Existence by Picard Iteration

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The Toolkit The Iteration Pathwise Existence Beyond Lipschitz

The Toolkit

The previous page stated the existence and uniqueness theorem and proved its uniqueness half. Under the linear growth condition (H1) and the Lipschitz condition (H2), two solutions with the same initial value are indistinguishable. This page proves the other half. A solution must now be manufactured from the coefficients, the noise, and the initial value alone, and the whole of it is a construction.

The setting carries over unchanged. Throughout, \(w_t\) is the standard one-dimensional Brownian motion, \(Z\) is a square-integrable initial value independent of the entire motion, \(\{\mathcal{H}_t\}\) is the enlarged filtration built from \(Z\), the motion, and the null sets, and \(\mathcal{V}(0, T)\) is the integrand class over this filtration, with every quoted result taken in its enlarged version, as the previous page declared once and for all. The coefficients \(b\) and \(\sigma\) satisfy (H1) and (H2) with the constants \(C\) and \(D\) fixed there, and the target is a process satisfying clauses (S1) through (S4) of the solution definition. The horizon \(T \gt 0\) stays fixed, and \(\lambda\) denotes Lebesgue measure on \([0, T]\), so that \(L^2(\lambda \otimes \mathbb{P})\) is the space in which integrands and processes are measured over the whole strip \([0, T] \times \Omega\).

The construction runs in two stages, and each stage runs on inequalities. The first stage iterates the equation and shows that successive iterates contract in mean square at factorial speed, by the same isometry-plus-Gronwall mechanism that drove the uniqueness proof. The second stage upgrades this mean-square convergence to convergence of the paths themselves, uniformly in time, and here a new kind of statement is needed. A mean square is an average over \(\Omega\), while a statement about paths concerns the probability of an event. The bridge between the two is one inequality, used everywhere in probability, that this track has never owned. We make it ours first.

Tail Bounds from Moments

The observation is that a nonnegative random variable cannot be large with large probability without its expectation noticing. Cutting the variable down at a threshold can only shrink it, and what remains still dominates the threshold times the indicator of the exceedance.

Theorem: The Markov and Chebyshev Inequalities

(i)
(Markov) Let \(Y\) be a nonnegative random variable and \(a \gt 0\). Then

\[ \mathbb{P}\bigl[ Y \geq a \bigr] \leq \frac{1}{a}\, \mathbb{E}[Y] . \]

(ii)
(Chebyshev) Let \(X\) be any random variable and \(a \gt 0\). Then

\[ \mathbb{P}\bigl[ |X| \geq a \bigr] \leq \frac{1}{a^2}\, \mathbb{E}\bigl[ X^2 \bigr] . \]

Proof.

(i).
The pointwise inequality

\[ a\, \mathbf{1}_{\{ Y \geq a \}}(\omega) \leq Y(\omega) \]

holds for every \(\omega\). On the event \(\{Y \geq a\}\) it reads \(a \leq Y(\omega)\), which is the membership condition itself, and off the event it reads \(0 \leq Y(\omega)\), which is nonnegativity. The event is measurable because \(Y\) is, expectation is monotone, and the expectation of \(a\) times an indicator is \(a\) times the probability of its event, so \(a\, \mathbb{P}[Y \geq a] \leq \mathbb{E}[Y]\). Dividing by \(a \gt 0\) finishes. No integrability is assumed. When \(\mathbb{E}[Y]\) is infinite the bound holds and says nothing.

(ii).
Squaring is strictly increasing on \([0, \infty)\), so the events \(\{ |X| \geq a \}\) and \(\{ X^2 \geq a^2 \}\) are equal. Applying (i) to the nonnegative variable \(X^2\) at the threshold \(a^2\) gives \(\mathbb{P}[X^2 \geq a^2] \leq a^{-2}\, \mathbb{E}[X^2]\).

Two remarks position the theorem for use. Since \(\{Y \gt a\}\) is contained in \(\{Y \geq a\}\), the same bounds hold for strict exceedances, and it is the strict form that the arguments ahead consume. And applying (ii) to \(X - \mathbb{E}[X]\), when the mean exists, bounds the probability of a deviation from the mean by the variance over the threshold squared, which is the statement the name Chebyshev most often attaches to. This page needs the uncentered form.

What Is Already in Hand

The rest of the kit exists, and it is worth laying out on the bench in the order of use. The mean-square stage runs on three estimates and one completeness theorem. The Cauchy-Schwarz case of Hölder's inequality controls the drift, the Itô isometry converts the mean square of the noise term into a time integral, and Gronwall's inequality from the previous page turns the resulting self-referential estimates into explicit bounds. The division of labor is exactly that of the uniqueness proof, and the membership checks that license the isometry will be repeated, not inherited, since the iterates are new processes. When the mean-square estimates are in place, the iterates form a Cauchy sequence in \(L^2(\lambda \otimes \mathbb{P})\), and the Riesz-Fischer theorem supplies a limit in that space, since the product measure is a measure like any other.

The pathwise stage runs on three more tools. The Chebyshev inequality just proved converts the factorial-speed mean-square estimates into summable bounds on tail probabilities of the drift differences. The maximal bound for the Itô integral does the same for the noise differences, and does something Chebyshev alone cannot. For an integrand in \(\mathcal{V}(0, T)\) and a continuous version of its integral process, it bounds the tail of the running supremum over the whole horizon by the mean energy of the integrand over the threshold squared, which is exactly the uniform-in-time control that path convergence requires. Finally, the Borel-Cantelli lemma converts summable tail probabilities into an almost sure statement, that only finitely many of the exceedances ever happen. Chained together, these three turn factorial decay of mean squares into uniform convergence of almost every path.

