Existence and Uniqueness for SDEs

Knowledge Map Atlas
The Solution Concept Gronwall's Inequality The Main Theorem Uniqueness

The Solution Concept, Completed

The example pages solved three equations by producing formulas and verifying them against the integral equation, and every verification ended with the same reservation. The candidate was a solution; whether it was the solution stayed open. A second reservation traveled alongside it. Every initial value was a constant, because the machinery for genuinely random initial data was not yet on the table. This page and the next retire both reservations. This page completes the solution concept, builds the one inequality the whole theory runs on, states the main theorem, and proves uniqueness. The next page constructs the solution.

Throughout, \(w_t\) is the standard one-dimensional Brownian motion of the preceding pages, started at \(w_0 = 0\) and carried on \((\Omega, \mathcal{F}, \mathbb{P})\), and \(T \gt 0\) is a fixed horizon. Everything on this page is one-dimensional. The systems version repeats the arguments coordinatewise and is recorded at statement level at the end of the next page.

Room for a Random Start

The obstruction to random initial values is informational, not computational. Under the Brownian filtration, \(\mathcal{F}_0\) consists of null sets and their complements, so condition (I1) of the Itô process definition admits only almost-surely-constant starts. The definition page recorded that richer filtrations, where a genuinely random initial condition is the point, were the intended destination. This is that destination. Let \(Z\) be a random variable independent of the entire motion, that is, independent of \(\sigma(w_s : s \geq 0)\), and enlarge the information flow to

\[ \mathcal{H}_t = \sigma\bigl( \sigma(Z) \cup \sigma(w_s : s \leq t) \cup \mathcal{N} \bigr) , \quad t \geq 0 , \]

where \(\mathcal{N}\) is the family of \(\mathbb{P}\)-null sets. The starting \(\sigma\)-algebra \(\mathcal{H}_0\) now contains \(\sigma(Z)\), so a start distributed like \(Z\) is measurable at time zero. What must be checked is that the enlargement does not damage the noise.

Lemma: The Enlarged Brownian Filtration

Let \(Z\) be a random variable independent of \(\sigma(w_s : s \geq 0)\), and let \(\{\mathcal{H}_t\}_{t \geq 0}\) be as above. Then:

(i)
\(\{\mathcal{H}_t\}\) is a filtration containing every \(\mathbb{P}\)-null set, and \(w\) is adapted to it.

(ii)
For all \(0 \leq a \lt b\), the increment \(w_b - w_a\) is independent of \(\mathcal{H}_a\).

Proof.

Part (i) is bookkeeping. Each \(\mathcal{H}_t\) is a \(\sigma\)-algebra by construction, the family increases in \(t\) because the generating families do, every null set is a generator, and \(w_s\) is \(\mathcal{H}_t\)-measurable for \(s \leq t\) because it is a generator.

For (ii), write \(\mathcal{F}_a^0 = \sigma(w_s : s \leq a)\) for the raw past of the motion and fix events \(A \in \sigma(Z)\), \(B \in \mathcal{F}_a^0\), and \(C \in \sigma(w_b - w_a)\). Both \(B\) and \(C\) belong to \(\sigma(w_s : s \geq 0)\), so the independence of \(Z\) from the entire motion gives \(\mathbb{P}[A \cap B \cap C] = \mathbb{P}[A]\, \mathbb{P}[B \cap C]\). The increment is independent of the past of the motion, and \(\mathcal{F}_a^0\) is contained in that past, so \(\mathbb{P}[B \cap C] = \mathbb{P}[B]\, \mathbb{P}[C]\). Multiplying,

\[ \mathbb{P}[ A \cap B \cap C ] = \mathbb{P}[A]\, \mathbb{P}[B]\, \mathbb{P}[C] = \mathbb{P}[ A \cap B ]\, \mathbb{P}[C] . \]

The events \(A \cap B\) form a family closed under intersection that generates \(\sigma\bigl( \sigma(Z) \cup \mathcal{F}_a^0 \bigr)\), and the extension of independence from such a family to the \(\sigma\)-algebra it generates is the monotone-class step this track has taken on faith since the construction of the integral. We extend the same trust to this one application. Finally, every event of \(\mathcal{H}_a\) differs from an event of \(\sigma\bigl( \sigma(Z) \cup \mathcal{F}_a^0 \bigr)\) by a null set, by the same symmetric-difference argument that the representation page ran for its augmented \(\sigma\)-algebra, and altering an event by a null set changes no probability. Independence therefore passes to \(\mathcal{H}_a\).

