Beyond a Fixed Horizon
The stochastic integral was built with a horizon fixed in advance, and
until now nothing has minded. Every use it has been put to came with a
horizon of its own already in place. The formula for a transformed
process runs on \([0, T]\); the solution theory solves equations up to
a prescribed final time; the representation theorem represents a
variable measurable at a prescribed final time. A stopping time is the
first object on this track that refuses to name a horizon in advance.
It is read off the path as the path unfolds, and the natural hypotheses
about it bound its mean rather than its values.
The
stopped Itô integral
handled this by refusing to leave the horizon: it reads the integral at
\(\tau \wedge T\), the moment the clock stops or the horizon expires,
whichever comes first. That is enough when the horizon can be chosen
large enough to be irrelevant, and not enough when the stopping time is
merely integrable, since no finite \(T\) then dominates it. This page
removes the horizon, defines the integral at an almost surely finite
stopping time, and shows that the two properties which make the
integral worth having survive the removal. The reward is collected in
the final section, where Itô's formula is averaged at a stopping
time and yields an identity that the rest of the track will use
repeatedly.
The construction of the Itô integral fixed a horizon \(T\) and
admitted integrands in
the class \(\mathcal{V}(0, T)\),
and the
continuous version
was likewise produced on \([0, T]\). The
properties page did consider
relaxing the class, and declined. Dropping the square-integrability
condition (V3) in favour of an almost sure finiteness costs the
isometry, the zero mean, and the martingale property, and buys only a
local martingale, so that extension was recorded and left unbuilt. The
move made here is not that one. Condition (V3) is kept in full, and
only the horizon is released, so nothing that made the integral useful
is given up.
Releasing it asks for a hypothesis, not for a new theorem. Write
\[
\mathcal{V}( 0, \infty ) = \bigcap_{T \gt 0} \mathcal{V}( 0, T )
\]
for the integrands admissible at every horizon. An integrand in this
class carries a compatible family of integrals, one for each horizon,
and a compatible family of continuous processes on growing intervals is
a single continuous process on the half-line. That is the whole
argument, and the two theorems below are its two halves.
The class deserves one remark, because the notation invites a
misreading. Membership in \(\mathcal{V}( 0, \infty )\) is not square
integrability over the half-line. A bounded adapted integrand belongs
to the class, since \(\mathbb{E}[ \int_0^T f^2\, ds ]\) is finite for
every finite \(T\), while \(\mathbb{E}[ \int_0^\infty f^2\, ds ]\) is
infinite for such an \(f\) unless it decays. The intersection asks for
finiteness at each horizon separately and never for finiteness in the
limit, and that is exactly the strength the integrands of this page
will have.
Theorem: A Continuous Version on the Half-Line
Let \(f \in \mathcal{V}( 0, \infty )\). There exists a stochastic
process \(\{ J_t \}_{t \geq 0}\), adapted to
\(\{\mathcal{F}_t\}\), with every path continuous,
such that
\[
\mathbb{P} \Bigl[ J_t = \int_0^t f\, d w_s \Bigr] = 1
\quad \text{for every } t \geq 0 ,
\]
and any process with almost surely continuous paths satisfying
the same identities is indistinguishable from it.
Proof.
The finite pieces are compatible.
For each positive integer \(k\) membership in the intersection
gives \(f \in \mathcal{V}( 0, k )\), so a continuous version
\(J^{(k)}\) exists on \([0, k]\). For \(t \leq k\) the symbol
\(\int_0^t f\, d w_s\) denotes the integral of \(f\) read as an
element of \(\mathcal{V}( 0, t )\). This reading does not depend on
the horizon it is taken from, the construction using only the
restriction of \(f\) to \([0, t]\), and the
properties theorem
keeps the readings consistent across horizons: it records that an
integrand admissible up to a
horizon is admissible up to any earlier one, and it states
additivity over adjacent intervals as an identity among integrals
read at different horizons. The identification is therefore the one
already in force. Hence for \(k \lt l\) and each fixed
\(t \in [0, k]\) the variables \(J^{(k)}_t\) and \(J^{(l)}_t\) are
versions of one and the same random variable, and agree almost
surely.