The kit is complete. The scheme that consumes it comes next.

The Iteration

The scheme is the stochastic version of a classical idea. Feed a guess for the solution through the right side of the integral equation, and the output is a new process, hopefully a better guess. If the coefficients do not amplify differences too violently, which is exactly what (H2) forbids, successive outputs should huddle together, and their limit should pass through the equation unchanged. The deterministic ancestor of this scheme built the solutions of ordinary differential equations, and the noise term will turn out to be no obstacle, because the isometry prices it in mean square exactly as Cauchy-Schwarz prices the drift.

We prove the existence half of the main theorem. Under (H1) and (H2), with \(Z\) square integrable and independent of the motion, there is a process satisfying clauses (S1) through (S4). The proof occupies this section and the next. This section builds the iterates and proves that they contract in mean square at factorial speed, ending with a limit in \(L^2(\lambda \otimes \mathbb{P})\). The next section upgrades the convergence to almost sure uniform convergence of paths and verifies that the limit solves.

Proof of the existence half, part 1: mean-square convergence.

Step 1 (The scheme, and that it is well defined).
Define \(Y^{(0)}_t = Z\) for all \(t \in [0, T]\), and inductively

\[ Y^{(k+1)}_t = Z + \int_0^t b\bigl( s, Y^{(k)}_s \bigr)\, ds + \int_0^t \sigma\bigl( s, Y^{(k)}_s \bigr)\, dw_s , \quad 0 \leq t \leq T , \]

with the \(dw\)-term taken, here and throughout, in its everywhere-continuous realization. For the induction to make sense, each iterate must qualify as raw material for the next, and we carry the invariant through the induction: \(Y^{(k)}\) is \(\{\mathcal{H}_t\}\)-adapted with continuous paths, and \(m_k = \sup_{0 \leq t \leq T} \mathbb{E}\bigl[ |Y^{(k)}_t|^2 \bigr]\) is finite. The start \(Y^{(0)} = Z\) is \(\mathcal{H}_0\)-measurable with constant paths and \(m_0 = \mathbb{E}[Z^2] \lt \infty\).

Assume the invariant for \(Y^{(k)}\). Adaptedness with continuous paths makes \(Y^{(k)}\) progressively measurable by the continuity criterion, and composing \((s, \omega) \mapsto (s, Y^{(k)}_s(\omega))\) with the Borel functions \(b\) and \(\sigma\) keeps progressive measurability. The drift entry is pathwise integrable, since (H1) bounds \(|b(s, Y^{(k)}_s)| \leq C(1 + |Y^{(k)}_s|)\) and each continuous path of \(Y^{(k)}\) is bounded on \([0, T]\), so the pathwise time integral applies over \(\{\mathcal{H}_t\}\), with the integrand extended by zero beyond the horizon, and returns a process that is adapted, continuous for every \(\omega\), and progressively measurable. The noise entry lies in \(\mathcal{V}(0, T)\). It is progressively measurable as just noted, and by (H1), \((1 + x)^2 \leq 2 (1 + x^2)\), and Tonelli's theorem on the product space, whose joint-measurability hypothesis progressive measurability supplies,

\[ \begin{align*} \mathbb{E}\Bigl[ \int_0^T \sigma\bigl( s, Y^{(k)}_s \bigr)^2\, ds \Bigr] &\leq 2 C^2 \int_0^T \Bigl( 1 + \mathbb{E}\bigl[ |Y^{(k)}_s|^2 \bigr] \Bigr) ds \\\\ &\leq 2 C^2 T ( 1 + m_k ) \lt \infty . \end{align*} \]

So \(Y^{(k+1)}\) is defined. It is adapted, the constant \(Z\) by \(\mathcal{H}_0\)-measurability, the time integral by the pathwise theorem, and the stochastic integral because adaptedness in the properties theorem holds for every representative, the continuous realization included, and its paths are sums of three continuous functions. For the moment bound, split by \((x + y + z)^2 \leq 3 (x^2 + y^2 + z^2)\), control the drift term pathwise by the Cauchy-Schwarz case of Hölder's inequality and then in the mean by Tonelli, and compute the noise term by the Itô isometry, whose membership hypothesis was verified above. With the display just proved,

\[ \begin{align*} \mathbb{E}\bigl[ |Y^{(k+1)}_t|^2 \bigr] &\leq 3\, \mathbb{E}[Z^2] + 3 t\, \mathbb{E}\Bigl[ \int_0^t b\bigl( s, Y^{(k)}_s \bigr)^2 ds \Bigr] + 3\, \mathbb{E}\Bigl[ \int_0^t \sigma\bigl( s, Y^{(k)}_s \bigr)^2 ds \Bigr] \\\\ &\leq 3\, \mathbb{E}[Z^2] + 3\, ( T + 1 ) \cdot 2 C^2 T ( 1 + m_k ) , \end{align*} \]

uniformly in \(t\), so \(m_{k+1}\) is finite and the invariant survives.

