingale” without indicating the
filtration, we mean that it is with respect to it own “past” (and in the case
of a backward martingale, with respect to its own “future”).
Consider an Itˆo diffusion type process ξ(t) (see Definition 6.16)
ξ(t)=ξ
0
+
t
0
a(s)ds +
t
0
A(s)dw(s) (8.11)
and a diffusion process (see Definition 6.17)
η(t)=η
0
+
t
0
a(s, η(s))ds +
t
0
A(s, η(s))dw(s). (8.12)
It should be noted that ξ(t) can neither be a diffusion nor a Markov pro-
cess.
8.1 General Definitions and Results 191
Theorem 8.7 For a process ξ(t) of type (8.11) Dξ(t) exists and is equal to
E
ξ
t
(a(t)). For a diffusion process η(t) of type (8.12) Dη(t)=a(t, η(t)).
Proof. Evidently
D
ξ
0
+
t
0
a(s)ds +
t
0
A(s)dw(s)
= D
ξ
t
0
a(s)ds
+ D
ξ
t
0
A(s)dw(s)
.
Since
t
0
A(s)dw(s) is a martingale with respect to P
ξ
t
(see Theorem
6.11), from Corollary 8.6 it follows that D
ξ
(
t
0
A(s)dw(s)) = 0. Then
D
ξ
(
t
0
a(s)ds)=E
ξ
t
(a(t)) and Dη(t)=a(t, η(t)) since a(t, η(t)) is measur-
able with respect to N
η
t
.
Theorem 8.8 For diffusion type process (8.11) the derivative D
P
ξ(t) exists
and takes the form D
P
ξ(t)=a(t).
Proof. Note that ξ(t+t)−ξ(t)=
t+t
t
a(s)ds+
t+t
t
A(s)dw(s). Since the
Itˆo integral is a martingale with respect to P
ξ
t
, E(
t+t
t
A(s)dw(s) |P
ξ
t
)=0
and so we get
E(ξ(t + t) − ξ(t) |P
ξ
t
)=E
⎛
⎝
t+t
t
a(t)dt
P
ξ
t
⎞
⎠
=
t+t
t
E
a(t)
P
ξ
t
dt.
Applying formula (8.3) we obtain that D
P
ξ(t)=E(a(t) |P
ξ
t
). Since, by
definition of a diffusion type process, a(t) is measurable with respect to P
ξ
t
,
E(a(t) |P
ξ
t
)=a(t).
Theorem 8.9 For a backward Itˆoprocess
ξ(t)=ξ
0
+
t
0
a(s)ds +
t
0
A(s)d
∗
w(s)
given on an interval [0,T] and such that A(t) is measurable with respect to
N
ξ
t
for all t, the derivative D
∗
ξ(t) at t ∈ (0,T] exists and equals E
ξ
t
(a(t)) +
A(t)D
ξ
∗
w(t).
Proof. Indeed, as in Theorem 8.7, D
ξ
∗
(
t
0
a(s)ds)=E
ξ
t
(a(t)). Approximate
the backward increment of
t
0
A(s)d
∗
w(s) by a summand of the backward
integral sum (6.4)oftheformA(t)(w(t) − w(t −t)). Since A(t)ismeasur-
able with respect to N
ξ
t
, lim
t→+0
E
ξ
t
(A(t)
(w(t)−w(t−t))
t
)=A(t)D
ξ
∗
w(t). The
Theorem follows.
192 8 Mean Derivatives in Linear Spaces
By Theorem 8.7 the forward mean derivative gives information about the
drift of an Itˆo process. Following [6, 7] we introduce a new mean derivative
D
2
, called the quadratic mean derivative, that is responsible for the diffusion
term of a process. This is a slight modification of an idea of Nelson from
[188, 190].
Definition 8.10. For an L
1
-stochastic process ξ(t), t ∈ [0,T], its quadratic
mean derivative D
2
ξ(t) is defined by the formula
D
2
ξ(t) = lim
t→+0
E
ξ
t
(ξ(t + t) − ξ(t)) ⊗ (ξ(t + t) −ξ(t))
t
, (8.13)
where ⊗ denotes the tensor product and the limit is assumed to exist in
L
1
(Ω,F, P).
Note that here the tensor product of two vectors in R
n
is the n ×n matrix
formed by the products of every component of the first vector with every
component of the second one. Note also that for column vectors X, Y ∈ R
n
their tensor product X ⊗ Y equals the matrix product XY
∗
of the column
vector X and the row vector Y
∗
(the transpose of column Y ).
As in the case of forward mean derivatives, if ξ(t) is not Markovian, the
quadratic mean derivative with respect to the past differs from that in Defi-
nition 8.10. To distinguish these constructions we introduce:
Definition 8.11. The quadratic mean derivative relative to the past (for
short, quadratic P-mean derivative) D
P
2
ξ(t)ofξ(t)att is the L
1
-random
element of the form
D
P
2
ξ(t) = lim
t→+0
E
(ξ(t + t) − ξ(t)) ⊗ (ξ(t + t) −ξ(t))
t
P
ξ
t
, (8.14)
where the limit is assumed to exist in L
1
, t → +0 means that t tends to
0andt>0and⊗ denotes the tensor product in R
n
.