Compatibility upgrades to a single null set.
Fix \(k \lt l\). The two processes agree almost surely at each
rational \(t \in [0, k]\), hence, the rationals being countable,
simultaneously at all of them outside one null set
\(N_{k, l}\); off \(N_{k, l}\) and off the null sets where either
path fails to be continuous on \([0, k]\), continuity extends the
agreement to every \(t\). Let \(N\) be the union of the
countably many sets \(N_{k, l}\) together with the null sets on
which some \(J^{(k)}\) fails to have a continuous path.
Gluing.
For \(\omega \notin N\) and \(t \geq 0\) set
\(J_t( \omega ) = J^{(k)}_t( \omega )\) for any integer
\(k \geq t\); the previous step makes the value independent of the
choice, and on \(N\) set \(J \equiv 0\). For fixed \(t\) the
variable \(J_t\) agrees almost surely with
\(\int_0^t f\, d w_s\) because \(J^{(k)}_t\) does, so \(J_t\) is a
representative of that class and is therefore
\(\mathcal{F}_t\)-measurable by
the measurability property
read at horizon \(t\), whose every-representative form, resting on
the null-set convention, was built for exactly this transfer. The
process is thus adapted. Every path of \(J\) is
continuous on each \([0, k]\) and hence on \([0, \infty)\). If \(\tilde{J}\)
is another such process, the rational argument applied once more
makes the two agree at all times outside a null set, that set
including the one where \(\tilde{J}\) fails to be
continuous, which is indistinguishability.
The Integral at a Stopping Time
With the continuous version available at every time, a stopping time
that is almost surely finite can simply be substituted into it. That
substitution is what the symbol below will mean, and the theorem
records that the two properties of the integral that matter survive it.
Theorem: The Integral at a Stopping Time
Let \(f \in \mathcal{V}( 0, \infty )\), let \(J\) be its continuous
version on the half-line, and let \(\tau\) be a stopping time with
\(\tau \lt \infty\) almost surely. Define
\[
\int_0^\tau f\, d w_s := J_\tau .
\]
The substitution is legitimate: \(J\) is continuous and adapted,
hence
progressively measurable,
so
item (v) of the calculus
makes \(J_\tau\) measurable on \(\{\tau \lt \infty\}\).
If moreover
\(\mathbb{E} [ \int_0^\tau f( s, \omega )^2\, ds ] \lt \infty\),
then \(J_\tau \in L^2( \mathbb{P} )\) and
\[
\begin{align*}
\mathbb{E} \bigl[ J_\tau \bigr] &= 0 , \\\\
\mathbb{E} \bigl[ J_\tau^2 \bigr]
&= \mathbb{E} \Bigl[ \int_0^\tau f( s, \omega )^2\, ds \Bigr] .
\end{align*}
\]
Proof.
The truncations form a Cauchy sequence.
Let \(k \lt l\) be positive integers. Both \(\tau\) and
\(\tau \wedge k\) are stopping times, and at the horizon \(l\) the
second satisfies \(( \tau \wedge k ) \wedge l = \tau \wedge k\), so
the
stopped Itô integral
identifies both stopped values as ordinary integrals over
\([0, l]\) with switched-off integrands, the half-line
version agreeing with the version on \([0, l]\) by
construction. Subtracting the two identifications, linearity of
the integral gives
\[
J_{\tau \wedge l} - J_{\tau \wedge k}
= \int_0^l \mathbf{1}_{[ \tau \wedge k,\, \tau )}( s )\,
f( s, \omega )\, d w_s ,
\]
and the
Itô isometry
turns the second moment of the difference into
\[
\mathbb{E} \bigl[ ( J_{\tau \wedge l} - J_{\tau \wedge k} )^2 \bigr]
= \mathbb{E} \Bigl[ \int_{\tau \wedge k}^{\tau \wedge l}
f^2\, ds \Bigr]
\leq \mathbb{E} \Bigl[ \int_{\tau \wedge k}^{\tau} f^2\, ds \Bigr] .