Step 2 (The first gap).
Write \(d_k(t) = \mathbb{E}\bigl[ |Y^{(k+1)}_t - Y^{(k)}_t|^2 \bigr]\) for the mean-square gap between consecutive iterates, a measurable function of \(t\), as Tonelli certifies from joint measurability. The first gap is

\[ Y^{(1)}_t - Y^{(0)}_t = \int_0^t b(s, Z)\, ds + \int_0^t \sigma(s, Z)\, dw_s , \]

and splitting by \((x + y)^2 \leq 2 (x^2 + y^2)\), then treating drift and noise exactly as in Step 1, with (H1) applied at the constant argument \(Z\),

\[ \begin{align*} d_0(t) &\leq 2 t\, \mathbb{E}\Bigl[ \int_0^t b(s, Z)^2\, ds \Bigr] + 2\, \mathbb{E}\Bigl[ \int_0^t \sigma(s, Z)^2\, ds \Bigr] \\\\ &\leq 2\, ( T + 1 )\, C^2\, \mathbb{E}\bigl[ ( 1 + |Z| )^2 \bigr]\, t \\\\ &= A_1\, t , \end{align*} \]

where \(A_1 = 2 ( T + 1 )\, C^2\, \mathbb{E}[ ( 1 + |Z| )^2 ]\) is finite because \(Z\) is square integrable.

Step 3 (One turn of the crank).
Fix \(k \geq 1\) and \(t \in [0, T]\). Each iterate satisfies its defining identity, so subtracting, and using linearity of both integrals, the stochastic one by part (i) of the properties theorem, gives, almost surely, with \(\beta^{(k)}_s = b(s, Y^{(k)}_s) - b(s, Y^{(k-1)}_s)\) and \(\gamma^{(k)}_s = \sigma(s, Y^{(k)}_s) - \sigma(s, Y^{(k-1)}_s)\),

\[ Y^{(k+1)}_t - Y^{(k)}_t = \int_0^t \beta^{(k)}_s\, ds + \int_0^t \gamma^{(k)}_s\, dw_s . \]

The initial slot is empty, since every iterate shares the start \(Z\). Both differences are progressively measurable, (H2) gives \(|\beta^{(k)}_s| \leq D\, |Y^{(k)}_s - Y^{(k-1)}_s|\), bounded along each pair of continuous paths, and \(\mathbb{E}\bigl[ \int_0^T (\gamma^{(k)}_s)^2\, ds \bigr] \leq 2 D^2 T ( m_k + m_{k-1} ) \lt \infty\) by (H2), \((x - y)^2 \leq 2 x^2 + 2 y^2\), and Tonelli, so \(\gamma^{(k)} \in \mathcal{V}(0, T)\). Splitting as in the uniqueness proof, with the empty slot retained so that the same constant serves both pages, and running Cauchy-Schwarz, Tonelli, the isometry, and (H2) exactly as there,

\[ \begin{align*} d_k(t) &\leq 3 t\, \mathbb{E}\Bigl[ \int_0^t \bigl( \beta^{(k)}_s \bigr)^2 ds \Bigr] + 3\, \mathbb{E}\Bigl[ \int_0^t \bigl( \gamma^{(k)}_s \bigr)^2 ds \Bigr] \\\\ &\leq 3\, ( 1 + t )\, D^2 \int_0^t d_{k-1}(s)\, ds \\\\ &\leq A \int_0^t d_{k-1}(s)\, ds , \end{align*} \]

where the last line absorbs \(t \leq T\) into \(A = 3\, ( 1 + T )\, D^2\), the constant of the uniqueness proof.

Step 4 (Factorial decay).
We claim

\[ d_k(t) \leq A_1\, A^k\, \frac{t^{k+1}}{(k+1)!} , \quad 0 \leq t \leq T ,\ k \geq 0 . \]

Step 2 is the case \(k = 0\). If the bound holds for \(k - 1\), Step 3 gives

\[ d_k(t) \leq A \int_0^t A_1\, A^{k-1}\, \frac{s^{k}}{k!}\, ds = A_1\, A^{k}\, \frac{t^{k+1}}{(k+1)!} . \]

The factorial is the entire story. Each turn of the crank costs one factor of \(A\) and buys one more integration of a power, and the powers' denominators grow faster than any geometric factor can spend.

Step 5 (Two Cauchy conclusions).
Integrating the claim over the horizon, by Tonelli once more,

\[ \bigl\| Y^{(k+1)} - Y^{(k)} \bigr\|_{L^2(\lambda \otimes \mathbb{P})}^2 = \int_0^T d_k(t)\, dt \leq A_1\, A^{k}\, \frac{T^{k+2}}{(k+2)!} . \]

The square roots of these bounds form a convergent series. The ratio of consecutive terms is \(( A T / (k + 3) )^{1/2}\), which falls below \(\tfrac{1}{2}\) once \(k \geq 4 A T\), so the tail is dominated by a halving geometric series. For \(m \gt n\), the triangle inequality in \(L^2(\lambda \otimes \mathbb{P})\) gives

\[ \bigl\| Y^{(m)} - Y^{(n)} \bigr\|_{L^2(\lambda \otimes \mathbb{P})} \leq \sum_{k=n}^{m-1} \Bigl( A_1\, A^{k}\, \frac{T^{k+2}}{(k+2)!} \Bigr)^{1/2} \longrightarrow 0 \quad \text{as } n \to \infty , \]

a tail of a convergent series. The iterates are Cauchy in \(L^2(\lambda \otimes \mathbb{P})\), the product measure is a measure like any other, and the Riesz-Fischer theorem supplies a limit: a jointly measurable process \(X\) on \([0, T] \times \Omega\) with

\[ \mathbb{E}\Bigl[ \int_0^T X_t^2\, dt \Bigr] \lt \infty \quad \text{and} \quad \bigl\| Y^{(n)} - X \bigr\|_{L^2(\lambda \otimes \mathbb{P})} \longrightarrow 0 , \]

determined up to a \(\lambda \otimes \mathbb{P}\)-null set. The same comparison also holds at each fixed time. From Step 4, \(\sum_k d_k(t)^{1/2} \leq \sum_k ( A_1 A^k T^{k+1} / (k+1)! )^{1/2} \lt \infty\), so \(\{ Y^{(n)}_t \}_n\) is Cauchy in \(L^2(\mathbb{P})\) for every single \(t \in [0, T]\). We record both conclusions and pause part 1 here.