Denote by S
+
(n) the set of symmetric positive definite n ×n matrices and
by
¯
S
+
(n) the set of symmetric positive semi-definite matrices (the closure of
S
+
(n) in the space of all symmetric matrices S(n)).
We emphasize that the tensor product in (8.13) is a symmetric positive
semi-definite matrix so that D
2
ξ(t) is a function with values in
¯
S
+
(n).
From the properties of conditional expectation (see, e.g., [194]) it follows
that there exists a Borel mapping α(t, x)from[0,T] ×R
n
to
¯
S
+
(n) such that
D
2
ξ(t)=α(t, ξ(t)). As above, following [194], we call α(t, x)theregression.
Theorem 8.12 Let ξ(t) be a diffusion type process of the form (8.11).Then
D
2
ξ(t)=E
ξ
t
[α(t)] where α(t)=A(t)A
∗
(t), A
∗
(t) is the transposed ma-
trix A(t) and A(t)A
∗
(t) is the matrix product. If ξ(t) is a diffusion pro-
cess, D
2
ξ(t)=α(t, ξ(t)) where α is the diffusion coefficient. In particular,
D
2
ξ(t)=0if and only if ξ(t) a.s. has C
1
-smooth sample paths.
8.1 General Definitions and Results 193
Proof. By direct calculation it follows that the components of (ξ(t + t) −
ξ(t)) ⊗(ξ(t + t) − ξ(t)) are elements of the matrix (ξ(t + t) − ξ(t))(ξ(t +
t) − ξ(t))
∗
where we use the matrix multiplication of the column-vector
(ξ(t + t) − ξ(t)) and the row-vector (ξ(t + t) − ξ(t))
∗
(i.e., the transpose
of (ξ(t + t) − ξ(t))). The product is a symmetric positive semi-definite
matrix. Since ξ(t + t) −ξ(t)=
t+t
t
a(s)ds +
t+t
t
A(s)dw(s), taking into
account the properties of the Lebesgue and Itˆo integrals one can see that
(ξ(t + t) − ξ(t))(ξ(t + t) − ξ(t))
∗
is approximated by a(t)a(t)
∗
(Δt)
2
+
(a(t)Δt)(A(t)Δw(t))
∗
+(A(t)Δw(t))(a(t)Δt)
∗
+ A(t)A(t)
∗
Δt.Thuswesee
that only A(t)A(t)
∗
Δt is infinitesimal of the same order as Δt while the
other summands are infinitesimals of order higher than Δt. Applying formula
(8.13), we obtain the assertion of Theorem since AA
∗
= α (see above). If ξ(t)
a.s. has C
1
-sample paths, A = 0 a.s. and so D
2
ξ(t) = 0. On the other hand,
if D
2
ξ(t) = 0, this means that in the expression for (ξ(t + t) −ξ(t)) ⊗(ξ(t +
t) − ξ(t)) there is no term with the same infinitesimal order as t. Hence,
a.s. A =0.
Theorem 8.13 For the above-mentioned Itˆo diffusion type process ξ(t) the
derivative D
P
2
ξ(t) exists and takes the form D
P
2
ξ(t)=A(t)A
∗
(t) where A
∗
(t)
is the transpose of the matrix A(t).
The proof of Theorem 8.13 is a simple modification of that for Theo-
rem 8.12 based on the fact that A(t) is measurable with respect to P
ξ
t
.
Below we will often deal with the particular case of the process (8.11)for
F = R
n
with A = σI,whereσ>0 is a real constant and I is the identity
operator; i.e., with diffusion type processes in R
n
of the form
ξ(t)=ξ
0
+
t
0
a(s)ds + σw(t). (8.15)
Note that Theorem 8.7 is valid for processes of the form (8.15).
For a process of type (8.15) we can obtain from the Itˆo formulae (6.10)
and (6.13) and from formulae (8.9)and(8.10) two important relations.
Lemma 8.14 Let f(t, x) be a smooth mapping. For a process of type (8.15)
for every t>sthe relations
E
ξ
s
(f(t, ξ(t)) − f(s, ξ(s))) (8.16)
= E
ξ
s
t
s
∂f
∂τ
dτ +
t
s
f
(Y
0
(τ,ξ(τ)))dτ +
σ
2
2
t
s
∇
2
f(τ, ξ(τ))dτ
and
E
ξ
t
(f(t, ξ(t)) − f(s, ξ(s))) (8.17)
= E
ξ
t
t
s
∂f
∂τ
dτ +
t
s
f
(Y
0
∗
(τ,ξ(τ)))dτ −
σ
2
2
t
s
∇
2
f(τ, ξ(τ))dτ
hold.