\]
The right side tends to zero as \(k \to \infty\). The inner
integral vanishes for all large \(k\) at almost every
\(\omega\), since \(\tau \wedge k = \tau\) once \(k\)
exceeds \(\tau( \omega )\), and it is dominated by
\(\int_0^\tau f^2\, ds\), which
is integrable by hypothesis, so
dominated convergence
applies. The sequence \(( J_{\tau \wedge k} )_k\) is therefore
Cauchy in \(L^2( \mathbb{P} )\).
The limit is the value at \(\tau\).
Since \(\tau\) is almost surely finite, \(\tau \wedge k = \tau\)
for all large \(k\) at almost every \(\omega\), so
\(J_{\tau \wedge k} \to J_\tau\) almost surely. By the
completeness of \(L^2\)
the Cauchy sequence converges in \(L^2( \mathbb{P} )\) to some
limit \(X\), and
Fatou's lemma
applied to the nonnegative variables
\(( J_{\tau \wedge k} - X )^2\), whose almost sure limit is
\(( J_\tau - X )^2\), gives
\[
\begin{align*}
\mathbb{E} \bigl[ ( J_\tau - X )^2 \bigr]
&\leq \liminf_{k \to \infty}
\mathbb{E} \bigl[ ( J_{\tau \wedge k} - X )^2 \bigr] \\\\
&= 0 .
\end{align*}
\]
Hence \(J_\tau = X\) almost surely, so
\(J_\tau \in L^2( \mathbb{P} )\) and
\(J_{\tau \wedge k} \to J_\tau\) in \(L^2\).
Passing the two identities to the limit.
For each \(k\) the stopped integral has mean zero and second moment
\(\mathbb{E}[ \int_0^{\tau \wedge k} f^2\, ds ]\). Convergence in
\(L^2\) carries both moments along: the first since
\(| \mathbb{E}[ Y ] | \leq \| Y \|_{L^2}\) on a probability
space by
Hölder's inequality
against the constant function \(1\), the second since \(L^2\)
norms converge along the sequence. Hence
\(\mathbb{E}[ J_\tau ] = 0\) and
\(\mathbb{E}[ J_\tau^2 ] = \lim_k \mathbb{E}[ \int_0^{\tau \wedge k}
f^2\, ds ]\). The integrands increase to
\(\int_0^\tau f^2\, ds\) pointwise, so
monotone convergence
identifies the limit as
\(\mathbb{E}[ \int_0^\tau f^2\, ds ]\).
Averaging Itô's Formula
Here is the question the identity answers. Take an Itô process, a
smooth compactly supported function of its state, and a stopping time
with finite mean, and ask for the average value of the function at the
moment the clock stops. The answer is that this average
exceeds the value at the start by the expected accumulation, along the
path and up to the stopping time, of a second-order expression built
from the coefficients of the process and the derivatives of the
function.
Nothing in that is surprising once Itô's formula is in hand.
Applying the formula produces exactly such an accumulation, together
with a stochastic integral against the driving motion, and the whole
content of the identity is that the stochastic integral averages to
zero even when its upper limit is random. The two theorems just proved
are what make that last clause both meaningful and true.