What the mean-square stage delivers is deliberately less than a solution. The limit \(X\) is so far an equivalence class on the strip, defined up to a joint null set, with no distinguished paths, no continuity, and no verified relation to the equation. Candidates for all three exist, because the iterates themselves are continuous and adapted, and the factorial decay is strong enough to make them converge not just on average but uniformly along almost every path. Extracting that convergence is the business of the next section, and it is where the tail bounds of the toolkit earn their place.

Pathwise Existence

The factorial decay of the mean-square gaps is far stronger than the Cauchy property alone required, and the surplus is now spent on the paths. The plan compares each uniform gap \(\sup_{0 \leq t \leq T} |Y^{(k+1)}_t - Y^{(k)}_t|\) against the threshold \(2^{-k}\). The probability of exceeding the threshold will be summable in \(k\), because the mean squares shrink factorially while the thresholds shrink only geometrically, and the Borel-Cantelli lemma then confines almost every \(\omega\) to finitely many exceedances. Beyond them, the path gaps are dominated by a convergent geometric series, and the iterates converge uniformly. This is the argument for which the toolkit was assembled, and every tool is consumed exactly once per gap: Chebyshev on the drift, the maximal bound on the noise, Borel-Cantelli on the sum.

Proof of the existence half, part 2: pathwise convergence and verification.

We keep the notation of part 1, including the gaps \(d_k\), the constants \(A_1\) and \(A\), and, for \(k \geq 1\), the coefficient differences \(\beta^{(k)}\) and \(\gamma^{(k)}\) of Step 3 there.

Step 1 (Tail bounds for the uniform gaps).
Fix \(k \geq 1\). The two iteration identities subtract as processes. For every \(\omega\), the gap \(Y^{(k+1)}_t - Y^{(k)}_t\) is the sum of the difference of the two time integrals and the difference \(U_t\) of the two continuous realizations of the stochastic integrals. On a single full-measure event, the time integrals obey their pathwise identities simultaneously for all \(t\), so there

\[ \sup_{0 \leq t \leq T} \bigl| Y^{(k+1)}_t - Y^{(k)}_t \bigr| \leq \int_0^T \bigl| \beta^{(k)}_s \bigr|\, ds + \sup_{0 \leq t \leq T} \bigl| U_t \bigr| . \]

The process \(U\) is continuous, and at each \(t\) it is almost surely a representative of \(\int_0^t \gamma^{(k)}_s\, dw_s\), by linearity from the properties theorem, so \(U\) is a continuous version of that integral process, which is precisely what the maximal bound asks for. If the right side above stays at or below \(2^{-k-1}\) in each of its two terms, the left side stays at or below \(2^{-k}\), so, up to a null set,

\[ \Bigl\{ \sup_{t \leq T} \bigl| Y^{(k+1)}_t - Y^{(k)}_t \bigr| \gt 2^{-k} \Bigr\} \subseteq \Bigl\{ \int_0^T \bigl| \beta^{(k)}_s \bigr| ds \gt 2^{-k-1} \Bigr\} \cup \Bigl\{ \sup_{t \leq T} | U_t | \gt 2^{-k-1} \Bigr\} , \]

and all three events are measurable, the suprema as countable suprema over rational times of continuous paths. Subadditivity bounds the left probability by the sum of the right two. For the drift, the strict form of the Chebyshev inequality, then Cauchy-Schwarz pathwise and Tonelli in the mean, then (H2) and the factorial bound of part 1,

\[ \begin{align*} \mathbb{P}\Bigl[ \int_0^T \bigl| \beta^{(k)}_s \bigr| ds \gt 2^{-k-1} \Bigr] &\leq 2^{2(k+1)}\, T\, \mathbb{E}\Bigl[ \int_0^T \bigl( \beta^{(k)}_s \bigr)^2 ds \Bigr] \\\\ &\leq 2^{2(k+1)}\, T\, D^2 \int_0^T d_{k-1}(s)\, ds \\\\ &\leq 4^{k+1}\, T\, D^2\, A_1 A^{k-1}\, \frac{T^{k+1}}{(k+1)!} . \end{align*} \]

For the noise, the maximal bound applied to the integrand \(\gamma^{(k)} \in \mathcal{V}(0, T)\), whose membership part 1 verified, at the strict threshold, harmless because the strict event is the smaller one, gives, with (H2), Tonelli, and the factorial bound folding the energy exactly as for the drift,

\[ \mathbb{P}\Bigl[ \sup_{t \leq T} | U_t | \gt 2^{-k-1} \Bigr] \leq 2^{2(k+1)}\, \mathbb{E}\Bigl[ \int_0^T \bigl( \gamma^{(k)}_s \bigr)^2 ds \Bigr] \leq 4^{k+1}\, D^2\, A_1 A^{k-1}\, \frac{T^{k+1}}{(k+1)!} . \]

Writing \(p_k\) for the sum of the two right sides,

\[ \mathbb{P}\Bigl[ \sup_{t \leq T} \bigl| Y^{(k+1)}_t - Y^{(k)}_t \bigr| \gt 2^{-k} \Bigr] \leq p_k = 4^{k+1}\, ( 1 + T )\, D^2\, A_1 A^{k-1}\, \frac{T^{k+1}}{(k+1)!} , \]

and the ratio \(p_{k+1} / p_k = 4 A T / (k + 2)\) falls below \(\tfrac{1}{2}\) for all large \(k\), so \(\sum_k p_k \lt \infty\) by comparison with a geometric series. Factorial beats geometric, once again.