194 8 Mean Derivatives in Linear Spaces
Proof. Note that by Theorem 8.7 and by formula (8.5) defining the regression
Y
0
, the equality E
ξ
t
(a(t)) = Y
0
(t, ξ(t)) holds. Also, trf
(·, ·)=∇
2
f for a
smooth mapping f(t, x)where∇
2
is the Laplacian. Then for t>sand for a
smooth mapping f(t, x) one can easily derive from (6.10)and(8.9)that
E
ξ
s
(f(t, ξ(t)) − f(s, ξ(s)))
= E
ξ
s
t
s
D
ξ
f(τ, ξ(τ))dτ
= E
ξ
s
t
s
∂f
∂τ
dτ +
t
s
f
(Y
0
(τ,ξ(τ)))dτ +
σ
2
2
t
s
∇
2
f(τ, ξ(τ))dτ
since E
ξ
s
(
t
s
f
(dw(τ)) = 0.
For the process (8.15) the regression Y
0
∗
introduced by formula (8.6)
takes the form Y
0
∗
= Y
0
+ Y
1
where Y
1
is the regression for D
ξ
∗
w(t), i.e.,
Y
1
(t, ξ(t)) = D
ξ
∗
w(t). Obviously, the backward Itˆo formula (6.13) is applica-
ble to ξ(t)aswellas(6.10). Then for t>sand for a smooth mapping f(t, x)
one can easily derive from (6.13), (8.10) and Theorem 8.9 that
E
ξ
t
(f(t, ξ(t)) − f(s, ξ(s)))
= E
ξ
t
t
s
D
ξ
f(τ, ξ(τ))dτ
= E
ξ
t
t
s
∂f
∂τ
dτ +
t
s
f
(Y
0
(τ,ξ(τ)))dτ −
σ
2
2
t
s
∇
2
f(τ, ξ(τ))dτ
+
t
s
f
(D
ξ
∗
w(τ))dτ
= E
ξ
t
t
s
∂f
∂τ
dτ +
t
s
f
(Y
0
(τ,ξ(τ)))dτ −
σ
2
2
t
s
∇
2
f(τ, ξ(τ)) dτ
+
t
s
f
(Y
1
(τ,ξ(τ)))dτ
= E
ξ
t
t
s
∂f
∂τ
dτ +
t
s
f
(Y
0
∗
(τ,ξ(τ)))dτ −
σ
2
2
t
s
∇
2
f(τ, ξ(τ))dτ
.
Definition 8.15. The derivative D
S
=
1
2
(D + D
∗
) is called the symmetric
mean derivative. The derivative D
A
=
1
2
(D −D
∗
) is called the antisymmetric
mean derivative.
Consider the vectors v
ξ
(t, x)=
1
2
(Y
0
(t, x)+Y
0
∗
(t, x)) and u
ξ
(t, x)=
1
2
(Y
0
(t, x) − Y
0
∗
(t, x)).
8.1 General Definitions and Results 195
Definition 8.16. v
ξ
(t)=v
ξ
(t, ξ(t)) = D
S
ξ(t) is called the current velocity
of the process ξ(t)andu
ξ
(t)=u
ξ
(t, ξ(t)) = D
A
ξ(t) is called the osmotic
velocity of the process ξ(t).
In Nelson’s works it is shown that in many natural problems the current
velocity plays the same role as the ordinary velocity of a deterministic process.
The osmotic velocity measures the variation of randomness of a process. The
precise meaning of osmotic and current velocities is clarified by the following.
Denote by ρ
ξ
(t, x) the density of the process (8.15) with respect to
Lebesgue measure λ on [0,l] ×R
n
. This means that for any continuous inte-
grable function f(t, x)on[0,l] × R
n
the following equality holds
[0,l]×R
n
f(t, x)ρ
ξ
(t, x)dλ =
Ω×[0,l]
f(t, ξ(t))dPdt.
Lemma 8.17 For the process (8.15) in R
n
the vector field u
ξ
(t, x) is pre-
sented in the form
u
ξ
(t, x)=
1
2
σ
2
grad log ρ
ξ
(t, x)=σ
2
grad log
ρ
ξ
(t, x). (8.18)
Proof. We shall prove (8.18) using an idea developed in [65] for more compli-
cated processes. For an alternative proof see, e.g., [187, 188, 190] where only
Markovian diffusion processes are considered.
Let f be a smooth function on R
n
with compact support. Since f(ξ(t)) is
N
ξ
t
-measurable,
E[f (ξ(t))E
ξ
t
(w(t) − w(t − Δt))] = E[f(ξ(t))(w(t) − w(t − Δt))]
(see the properties of conditional expectations in Section 6.1.2). Since f(ξ(t−
Δt)) and w(t) −w(t −Δt) are independent and E(w(t) −w(t −Δt)) = 0 (see
Theorem 6.8), we obtain the equality E[f(ξ(t −Δt))(w(t) −w(t −Δt))] = 0.
Thus E[f(ξ(t))E
ξ
t
(w(t) − w(t − Δt))] = E[(f(ξ(t)) − f(ξ(t − Δt)))(w(t) −
w(t−Δt))]. Using the Itˆo formula (6.10) and taking into account that f
(a)=
gradf ·a, we obtain the presentation
f(ξ(t)) −f (ξ(t − Δt))
=
t
t−Δt
(gradf(ξ(s)) · a(s))ds +
σ
2
2
t
t−Δt
trf
(ξ(s))ds
+
t
t−Δt
σgradf(ξ(s)) ·dw(s).
Hence,