Convergence of Picard Iterates on a Finite Horizon

From Factorial Bounds to Convergence

Picard iteration builds a sequence of adapted processes by substituting the previous process into the drift and diffusion coefficients of an SDE. A bound on each successive difference is useful only if the sum of the differences is controlled: that is what lets us prove the sequence is Cauchy. On a fixed finite horizon, the factorial in the denominator supplies precisely this control, even after taking square roots to obtain the \(L^2\) supremum norm.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\). Fix \(T>0\), and consider the scalar SDE with initial value \(\xi\), where \(\xi\) is \(\mathcal{F}_0\)-measurable and \(\mathbb{E}[\xi^2]<\infty\). Assume that \(\mu,\sigma:[0,T]\times\mathbb{R}\to\mathbb{R}\) are jointly Borel measurable and that, for constants \(L,A<\infty\),

for every \(t\in[0,T]\) and \(x,y\in\mathbb{R}\). The last two assumptions imply linear growth: \(|\mu(t,x)|\le A+L|x|\) and \(|\sigma(t,x)|\le A+L|x|\).

Joint Borel measurability matters: since each Picard iterate is continuous and adapted, it is progressively measurable, and composing it with the jointly Borel coefficients gives progressively measurable coefficient processes. The drift integrals below are defined as time integrals. Every Ito integral below has a progressively measurable integrand \(H\) in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). The linear growth bounds and the moment estimates in the construction ensure these integrability conditions.

The Difference Estimate and the Cauchy Argument

Set \(X^0_t=\xi\), and define the Picard iterates by

Write \(\Delta_n=X^{n+1}-X^n\) and \(D_n(t)=\mathbb{E}[\sup_{0\le u\le t}|\Delta_n(u)|^2]\). The estimate below uses the Ito isometry and the standard Doob \(L^2\) maximal inequality, \(\mathbb{E}[\sup_{u\le t}|M_u|^2]\le4\mathbb{E}[|M_t|^2]\) for a square-integrable martingale \(M\). These are standard stochastic-integral norm estimates.

Proof. For \(n\ge1\), subtract the formulas for \(X^{n+1}\) and \(X^n\). The drift integrand is \(\mu(s,X^n_s)-\mu(s,X^{n-1}_s)\), and the Ito integrand is \(\sigma(s,X^n_s)-\sigma(s,X^{n-1}_s)\). The Lipschitz bounds control the squared absolute value of either difference by \(L^2|\Delta_{n-1}(s)|^2\). By Cauchy–Schwarz, the squared supremum of the drift integral up to \(t\) is at most \(t\int_0^t L^2|\Delta_{n-1}(s)|^2\,ds\). The Ito integral is a square-integrable martingale: its integrand is progressively measurable and square-integrable by the preceding moment bounds, and its conditional expectation increments are zero. The Ito isometry and Doob inequality therefore bound its expected squared supremum by \(4L^2\int_0^t\mathbb{E}[|\Delta_{n-1}(s)|^2]\,ds\). Using \(|a+b|^2\le2|a|^2+2|b|^2\) gives $$ D_n(t)\le 2L^2(t+4)\int_0^t D_{n-1}(s)\,ds \le K\int_0^t D_{n-1}(s)\,ds. $$ The initial difference has finite \(D_0(T)\). Indeed, \(\Delta_0=X^1-X^0\), and the same integral estimates, now using linear growth at \(\xi\), give $$ D_0(T)\le (4T^2+16T)\big(A^2+L^2\mathbb{E}[\xi^2]\big)<\infty. $$ Induction in the recursive inequality yields the factorial bound: it holds for \(n=0\); if it holds for \(n-1\), then $$ D_n(t)\le K\int_0^t D_0(T)\frac{(Ks)^{n-1}}{(n-1)!}\,ds =D_0(T)\frac{(Kt)^n}{n!}. $$

It remains to pass from differences to the Cauchy property. For \(n\ge0\), the norm of \(\Delta_n\) is at most \(\sqrt{D_0(T)}(KT)^{n/2}/\sqrt{n!}\). The series of these bounds converges: when \(K T>0\), the ratio of consecutive terms is \(\sqrt{KT/(n+1)}\), which is at most \(1/2\) for all sufficiently large \(n\). If \(KT=0\), the recursive estimate makes all differences after the initial one zero. For \(m>n\), the triangle inequality gives $$ \|X^m-X^n\|_{2,\infty;T} \le\sum_{j=n}^{m-1}\|\Delta_j\|_{2,\infty;T}. $$ The right-hand side tends to zero uniformly in \(m>n\) as \(n\to\infty\), because it is bounded by a tail of the convergent series. This proves the Cauchy assertion. \(\square\)

This argument proves Cauchy convergence in the stated norm; it does not itself identify a limit process in the space. That requires a completeness argument. Nor does it yet show that any limit satisfies the SDE: passing the limit through the drift and Ito integrals is a separate step.

Worked Difference Estimates

Why the Horizon and the Norm Matter

The constants in the recursive estimate depend on the chosen horizon: \(K=2L^2(T+4)\). The conclusion is therefore a finite-horizon statement, valid for each fixed \(T\). It does not assert a single bound uniform over all \(T\ge0\). On any specified finite interval, however, the factorial dominates the fixed exponential factor \((KT)^n\), and the square-root factorial still leaves a summable series.

A common mistake is to stop after observing \(D_n(T)\to0\). That only says the individual differences become small; it does not by itself prove that the tails \(\sum_{j=n}^{m-1}\Delta_j\) become small uniformly over \(m\). The correction is to bound the norms of the differences and show their series converges, as in the ratio argument. Another pitfall is to omit time measurability: Lipschitz and growth bounds in \(x\) alone do not ensure the coefficient processes are valid integrands. Joint Borel measurability supplies the needed measurability under the stated construction.