The lemma is exactly the entry ticket that the earlier pages priced. The pathwise time integral is stated for any filtration containing the null sets, which (i) supplies. The weighted increment lemma asks that the motion be adapted and its increments be independent of the filtration, which (i) and (ii) supply. The construction of the Itô integral and its four properties consume, as the extension survey of the properties page recorded, adaptedness and the independence of increments from the enlarged past, and nothing else. Every hypothesis is met. From here on, \(\mathcal{V}(0, T)\) denotes the integrand class of the construction taken over \(\{\mathcal{H}_t\}\), and every result quoted from the integral pages is quoted in this enlarged version. When \(Z\) is a constant the enlargement collapses to the Brownian filtration, and nothing on the earlier pages changes.

What It Means to Solve, in Full

With the information flow settled, the working notion of the example pages can be upgraded to a definition that carries uniqueness questions. Fix measurable coefficients and a square-integrable start.

Definition: Solution of a Stochastic Differential Equation

Let \(T \gt 0\), let \(b, \sigma : [0, T] \times \mathbb{R} \to \mathbb{R}\) be jointly Borel measurable, and let \(Z\) be a random variable independent of \(\sigma(w_s : s \geq 0)\) with \(\mathbb{E}[Z^2] \lt \infty\). A solution of

\[ dX_t = b(t, X_t)\, dt + \sigma(t, X_t)\, dw_t , \quad 0 \leq t \leq T , \quad X_0 = Z , \]

is a stochastic process \(\{X_t\}_{0 \leq t \leq T}\) such that:

(S1)
\(X\) is adapted to \(\{\mathcal{H}_t\}\) with continuous paths, hence progressively measurable by the continuity criterion;

(S2)
\(X\) is square integrable in the mean over the horizon,

\[ \mathbb{E}\Bigl[ \int_0^T X_t^2\, dt \Bigr] \lt \infty ; \]

(S3)
the substituted coefficients qualify as integrands: \(\mathbb{P}\bigl[ \int_0^T |b(s, X_s)|\, ds \lt \infty \bigr] = 1\) and \(\bigl( (s, \omega) \mapsto \sigma(s, X_s(\omega)) \bigr) \in \mathcal{V}(0, T)\);

(S4)
almost surely, simultaneously for every \(t \in [0, T]\),

\[ X_t = Z + \int_0^t b(s, X_s)\, ds + \int_0^t \sigma(s, X_s)\, dw_s , \]

the \(dw\)-term taken in its everywhere-continuous realization.

Conditions (S2) and (S3) are meaningful because of (S1). Progressive measurability makes \((t, \omega) \mapsto X_t^2\) jointly measurable, so the double integral of (S2) is a legitimate expectation, and the map \((s, \omega) \mapsto (s, X_s(\omega))\) composed with the Borel functions \(b\) and \(\sigma\) keeps progressive measurability, so the two integrals of (S4) are the pathwise time integral and the Itô integral of the enlarged construction. Condition (S2) does different work. It names the class inside which uniqueness will be asserted, and the main theorem produces its solution inside this class, so the two halves of the theory meet on common ground. When \(Z\) is a constant, the definition reduces to the working notion of the example pages together with (S2), and each of the three equations verified there already met (S2), since their membership computations bounded exactly these mean squares. Everything solved so far remains a solution in the present sense.

Constants Come Out of the Integral

One lemma completes the upgrade of the example pages themselves. Their closed forms multiply the noise by the initial value, and for a random start this requires pulling an \(\mathcal{H}_0\)-measurable factor out of a stochastic integral. The operation is plausible, because the factor is known before the integration begins, and it is not a triviality, because the integral is an \(L^2\) limit rather than a pathwise sum.