Step 2 (Borel-Cantelli, and the limit acquires paths).
By the Borel-Cantelli lemma, almost every \(\omega\) exceeds its threshold for only finitely many \(k\). For such \(\omega\) the series \(\sum_k \sup_{t \leq T} |Y^{(k+1)}_t(\omega) - Y^{(k)}_t(\omega)|\) converges. Its early terms are finite, because continuous paths are bounded on \([0, T]\), and its tail is dominated by \(\sum_k 2^{-k}\). For \(m \gt n\), the triangle inequality bounds \(\sup_{t} |Y^{(m)}_t(\omega) - Y^{(n)}_t(\omega)|\) by the tail of this series from \(n\), so the paths \(Y^{(n)}(\omega)\) form a uniformly Cauchy sequence of functions on \([0, T]\). Pointwise limits exist by completeness of the reals, and the uniform Cauchy bounds transfer to the limit, so the convergence is uniform. Let \(G \in \mathcal{F}\) be the event on which this happens, \(\mathbb{P}[G] = 1\), and define

\[ \tilde{X}_t(\omega) = \lim_{n \to \infty} Y^{(n)}_t(\omega)\, \mathbf{1}_{G}(\omega) , \quad 0 \leq t \leq T . \]

Every path of \(\tilde{X}\) is continuous. Off \(G\) the path is the zero function, and on \(G\), for \(s\) near \(t\), the gap \(|\tilde{X}_s - \tilde{X}_t|\) is at most twice the uniform distance from \(\tilde{X}\) to \(Y^{(n)}\) plus \(|Y^{(n)}_s - Y^{(n)}_t|\), and choosing \(n\) large and then \(s\) close makes all three terms small. Adaptedness is the delicate point, and it is exactly where the enlarged filtration pays. The complement of \(G\) is a null set, so \(\mathbf{1}_{G}\) is \(\mathcal{H}_0\)-measurable, because \(\mathcal{H}_0\) contains \(\mathcal{N}\) by construction. Each \(Y^{(n)}_t\) is \(\mathcal{H}_t\)-measurable, hence so is the limit of the products, and \(\tilde{X}\) is adapted with continuous paths. Clause (S1) holds.

Step 3 (One limit in two modes).
Set \(\varepsilon_n = \sum_{k \geq n} \bigl( A_1 A^k T^{k+1} / (k+1)! \bigr)^{1/2}\), the tail of the convergent series of part 1, so \(\varepsilon_n \to 0\). At each fixed \(t\), for \(m \gt n\), the triangle inequality in \(L^2(\mathbb{P})\) gives \(\mathbb{E}[ |Y^{(m)}_t - Y^{(n)}_t|^2 ] \leq \varepsilon_n^2\). Letting \(m \to \infty\), the integrands converge to \(|\tilde{X}_t - Y^{(n)}_t|^2\) almost surely on \(G\), and Fatou's lemma preserves the bound:

\[ \sup_{0 \leq t \leq T}\, \mathbb{E}\bigl[ | \tilde{X}_t - Y^{(n)}_t |^2 \bigr] \leq \varepsilon_n^2 \longrightarrow 0 . \]

Two consequences follow at once. First, by the triangle inequality, \(\mathbb{E}[ \tilde{X}_t^2 ]^{1/2} \leq \mathbb{E}[ Z^2 ]^{1/2} + \varepsilon_0 = M\) uniformly in \(t\), so integrating over the horizon, with Tonelli and the progressive measurability that (S1) supplies, \(\mathbb{E}\bigl[ \int_0^T \tilde{X}_t^2\, dt \bigr] \leq M^2 T \lt \infty\), which is clause (S2). Second, integrating the display over \([0, T]\) shows \(Y^{(n)} \to \tilde{X}\) in \(L^2(\lambda \otimes \mathbb{P})\), so \(\tilde{X}\) is a representative of the Riesz-Fischer limit \(X\) of part 1. The equivalence class has acquired a distinguished member, adapted and continuous along every path.

Step 4 (The limit solves).
The substituted coefficients of \(\tilde{X}\) qualify exactly as in Step 1 of part 1. Progressive measurability passes through composition, (H1) and continuity of paths make \(s \mapsto b(s, \tilde{X}_s)\) pathwise integrable, and \(\mathbb{E}[ \int_0^T \sigma(s, \tilde{X}_s)^2 ds ] \leq 2 C^2 T ( 1 + M^2 ) \lt \infty\), so \(\sigma(\cdot, \tilde{X}) \in \mathcal{V}(0, T)\). Clause (S3) holds, and the process

\[ R_t = Z + \int_0^t b\bigl( s, \tilde{X}_s \bigr)\, ds + \int_0^t \sigma\bigl( s, \tilde{X}_s \bigr)\, dw_s , \]

with the \(dw\)-term in its everywhere-continuous realization, is defined, adapted, and continuous along every path, for the same reasons as every iterate. Fix \(t\). By Cauchy-Schwarz, Tonelli, and (H2), then the display of Step 3,