Throughout this section \(Y = Y^x\) is an
\(m\)-dimensional Itô process
started at \(x\), written in integral form as
\[
Y_t
= x
+ \int_0^t \mathbf{u}( s, \omega )\, ds
+ \int_0^t \mathbf{v}( s, \omega )\, d w_s ,
\quad t \geq 0 ,
\]
with \(\mathbf{u}\) taking values in \(\mathbb{R}^m\), \(\mathbf{v}\)
in \(\mathbb{R}^{m \times n}\), and \(w\) the \(n\)-dimensional driving
motion carrying the filtration
\(\{ \mathcal{F}_t^{(n)} \}\). The class \(C_0^2( \mathbb{R}^m )\)
consists of the twice continuously differentiable functions on
\(\mathbb{R}^m\) with compact support.
One piece of notation is carried over from the
study of diffusions as a family,
where averages were indexed by the point the process starts from. For a
process whose starting point is displayed we write
\[
E^x \bigl[ \Phi( Y ) \bigr] := \mathbb{E} \bigl[ \Phi( Y^x ) \bigr]
\]
for any functional \(\Phi\) of the path for which the right side is
defined, and more generally for any random variable on the
underlying space. The superscript is a label on the process,
recording which
member of the family is being averaged, and not a change of the measure
on the underlying space; the expectation itself is the ordinary one
throughout.
Theorem: The Stopped Expectation Identity
Let \(Y\) be as above, let \(f \in C_0^2( \mathbb{R}^m )\), and let
\(\tau\) be a
stopping time
for \(\{ \mathcal{F}_t^{(n)} \}\) with
\(E^x[ \tau ] \lt \infty\). Assume that \(\mathbf{u}\) and
\(\mathbf{v}\) are bounded on the set of pairs \(( s, \omega )\)
for which \(Y_s( \omega )\) lies in the support of \(f\). Then
\[
E^x \bigl[ f( Y_\tau ) \bigr]
= f( x )
+ E^x \Bigl[ \int_0^\tau
\Bigl(
\sum_{i = 1}^{m} u_i\, \partial_i f
+ \tfrac{1}{2} \sum_{i, l = 1}^{m}
\bigl( \mathbf{v} \mathbf{v}^\top \bigr)_{il}\, \partial_{il} f
\Bigr)\, ds \Bigr] ,
\]
where the derivatives of \(f\) are evaluated at \(Y_s\) and the
coefficients at \(( s, \omega )\), and both expectations are
finite.
Write \(K\) for the support of \(f\) and abbreviate the two integrands
produced by Itô's formula,
\[
\begin{align*}
G( s, \omega )
&= \sum_{i = 1}^{m} u_i\, \partial_i f
+ \tfrac{1}{2} \sum_{i, l = 1}^{m}
\bigl( \mathbf{v} \mathbf{v}^\top \bigr)_{il}\, \partial_{il} f ,
\\\\
h_j( s, \omega )
&= \sum_{i = 1}^{m} \partial_i f\, v_{ij} ,
\quad j = 1, \ldots, n ,
\end{align*}
\]
with the same evaluation convention. These abbreviations are used
throughout the proof.
Proof.
The integrands are bounded, and the composite is an Itô process.
All derivatives of \(f\) vanish off \(K\): on the complement of
\(K\), which is open, \(f\) is identically zero, and by
continuity from there the derivatives vanish on the boundary
as well. Hence \(G\) and each
\(h_j\) vanish at every pair \(( s, \omega )\) with
\(Y_s( \omega ) \notin K\), while at the remaining pairs the
hypothesis bounds \(\mathbf{u}\) and \(\mathbf{v}\) and the
derivatives of \(f\) are bounded by compactness. There are
therefore constants \(M_0\) and \(M_1\) with
\(| G | \leq M_0\) and \(| h_j | \leq M_1\) everywhere. The
integrand \(h_j\) lies in the
admissible class
\(\mathcal{V}( 0, T )\) for every horizon \(T\). Each factor
\(\partial_i f( Y_\cdot )\) is continuous and adapted, hence
progressively measurable by
the criteria for progressive measurability
and in particular jointly measurable, while each \(v_{ij}\) is
jointly measurable and adapted because the
definition of an Itô process
places it in the admissible class at every horizon. Products
preserve joint measurability and adaptedness, so (V1) and (V2)
hold, and the square integrability (V3) is immediate from the
bound. This is the
membership condition of the
general Itô formula,
applied with the time-independent map \(f\) in the role of \(g\).