Lemma: Taking Out an Initially Known Factor

Let \(Y\) be \(\mathcal{H}_0\)-measurable and \(f \in \mathcal{V}(0, T)\), and suppose the product satisfies \(Y f \in \mathcal{V}(0, T)\). Then, almost surely,

\[ \int_0^T Y f(s, \omega)\, dw_s = Y \int_0^T f(s, \omega)\, dw_s , \]

and the same identity holds over every subinterval \([0, t]\).

Proof.

Step 1 (Bounded factor, elementary integrand).
Let \(|Y| \leq K\) and let \(\phi = \sum_j e_j \mathbf{1}_{[t_j, t_{j+1})}\) be a bounded elementary process in \(\mathcal{V}(0, T)\). Then \(Y \phi\) is again a bounded elementary process. Its levels \(Y e_j\) are \(\mathcal{H}_{t_j}\)-measurable, since \(Y\) is \(\mathcal{H}_0\)-measurable and \(\mathcal{H}_0 \subseteq \mathcal{H}_{t_j}\), and they are bounded, by \(K\) times any common bound for the levels of \(\phi\). The elementary integral is a finite sum, and the factor \(Y\) distributes over it,

\[ \int_0^T Y \phi\, dw_s = \sum_j Y e_j\, \bigl( w_{t_{j+1}} - w_{t_j} \bigr) = Y \sum_j e_j\, \bigl( w_{t_{j+1}} - w_{t_j} \bigr) = Y \int_0^T \phi\, dw_s . \]

Step 2 (Bounded factor, general integrand).
Keep \(|Y| \leq K\) and let \(f \in \mathcal{V}(0, T)\). The construction of the integral supplies bounded elementary \(\phi_n \to f\) in \(L^2(\lambda \otimes \mathbb{P})\). Then \(Y \phi_n \to Y f\) in the same space, since \(\mathbb{E}\bigl[ \int_0^T Y^2 (\phi_n - f)^2\, ds \bigr] \leq K^2\, \mathbb{E}\bigl[ \int_0^T (\phi_n - f)^2\, ds \bigr] \to 0\). By L² continuity of the integral, both \(\int_0^T Y \phi_n\, dw_s \to \int_0^T Y f\, dw_s\) and \(\int_0^T \phi_n\, dw_s \to \int_0^T f\, dw_s\) in \(L^2(\mathbb{P})\), and multiplying the second convergence by the bounded \(Y\) preserves it. Step 1 makes the two sequences equal term by term, and two \(L^2\) limits of the same sequence agree almost surely.

Step 3 (General factor by truncation).
For general \(Y\) with \(Y f \in \mathcal{V}(0, T)\), set \(Y_K = Y\, \mathbf{1}_{\{|Y| \leq K\}}\), which is bounded and \(\mathcal{H}_0\)-measurable, with \(|Y_K f| \leq |Y f|\), so \(Y_K f \in \mathcal{V}(0, T)\) and Step 2 applies to \(Y_K\). As \(K \to \infty\), the Itô isometry gives

\[ \mathbb{E}\Bigl[ \Bigl( \int_0^T ( Y_K - Y ) f\, dw_s \Bigr)^{2} \Bigr] = \mathbb{E}\Bigl[ \int_0^T Y^2\, \mathbf{1}_{\{|Y| \gt K\}}\, f^2\, ds \Bigr] \longrightarrow 0 \]

by the dominated convergence theorem on the product space, with dominator the integrable \(Y^2 f^2\). So \(\int_0^T Y_K f\, dw_s \to \int_0^T Y f\, dw_s\) in \(L^2(\mathbb{P})\), while \(Y_K \int_0^T f\, dw_s \to Y \int_0^T f\, dw_s\) pointwise almost surely.