\[ \mathbb{E}\Bigl[ \Bigl( \int_0^t \bigl( b( s, Y^{(n)}_s ) - b( s, \tilde{X}_s ) \bigr) ds \Bigr)^{\!2} \Bigr] \leq T D^2 \int_0^t \mathbb{E}\bigl[ | Y^{(n)}_s - \tilde{X}_s |^2 \bigr] ds \leq T^2 D^2 \varepsilon_n^2 , \]

and by the Itô isometry, whose membership hypothesis holds for the progressively measurable difference with \(\mathbb{E} \int_0^T ( \sigma( s, Y^{(n)}_s ) - \sigma( s, \tilde{X}_s ) )^2 ds \leq T D^2 \varepsilon_n^2 \lt \infty\), the noise terms are bounded by \(T D^2 \varepsilon_n^2\) and vanish in the same way. So the right side of the iteration identity converges to \(R_t\) in \(L^2(\mathbb{P})\). Its left side \(Y^{(n+1)}_t\) converges to \(\tilde{X}_t\) almost surely on \(G\). A sequence with an almost sure limit and an \(L^2\) limit has only one limit, by the same Fatou argument as in Step 3 applied to \(| Y^{(n+1)}_t - R_t |^2\), so \(\tilde{X}_t = R_t\) almost surely.

This holds at each fixed \(t\). A countable union of null sets is null, so almost surely \(\tilde{X}_t = R_t\) simultaneously for every rational \(t \in [0, T]\), and both processes are continuous along every path, so two paths agreeing on a dense set agree everywhere. Clause (S4) holds with the integral equation read simultaneously for all \(t\), and \(\tilde{X}\) is a solution. Together with the uniqueness half proved on the previous page, the theorem is proved in full.

One design decision has now paid its bill. The solution was manufactured as a limit, and limits are blind to individual points of \(\Omega\): the convergence event \(G\) is beyond the reach of any raw filtration at time zero, and a limit taken along it can fail to be measurable for filtrations that ignore null sets. The enlargement of the first page of this pair included \(\mathcal{N}\) in \(\mathcal{H}_0\) before any limit was in sight, and Step 2 above is the moment that choice was for. The event \(G\) slid into \(\mathcal{H}_0\) through its null complement, and adaptedness of the solution came out a one-line argument instead of an obstruction.

The theorem is now a closed account. Coefficients with linear growth and a Lipschitz modulus admit exactly one solution from each square-integrable independent start. The final section asks what lies beyond the account: what the iteration says about the deterministic theory it generalizes, how far the solution's moments can be pushed, and what solving even means when the Lipschitz hypothesis is dropped.

Beyond Lipschitz

The theorem is closed, and this section walks its perimeter. Three questions remain worth asking. What was the proof, secretly, in the language of fixed points. What does the theorem pay back to the deterministic theory it generalizes, and what does it say about the size of its own solutions. And what does solving even mean when the Lipschitz hypothesis, which powered both halves, is taken away.

The Fixed Point Behind the Iteration

The scheme was a fixed-point iteration in disguise. The map that sends a process \(X\) to \(Z + \int_0^t b(s, X_s)\, ds + \int_0^t \sigma(s, X_s)\, dw_s\) has the solutions of the equation as its fixed points, and part 1's contraction estimate says that one application of the map trades the mean-square distance for its own running time integral, the germ of a contraction. Two standard devices make it a literal one, either shrinking the horizon until \(A T \lt 1\) and patching intervals together, or weighting the mean-square norm by \(e^{-\kappa t}\) with \(\kappa\) large. The Banach fixed-point theorem then hands over existence, uniqueness, and, through its convergence rate, a geometric error bound. The route taken on this page skipped the packaging and ran the induction by hand, because the integral structure gives away more than any contraction constant records. The gaps decay factorially, not geometrically, and that surplus is what made every summability check of part 2 a one-line comparison. The fixed-point frame explains why the scheme had to work. The factorial explains why it worked so comfortably.

The Deterministic Dividend

Setting \(\sigma = 0\) makes the noise term vanish and the theorem collapse onto ordinary differential equations, and the collapse is worth recording, because it repays a debt.

Corollary: Global Existence and Uniqueness for ODEs

Let \(b : [0, T] \times \mathbb{R} \to \mathbb{R}\) be jointly Borel measurable and satisfy (H1) and (H2). Then for every \(x_0 \in \mathbb{R}\) there is exactly one continuous function \(x : [0, T] \to \mathbb{R}\) with

\[ x(t) = x_0 + \int_0^t b\bigl( s, x(s) \bigr)\, ds , \quad 0 \leq t \leq T . \]

Proof.

Apply the theorem with \(\sigma \equiv 0\) and \(Z \equiv x_0\), a square-integrable start independent of everything. The noise term is the integral of the zero integrand and vanishes by linearity. For uniqueness, any continuous candidate \(x\) defines a deterministic process, adapted to any filtration, whose clauses (S1) through (S4) are immediate, the integrand \(s \mapsto b(s, x(s))\) being Borel by composition and bounded by (H1) on the bounded range of \(x\). Two such candidates are solutions with the same start, agree almost surely by the uniqueness half, and, being deterministic, agree outright. For existence, the iterates of part 1 are deterministic by induction, since each is built from \(x_0\) and a time integral of a deterministic path, so their almost sure uniform convergence is plain uniform convergence, and the limit is a deterministic continuous function satisfying the equation at every \(t\), both sides of the almost sure identity of part 2 being deterministic.

