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 holds at each fixed time, with its integrals read up to that
time. The solution theory solves equations up to a prescribed final
time, and the representation theorem expresses 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 reading 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 whenever the
integrand is square integrable up to the stopping time. 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 favor of almost sure finiteness leaves the isometry
without a finite right side, lets the zero mean and the martingale
property fail, and buys only a local martingale. That extension was
therefore 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 the horizon 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 proof of the theorem below carries out its two
halves in turn.
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, jointly measurable,
adapted integrand belongs to the class, since
\(\mathbb{E}[ \int_0^T f^2\, ds ]\) is finite for every finite \(T\),
while for the constant integrand \(f \equiv 1\) the expectation
\(\mathbb{E}[ \int_0^\infty f^2\, ds ]\) is infinite. 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 \(0 \lt t \leq k\) the symbol
\(\int_0^t f\, d w_s\) denotes the integral of \(f\) read as an
element of \(\mathcal{V}( 0, t )\), and \(\int_0^0 f\, d w_s := 0\).
This reading does not depend on
the horizon it is taken from, since the construction uses only the
restriction of \(f\) to \([0, t]\). 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 this value independent of the
choice, and on \(N\) we 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. It is therefore \(\mathcal{F}_t\)-measurable by
the measurability property
read at horizon \(t\). That property is stated for every
representative of the class, a form that rests on the null-set
convention and 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 includes
the set where \(\tilde{J}\) fails to be continuous. The two
processes are therefore indistinguishable.
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
under one integrability condition.
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
respect to \(\{\mathcal{F}_t\}\) such that \(\tau \lt \infty\)
almost surely. Define
\[
\int_0^\tau f\, d w_s := J_\tau .
\]
The substitution is legitimate. Since \(J\) is continuous and
adapted, it is
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 \(( \tau \wedge k ) \wedge l = \tau \wedge k\).
The
stopped Itô integral
therefore identifies both stopped values as ordinary integrals
over \([0, l]\) with switched-off integrands, and the half-line
version agrees with the version on \([0, l]\) by construction.
Subtracting the two identifications and using the linearity of the
integral, we obtain
\[
J_{\tau \wedge l} - J_{\tau \wedge k}
= \int_0^l \mathbf{1}_{[ \tau \wedge k,\, \tau )}( s )\,
f( s, \omega )\, d w_s ,
\]
and the
Itô isometry
turns the second moment of the difference into
\[
\begin{align*}
\mathbb{E} \bigl[ ( J_{\tau \wedge l} - J_{\tau \wedge k} )^2 \bigr]
&= \mathbb{E} \Bigl[ \int_{\tau \wedge k}^{\tau \wedge l}
f^2\, ds \Bigr] \\\\
&\leq \mathbb{E} \Bigl[ \int_{\tau \wedge k}^{\tau} f^2\, ds \Bigr] .
\end{align*}
\]
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, \(\tau \wedge k = \tau\)
eventually along almost every path, 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
twice continuously differentiable, compactly supported function of its
state, and a stopping time with finite mean, assume the coefficients
bounded while the state lies in the support of the function, and ask
for the average
value of the function at the moment the clock stops. The answer is that
this average equals the value at the start plus 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
\[
Y_t
= x
+ \int_0^t \mathbf{u}( s, \omega )\, ds
+ \int_0^t \mathbf{v}( s, \omega )\, d \mathbf{w}_s ,
\quad t \geq 0 ,
\]
with \(\mathbf{u}\) taking values in \(\mathbb{R}^m\), \(\mathbf{v}\)
in \(\mathbb{R}^{m \times n}\), and \(\mathbf{w}\) 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 that
records which member of the family is being averaged. It does not
change the underlying measure, and 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
\(\partial_i f( Y_\cdot )\) has continuous paths and is adapted, so
the criteria for progressive measurability
make it progressively measurable and in particular jointly
measurable. 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.
The one-dimensional machinery may be run over the joint filtration
with any one component as integrator. No extension of the theory is
involved, since the required hypothesis was checked when
the integral acquired
several driving motions. Each stochastic integral therefore 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. As in the half-line
construction, two processes with almost surely continuous paths
that agree almost surely at each fixed time agree, by way of the
rational times, at all times outside a single null set. 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
\(\sup | f |\), and the time integral is bounded in absolute value
by \(M_0 \tau\), whose expectation is finite. 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, over the joint filtration as above. 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 to bound \(f\) and its derivatives, hence the integrands, and
through that bound to secure the membership condition of Itô's
formula. The integrability of \(\tau\) was used three times: to know
that the stopping time is finite, to supply the dominating function
\(M_0 \tau\), and to meet the square-integrability hypothesis of the
integral at a stopping time through the bound \(M_1^2\, E^x[ \tau ]\).
The boundedness of the coefficients over the support was used once, in
the bound on the integrands. 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.