To marry the two modes, pick \(K_1 \lt K_2 \lt \cdots\) along which the displayed expectation is at most \(4^{-j}\). By the monotone convergence theorem applied to the partial sums, the sum \(\sum_j \bigl( \int_0^T ( Y_{K_j} - Y ) f\, dw_s \bigr)^{2}\) has finite expectation, so it is finite almost surely, its infinity set otherwise forcing the expectation to be infinite. Terms of an almost surely convergent series tend to zero, so \(\int_0^T Y_{K_j} f\, dw_s \to \int_0^T Y f\, dw_s\) almost surely as well. Each term of this sequence equals the corresponding \(Y_{K_j} \int_0^T f\, dw_s\) almost surely by Step 2, and the two almost sure limits agree. The subinterval version replaces \(T\) by \(t\) throughout, or multiplies \(f\) by \(\mathbf{1}_{[0, t]}\).

The lemma and the independence of \(Z\) from the noise retire the constant-start restriction of the example pages at one stroke. Geometric Brownian motion with start \(Z\) is \(Z \exp\bigl( (r - \tfrac{1}{2} \alpha^2) t + \alpha w_t \bigr)\), the linear equation and the Ornstein-Uhlenbeck family carry \(Z\) through their ledgers unchanged, and the three verifications survive line by line with two upgrades. Membership computations factor through independence, as in \(\mathbb{E}\bigl[ Z^2 e^{2 \alpha w_s} \bigr] = \mathbb{E}[Z^2]\, \mathbb{E}\bigl[ e^{2 \alpha w_s} \bigr]\) by the product rule for expectations, and initially known factors pulled out of stochastic integrals are licensed by the lemma. Mean formulas gain an expectation, \(\mathbb{E}[Z] e^{r t}\) in place of \(N_0 e^{r t}\). We record the upgrade without repeating the verifications.

The solution concept is complete. What the theory needs next is not stochastic at all. Both proofs on these two pages run on a single deterministic inequality, and we build it first.

Gronwall's Inequality

Both proofs ahead produce estimates of the same shape. A quantity built from a solution turns out to be bounded by a constant plus a multiple of its own running integral, because the isometry and the Lipschitz hypothesis convert the integral equation into exactly such a self-bound. A self-referential estimate is not yet a bound, and the lemma that converts one into the other is deterministic. Nothing in this section mentions probability.

Lemma: Gronwall's Inequality

Let \(v : [0, T] \to [0, \infty)\) be measurable with \(\int_0^T v(s)\, ds \lt \infty\), and let \(C \geq 0\) and \(A \geq 0\) be constants such that

\[ v(t) \leq C + A \int_0^t v(s)\, ds \quad \text{for every } t \in [0, T] . \]

Then \(v(t) \leq C e^{A t}\) for every \(t \in [0, T]\).

Proof.

For \(A = 0\) the hypothesis is the conclusion, so let \(A \gt 0\). Write \(m(r) = \int_0^r v(s)\, ds\) for \(r \in [0, T]\), a finite quantity by assumption, and fix \(t \in [0, T]\). The letter \(w\) stays reserved for the Brownian motion, which this proof never touches.

The classical argument multiplies \(m\) by the integrating factor \(e^{-A t}\) and differentiates. We reach the same identity by an interchange of integration order instead, which asks nothing of \(v\) beyond integrability. The integrand \((s, u) \mapsto A e^{-A s}\, v(u)\, \mathbf{1}_{\{u \lt s\}}\) is nonnegative and jointly measurable on \([0, t] \times [0, t]\), so Tonelli's theorem permits either order of integration,

\[ \begin{align*} \int_0^t A e^{-A s}\, m(s)\, ds &= \int_0^t \int_0^t A e^{-A s}\, v(u)\, \mathbf{1}_{\{u \lt s\}}\, du\, ds \\\\ &= \int_0^t v(u) \int_u^t A e^{-A s}\, ds\, du \\\\ &= \int_0^t v(u) \bigl( e^{-A u} - e^{-A t} \bigr)\, du \\\\ &= \int_0^t e^{-A s}\, v(s)\, ds - e^{-A t}\, m(t) . \end{align*} \]