The debt this repays was recorded on the manifold track. The local existence theorem for integral curves was stated there on trust, as classical ODE theory, and a smooth vector field read in a chart is locally Lipschitz, so the engine behind that classical theory is exactly the iteration of this page, run locally. The corollary proves the globally Lipschitz instance from scratch. What remains on trust is only the localization, the passage from a Lipschitz bound on a neighborhood to a solution on a short interval, and not the engine itself.

How Big Can a Solution Grow

The theory downstream of existence integrates functions of \(X_t\) against its law, and every such computation begins by asking whether the integrals are finite. The construction already contains the answer. Running the self-bound of the solution itself, rather than of a gap, through Gronwall's inequality gives a moment estimate that is uniform over the horizon.

Theorem: Second-Moment Bound for Solutions

Let \(b\) and \(\sigma\) satisfy (H1), and let \(X\) be any solution, in the sense of the definition, with initial value \(Z\). Then

\[ \mathbb{E}\bigl[ X_t^2 \bigr] \leq K_1\, e^{K_2 t} , \quad 0 \leq t \leq T , \]

where \(K_1 = 3\, \mathbb{E}[Z^2] + 6 C^2 T ( T + 1 )\) and \(K_2 = 6\, ( 1 + T )\, C^2\).

Proof.

Write \(v(t) = \mathbb{E}[X_t^2]\), measurable in \(t\) with values in \([0, \infty]\) for now, as Tonelli certifies from the progressive measurability that (S1) supplies. Split the integral equation of (S4) by \((x + y + z)^2 \leq 3 (x^2 + y^2 + z^2)\), control the drift pathwise by Cauchy-Schwarz and in the mean by Tonelli, and compute the noise term by the Itô isometry, whose membership hypothesis is clause (S3). With (H1) and \((1 + x)^2 \leq 2 ( 1 + x^2 )\),

\[ \begin{align*} v(t) &\leq 3\, \mathbb{E}[Z^2] + 3 t\, \mathbb{E}\Bigl[ \int_0^t b( s, X_s )^2\, ds \Bigr] + 3\, \mathbb{E}\Bigl[ \int_0^t \sigma( s, X_s )^2\, ds \Bigr] \\\\ &\leq 3\, \mathbb{E}[Z^2] + 6 C^2\, ( T + 1 ) \int_0^t \bigl( 1 + v(s) \bigr)\, ds \\\\ &\leq K_1 + K_2 \int_0^t v(s)\, ds . \end{align*} \]

The middle line absorbs \(t \leq T\), and the last line spends \(6 C^2 ( T + 1 ) \int_0^t 1\, ds \leq 6 C^2 T ( T + 1 )\) into \(K_1\). Clause (S2) makes \(\int_0^T v\) finite, so the right side of the display is finite and every value \(v(t)\) is finite with it. The hypotheses of Gronwall's inequality are met, with \(K_1\) as the additive constant and \(K_2\) as the multiplier, and the conclusion follows.

The Lipschitz hypothesis plays no role. Growth alone controls growth, and the bound holds for every solution of every equation with linearly growing coefficients, unique or not. When later pages differentiate expectations of functions of \(X_t\) in time, this bound is what makes the integrals finite before any computation starts.

Solving Without Lipschitz: Strong and Weak

Both halves of the theorem ran on (H2), the uniqueness proof through the self-bound of the gap and the existence proof through the contraction of the scheme. Removing it does not merely break the proofs. It splits the notion of solving into two, according to what is given in advance and what is allowed to be built.

Definition: Strong Solution

Let the probability space, the Brownian motion \(w\), and the initial value \(Z\) be given in advance, with \(\{\mathcal{H}_t\}\) the enlarged filtration they generate. A strong solution of the equation with coefficients \(b\) and \(\sigma\) is a process satisfying clauses (S1) through (S4) of the solution definition with respect to \(\{\mathcal{H}_t\}\).

Definition: Weak Solution

Let coefficients \(b\) and \(\sigma\) and a Borel probability measure \(\mu\) on \(\mathbb{R}\) be given. A weak solution is a filtered probability space carrying a filtration \(\{\hat{\mathcal{H}}_t\}\), a standard Brownian motion \(\hat{w}\) that is adapted and whose increments are independent of the running past, the two properties of the enlargement lemma, and a continuous adapted process \(\hat{X}\) with \(\hat{X}_0\) distributed according to \(\mu\), such that clauses (S2) and (S3) of the solution definition hold, their wording unchanged over any filtration, and

\[ \hat{X}_t = \hat{X}_0 + \int_0^t b\bigl( s, \hat{X}_s \bigr)\, ds + \int_0^t \sigma\bigl( s, \hat{X}_s \bigr)\, d\hat{w}_s \]

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

A strong solution is asked to be a functional of prescribed randomness. A weak solution is allowed to bring its own space and its own noise, and only the law of the start and the shape of the equation are prescribed. Every strong solution is a weak solution, its given setting serving as the setting. The uniqueness notions split along the same seam. Pathwise uniqueness asks that two solutions on the same setting with the same start be indistinguishable, and it is what the uniqueness half proved under (H2). Uniqueness in law asks only that any two weak solutions share their finite-dimensional distributions. One statement bridges the two notions under the hypotheses of this pair of pages, and the construction just completed is most of its proof.

Lemma: Uniqueness in Law under the Standard Hypotheses

Let \(b\) and \(\sigma\) satisfy (H1) and (H2) on \([0, T]\), and let \(\mu\) be a Borel probability measure on \(\mathbb{R}\) with \(\int_{\mathbb{R}} x^2 \, d\mu \lt \infty\). Then any two solutions of

\[ d X_t = b(t, X_t)\, dt + \sigma(t, X_t)\, d w_t , \quad X_0 \sim \mu , \]

weak or strong, and living on possibly different spaces, share their finite-dimensional distributions.