The formula gives, for each fixed \(t \geq 0\) and almost surely,
\[
f( Y_t )
= f( x )
+ \int_0^t G( s, \omega )\, ds
+ \sum_{j = 1}^{n} \int_0^t h_j( s, \omega )\, d w_s^{(j)} .
\]
One null set serves every time at once.
The identity as stated holds, for each \(t\), outside a null set
that may depend on \(t\), and a random time cannot be substituted
into such a statement. Each \(h_j\) lies in
\(\mathcal{V}( 0, \infty )\), being admissible at every horizon,
so each stochastic integral has a
continuous version on the half-line
\(J^{(j)}\). The other two terms are continuous as well: the left
side because \(Y\) has continuous paths and \(f\) is continuous,
and the time integral because \(G\) is bounded. Two processes with
almost surely continuous paths which agree almost surely at each
fixed time agree at every rational time outside a single null set,
and hence, off that set and off the null sets where either path
fails to be continuous, at every time. Read with the continuous
versions, the identity therefore holds simultaneously for all
\(t \geq 0\) outside one null set.
Substituting the stopping time.
Since \(E^x[ \tau ] \lt \infty\), the stopping time is almost
surely finite, so it may be substituted for \(t\), giving
\[
f( Y_\tau )
= f( x )
+ \int_0^\tau G( s, \omega )\, ds
+ \sum_{j = 1}^{n} J^{(j)}_\tau
\]
almost surely. Each term is integrable. The left side is bounded by
\(\sup | f |\); the time integral is bounded in absolute value by
\(M_0 \tau\), whose expectation is finite; and each
\(J^{(j)}_\tau\) is square integrable by
the integral at a stopping time,
the hypothesis there being met because
\(\mathbb{E}[ \int_0^\tau h_j^2\, ds ] \leq M_1^2\, E^x[ \tau ]\)
is finite. Taking expectations is therefore legitimate.
The noise terms have mean zero.
Apply the
integral at a stopping time
once for each \(j\), with integrand \(h_j\), stopping time
\(\tau\), and the single component \(w^{(j)}\) in the role of
the driving motion. That the one-dimensional integral machinery may
be run over the joint filtration with any one component as
integrator is not an extension of anything but a hypothesis check
performed when
the integral acquired
several driving motions. Each of the
\(n\) terms therefore has expectation zero, and what survives is
the identity of the theorem.
It is worth recording where each hypothesis was spent, since the same
list governs how far the identity can later be pushed. Compact support
was used twice, once to bound the integrands and once, through that
bound, to secure the membership condition of Itô's formula. The
integrability of \(\tau\) was also used twice: to know that the
stopping time is finite, and to supply the dominating function
\(M_0 \tau\). The boundedness of the coefficients over the support was
used once, in the same bound. No hypothesis controls the process
outside the support of \(f\), and none needs to, since the integrands
are already zero there.
The identity is Itô's formula with the martingale part averaged
away, and what survives the averaging involves the coefficients only
through \(\mathbf{u}\) and the matrix
\(\mathbf{v} \mathbf{v}^\top\): the individual entries of
\(\mathbf{v}\) enter nowhere else. The wiring of the noise is not
erased outright, since the integrand is still evaluated along the
path, and the path remembers how the noise was wired. What the
identity does guarantee is that once the drift and
\(\mathbf{v} \mathbf{v}^\top\) are read along the path, no further
detail of \(\mathbf{v}\) matters to averages of this kind. The
special case in which the stopping time is a constant is what
attaches a second-order
differential operator to a diffusion, and there the reduction to
\(\mathbf{v} \mathbf{v}^\top\) becomes visible in the operator
itself.