Rearranging,

\[ e^{-A t}\, m(t) = \int_0^t e^{-A s}\, \bigl( v(s) - A m(s) \bigr)\, ds , \]

and both terms under the last integral are integrable over \([0, t]\), since \(e^{-A s} v(s) \leq v(s)\) and \(m\) is nondecreasing, hence bounded by \(m(T)\). The hypothesis reads \(v(s) - A m(s) \leq C\) for every \(s\), and multiplying by \(e^{-A s} \gt 0\) and integrating preserves the inequality,

\[ e^{-A t}\, m(t) \leq C \int_0^t e^{-A s}\, ds = \frac{C}{A} \bigl( 1 - e^{-A t} \bigr) . \]

So \(A m(t) \leq C ( e^{A t} - 1 )\), and one more use of the hypothesis finishes,

\[ v(t) \leq C + A m(t) \leq C + C \bigl( e^{A t} - 1 \bigr) = C e^{A t} . \]

The weak hypotheses are the point. The functions these two pages feed to the lemma are mean squares of stochastic processes, measurable and integrable in time by construction, and nothing forces them to be continuous. A version of the lemma that differentiates would demand regularity we would then have to manufacture. The version above consumes precisely what condition (S2) of the solution concept supplies. With the engine built, we can state what the theory delivers.

The Existence and Uniqueness Theorem

The pieces are assembled. The definition fixes what it means to solve, the lemma converts self-referential estimates into bounds, and the main theorem can now be stated in the generality this track will use. Its hypotheses are two inequalities on the coefficients, holding uniformly over the whole strip \([0, T] \times \mathbb{R}\).

Theorem: Existence and Uniqueness for SDEs

Let \(T \gt 0\), and let \(b, \sigma : [0, T] \times \mathbb{R} \to \mathbb{R}\) be jointly Borel measurable functions for which there exist constants \(C\) and \(D\) such that, for all \(t \in [0, T]\) and all \(x, y \in \mathbb{R}\):

(H1)
linear growth,

\[ |b(t, x)| + |\sigma(t, x)| \leq C \bigl( 1 + |x| \bigr) ; \]

(H2)
a Lipschitz bound in the state variable,

\[ |b(t, x) - b(t, y)| + |\sigma(t, x) - \sigma(t, y)| \leq D\, |x - y| . \]

Let \(Z\) be a random variable independent of \(\sigma(w_s : s \geq 0)\) with \(\mathbb{E}[Z^2] \lt \infty\). Then the stochastic differential equation

\[ dX_t = b(t, X_t)\, dt + \sigma(t, X_t)\, dw_t , \quad 0 \leq t \leq T , \quad X_0 = Z , \]

has a solution in the sense of the definition above, and the solution is unique: any two solutions \(X\) and \(\hat{X}\) of the same equation with the same initial value satisfy

\[ \mathbb{P}\bigl[ X_t = \hat{X}_t \ \text{for every } t \in [0, T] \bigr] = 1 . \]

The theorem is proved in full across two pages. Uniqueness closes this page, in the next section. Existence is the business of the next page, which builds the solution by an iteration of Picard type and extracts a pathwise limit. The constants \(C\) and \(D\) are fixed from here on, since the uniqueness proof consumes \(D\) by name.

What Each Hypothesis Buys

Condition (H2) says that \(x \mapsto b(t, x)\) and \(x \mapsto \sigma(t, x)\) are Lipschitz continuous, with one constant serving every time \(t\) at once. It is the uniqueness hypothesis. Two solutions driven by the same noise can separate only through the coefficients, and (H2) makes the separation feed back proportionally to itself, so the mean-square gap obeys precisely the self-bound that Gronwall's inequality collapses to zero. Without a Lipschitz bound, uniqueness genuinely fails, and the failure needs no randomness. An equation with \(\sigma = 0\) and infinitely many solutions sprouting from a single start is exhibited, with one-line verifications, in the closing section of the previous page.

Condition (H1) is the globality hypothesis. It caps how strongly the state can amplify itself, and this is what keeps solutions alive on all of \([0, T]\). The phenomenon it excludes is again already deterministic. A complete vector field is one whose flow exists for all time, and the manifold track's example of finite-time blow-up shows what happens when growth is superlinear. Linear growth is the probabilistic relative of the conditions that force completeness, and the same closing section of the previous page runs the blow-up computation for the equation read as an SDE with vanishing noise.