Proof Sketch.

The construction travels.
The two existence parts above and the uniqueness half of the main theorem consumed exactly two properties of the ambient filtration, adaptedness of the driving motion and independence of its increments from the running past, together with the membership clauses of the solution definition. The definition of a weak solution demands these same properties, and the finite second moment of \(\mu\) supplies what the constant \(A_1\) asked of the initial value. Both arguments therefore run verbatim on the space of any weak solution. Given a weak solution \(\bigl( \tilde{X}, \tilde{w} \bigr)\) over a filtration \(\{\tilde{\mathcal{H}}_t\}\), run the Picard iteration on that space, driven by \(\tilde{w}\) and started from \(\tilde{X}_0\). It produces a solution there, and pathwise uniqueness on that same space identifies the product with \(\tilde{X}\) up to indistinguishability. Every weak solution is thus the Picard limit over its own noise.

The inputs agree in law.
Let \(\bigl( \hat{X}, \hat{w} \bigr)\) over \(\{\hat{\mathcal{H}}_t\}\) be a second weak solution with the same initial law. On each space, independence of the increments from the running past telescopes into independence of the whole driving path from the time-zero \(\sigma\)-algebra, by the factorization argument that proved the enlargement lemma. The pair of initial value and driving path therefore carries the same law on both spaces, the product of \(\mu\) with the law of the Brownian path.

The law rides the iteration.
Write \(Y^{(k)}\) and \(\hat{Y}^{(k)}\) for the iterates on the two spaces. By induction, the pairs \(\bigl( Y^{(k)}, \tilde{w} \bigr)\) and \(\bigl( \hat{Y}^{(k)}, \hat{w} \bigr)\) have the same law for every \(k\). The inductive step is the one assertion this sketch takes on trust. Passing from \(k\) to \(k + 1\) applies a time integral and an Itô integral to the current iterate, the same limiting recipe on both spaces, and the law of the output of that recipe depends only on the joint law of its inputs. The machinery for pushing laws through the Itô construction has not been built on this track, and we do not build it here.

Pass to the limit.
On each space the iterates converge almost surely uniformly to the solution, by part 2 of the existence proof run on that space. Finite-dimensional distributions survive almost sure limits, so \(\bigl( \tilde{X}, \tilde{w} \bigr)\) and \(\bigl( \hat{X}, \hat{w} \bigr)\) agree in law, and in particular \(\tilde{X}\) and \(\hat{X}\) do.

The split is not decorative, and one equation shows it. Consider

\[ d X_t = \operatorname{sign}( X_t )\, d w_t , \quad X_0 = 0 , \quad \operatorname{sign}(x) = \begin{cases} +1 , & x \geq 0 , \\ -1 , & x \lt 0 , \end{cases} \]

whose diffusion coefficient jumps at the origin and satisfies no Lipschitz bound there, so the theorem is silent. Three statements describe it, and this track records them without proof, since their verification runs through the machinery of local time, which has not been built here. The equation has no strong solution at all. It has weak solutions, and in fact any Brownian motion serves as one once the driving noise is rebuilt from the solution itself. And it is weakly unique, every weak solution being a Brownian motion in law. Prescribed noise admits nothing. Self-built noise admits exactly one law. The weak concept is also the natural one for modelling, where an equation asserts how a system responds to white noise without asserting which white noise the world will supply.

Neural SDEs and Generative Diffusion

The hypotheses of the theorem are load-bearing in modern machine learning. A neural SDE takes the coefficients \(b\) and \(\sigma\) to be neural networks, and a network built from Lipschitz activations with norm-controlled weight matrices satisfies (H2) with a constant \(D\) read off from the product of the layer norms, while (H1) follows as soon as the coefficients at the origin stay bounded in time, which fixed network weights guarantee. The theorem is then the well-posedness certificate for the entire model class. It guarantees that the network-defined equation has exactly one solution per realization of the noise, so that the model is a genuine map from randomness to trajectories, which is what training and sampling implicitly assume. Generative diffusion models lean on the same foundation from the analytic side. Their samplers integrate a reverse-time equation whose drift contains a learned score, and existence and uniqueness are what make the generator a well-defined function of its noise seed, while a-priori estimates in the style of the second-moment bound are the standard opening move of their stability analyses. The theorem does not train anything. It is the reason the objects being trained exist.

One last account closes the pair. Everything on these two pages was one-dimensional, and the systems version, promised at the start of the previous page, is recorded here at statement level in this track's conventions: the state \(X_t\) takes values in \(\mathbb{R}^m\), the driving motion is \(n\)-dimensional with independent components, the coefficients are \(b : [0, T] \times \mathbb{R}^m \to \mathbb{R}^m\) and \(\sigma : [0, T] \times \mathbb{R}^m \to \mathbb{R}^{m \times n}\), and the hypotheses (H1) and (H2) read verbatim with Euclidean norms on vectors and \(|\sigma|^2 = \sum_{i,j} \sigma_{ij}^2\) on matrices. The theorem holds as stated. Each coordinate of the integral equation is a scalar identity of exactly the kind treated here, the mean-square estimates add across coordinates with constants that grow only by dimension-dependent factors, and Gronwall's inequality and the whole toolkit act on scalar functions of time and never notice the dimension. The account opened by the example pages, a formula here and a verification there, each with a reservation attached, is closed.