The two conditions carry independent information. A Lipschitz bound controls differences and says nothing about size at a single point, since \(|b(t, 0)|\) may still be unbounded in \(t\), so (H1) is not a consequence of (H2). Measurability is the third, silent hypothesis. It is what lets the substituted coefficients inherit progressive measurability, as recorded after the definition, and it costs nothing in practice.

Uniqueness goes first. The proof is where the enlarged filtration, the isometry, and the new inequality meet.

Uniqueness

Fix coefficients satisfying (H1) and (H2). The theorem's uniqueness half concerns two solutions of the same equation with the same initial value, but the estimate at the heart of the proof costs nothing extra in a slightly wider setting, and the extra generality pays a dividend at the end. Throughout this section, let \(\{\mathcal{H}_t\}\) be a filtration with the two properties of the enlargement lemma, containing the null sets and leaving the increments of \(w\) independent of the past. Let \(Z\) and \(\hat{Z}\) be \(\mathcal{H}_0\)-measurable and square integrable, and let \(X\) and \(\hat{X}\) solve the equation over \(\{\mathcal{H}_t\}\) in the sense of the definition, with \(X_0 = Z\) and \(\hat{X}_0 = \hat{Z}\). The definition's four clauses read verbatim over any such filtration. The theorem's setting is the case \(\hat{Z} = Z\) with \(\{\mathcal{H}_t\}\) the enlarged Brownian filtration of the first section.

Proof of the uniqueness half.

Step 1 (The gap is an Itô process).
Both solutions satisfy their integral equations almost surely, simultaneously for every \(t\), so on the intersection of the two full-measure events, writing \(\beta_s = b(s, X_s) - b(s, \hat{X}_s)\) and \(\gamma_s = \sigma(s, X_s) - \sigma(s, \hat{X}_s)\),

\[ X_t - \hat{X}_t = ( Z - \hat{Z} ) + \int_0^t \beta_s\, ds + \int_0^t \gamma_s\, dw_s , \quad 0 \leq t \leq T . \]

The two integrals are well posed. Both \(\beta\) and \(\gamma\) are progressively measurable as differences of progressively measurable processes. The Lipschitz bound (H2) gives \(|\beta_s| \leq D\, |X_s - \hat{X}_s|\), which is bounded on \([0, T]\) along each pair of continuous paths, and

\[ \begin{align*} \mathbb{E}\Bigl[ \int_0^T \gamma_s^2\, ds \Bigr] &\leq D^2\, \mathbb{E}\Bigl[ \int_0^T | X_s - \hat{X}_s |^2\, ds \Bigr] \\\\ &\leq 2 D^2\, \mathbb{E}\Bigl[ \int_0^T \bigl( X_s^2 + \hat{X}_s^2 \bigr) ds \Bigr] \\\\ &\lt \infty \end{align*} \]

by \((x - y)^2 \leq 2 x^2 + 2 y^2\) and condition (S2) of both solutions, so \(\gamma \in \mathcal{V}(0, T)\). This is the moment where the square-integrability clause of the definition earns its place.

Step 2 (Squaring and splitting).
Set \(v(t) = \mathbb{E}\bigl[ | X_t - \hat{X}_t |^2 \bigr]\), a measurable function of \(t\) with values in \([0, \infty]\) for now. From \((x + y + z)^2 \leq 3 ( x^2 + y^2 + z^2 )\), a consequence of \(2 x y \leq x^2 + y^2\) applied to each cross term,

\[ v(t) \leq 3\, \mathbb{E}\bigl[ | Z - \hat{Z} |^2 \bigr] + 3\, \mathbb{E}\Bigl[ \Bigl( \int_0^t \beta_s\, ds \Bigr)^{2} \Bigr] + 3\, \mathbb{E}\Bigl[ \Bigl( \int_0^t \gamma_s\, dw_s \Bigr)^{2} \Bigr] . \]

The drift term is controlled pathwise by the Cauchy-Schwarz case of Hölder's inequality on \([0, t]\), then in the mean by Tonelli's theorem and (H2),

\[ \begin{align*} \mathbb{E}\Bigl[ \Bigl( \int_0^t \beta_s\, ds \Bigr)^{2} \Bigr] &\leq t\, \mathbb{E}\Bigl[ \int_0^t \beta_s^2\, ds \Bigr] \\\\ &\leq t\, D^2 \int_0^t v(s)\, ds , \end{align*} \]

where Tonelli also certifies that \(s \mapsto v(s)\) is measurable, since \((s, \omega) \mapsto | X_s - \hat{X}_s |^2\) is jointly measurable by progressive measurability. The noise term is exactly what the Itô isometry computes. We quote the isometry in its enlarged-filtration version, using the membership from Step 1, then apply the same Tonelli interchange and (H2),

\[ \begin{align*} \mathbb{E}\Bigl[ \Bigl( \int_0^t \gamma_s\, dw_s \Bigr)^{2} \Bigr] &= \mathbb{E}\Bigl[ \int_0^t \gamma_s^2\, ds \Bigr] \\\\ &\leq D^2 \int_0^t v(s)\, ds . \end{align*} \]

Assembled,

\[ \begin{align*} v(t) &\leq 3\, \mathbb{E}\bigl[ | Z - \hat{Z} |^2 \bigr] + 3 ( 1 + t )\, D^2 \int_0^t v(s)\, ds \\\\ &\leq 3\, \mathbb{E}\bigl[ | Z - \hat{Z} |^2 \bigr] + A \int_0^t v(s)\, ds , \end{align*} \]

where the second line uses \(t \leq T\) and sets \(A = 3 ( 1 + T )\, D^2\).

Step 3 (Gronwall).
The hypotheses of Gronwall's inequality are met. The function \(v\) is measurable and nonnegative, its integral over \([0, T]\) is finite by the same (S2) computation as in Step 1, and the display above then makes every value \(v(t)\) finite, since its right side is. The lemma applies with \(3\, \mathbb{E}[ | Z - \hat{Z} |^2 ]\) as its additive constant and \(A\) as its multiplier, and yields

\[ \mathbb{E}\bigl[ | X_t - \hat{X}_t |^2 \bigr] \leq 3\, \mathbb{E}\bigl[ | Z - \hat{Z} |^2 \bigr]\, e^{A t} , \quad 0 \leq t \leq T . \]

Step 4 (Same start, same path).
Now let \(\hat{Z} = Z\). The estimate collapses to \(\mathbb{E}[ | X_t - \hat{X}_t |^2 ] = 0\) for every \(t\), so \(X_t = \hat{X}_t\) almost surely at each fixed time, because a nonnegative function with zero integral vanishes almost everywhere. A countable union of null sets is null, so

\[ \mathbb{P}\bigl[ X_t = \hat{X}_t \ \text{for every } t \in \mathbb{Q} \cap [0, T] \bigr] = 1 . \]

On this event the two paths are continuous functions agreeing on a dense subset of \([0, T]\), and two continuous functions that agree on a dense set agree everywhere, each point being a limit of points of agreement. The exceptional set has not grown, and \(\mathbb{P}[ X_t = \hat{X}_t \ \text{for every } t \in [0, T] ] = 1\). This is the uniqueness half of the theorem.

The wider setting now pays. For genuinely different starts, the Step 3 estimate says that solutions depend continuously on their initial data in mean square, with the explicit modulus \(3 e^{A T}\) over the whole horizon. The common filtration such a pair needs is built exactly as in the first section, enlarging by \(\sigma(Z, \hat{Z})\) instead of \(\sigma(Z)\), and the enlargement lemma's proof reads verbatim once the pair, rather than the single variable, is independent of the entire motion.

One half of the theorem is now a theorem. The other half is a construction. A solution must be produced from the coefficients and the noise alone, and the machine that produces it is an iteration of Picard type whose convergence runs through the toolkit of inequalities assembled so far. The construction is the subject of the next page.