The Global Lipschitz Existence and Uniqueness Theorem

The Global Lipschitz Theorem

For an SDE, global Lipschitz continuity controls how rapidly the coefficients can change with the state, while linear growth prevents them from becoming too large as the state grows. Together these assumptions let a Picard iteration converge on each finite time interval. The same estimates also force any two solutions driven by the same Brownian motion to agree.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) satisfying the usual conditions and carrying a standard Brownian motion \(B_t\). Consider the scalar SDE with deterministic initial value \(x_0\):

The stochastic integral here and below is the Itô integral. Its integrand must be progressively measurable and in \(\mathcal{L}^2\): for every finite \(T\), \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). A solution is a continuous adapted process satisfying the displayed integral equation almost surely for all \(t\le T\), with \(\sigma(s,X_s)\) in this integrand class. We assume \(\mu,\sigma\) are measurable in time and state and that constants \(L,K<\infty\) satisfy, for all \(t\) and \(x,y\in\mathbb{R}\),

Picard Iteration and Its Convergence

Define \(X^0_t=x_0\) and, for \(n\ge0\), set

Each stochastic integrand is progressively measurable because \(X^n\) is adapted and continuous and the coefficients are measurable. The growth bound ensures the integrals are defined once \(\mathbb{E}\sup_{s\le T}|X^n_s|^2<\infty\). This finiteness holds inductively: it holds for the constant \(X^0\); if it holds for \(X^n\), the growth assumption gives square-integrable drift and diffusion integrands, and the Itô isometry and the Doob \(L^2\) maximal inequality give a finite second moment for the supremum of \(X^{n+1}\). The Itô isometry and the martingale property of square-integrable Itô integrals are standard results; the maximal inequality is also standard.

For clarity, if \(H\in\mathcal{L}^2\), then \(M_t=\int_0^tH_s\,dB_s\) is a square-integrable martingale: \(\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s\) for \(s\le t\), and \(\mathbb{E}[M_t^2]=\mathbb{E}\int_0^tH_r^2\,dr\). The maximal inequality used below is \(\mathbb{E}\sup_{s\le t}|M_s|^2\le4\mathbb{E}|M_t|^2\). Its proof is a standard martingale inequality and is not repeated here.

Let \(D_n(t)=\mathbb{E}\sup_{u\le t}|X^{n+1}_u-X^n_u|^2\). For the first difference, \(X^1-X^0\), Cauchy–Schwarz for the drift and the maximal inequality for the stochastic integral give

For \(n\ge1\), apply the same estimates to the difference of two consecutive Picard equations. The drift difference is bounded by \(L|X^n-X^{n-1}|\), and so is the diffusion difference. Since \(t\le T\),

Iterating this integral inequality, starting from \(D_0(t)\le A\), yields \(D_n(t)\le A(Ct)^n/n!\). Indeed, integrating \(A(Cs)^{n-1}/(n-1)!\) from \(0\) to \(t\) gives \(A(Ct)^n/n!\). Therefore

The numerical series converges by the ratio test: the ratio of successive terms is \(\sqrt{CT/(n+1)}\), which tends to zero. Minkowski’s inequality now shows that the iterates are Cauchy in the norm \((\mathbb{E}\sup_{s\le T}|X_s|^2)^{1/2}\). Completeness gives a limit \(X\) in that norm. In particular, each \(X^n_t\) converges to \(X_t\) in \(L^2(\mathbb{P})\), so the limit is adapted. A subsequence converges uniformly almost surely: choose a subsequence whose successive differences have \(L^2\)-supremum norms at most \(2^{-2j}\); Markov’s inequality makes the probabilities that the corresponding supremum differences exceed \(2^{-j}\) summable. Borel–Cantelli then gives uniform convergence along that subsequence almost surely, so \(X\) has continuous paths.

It remains to check the equation. The Lipschitz assumption and convergence in the supremum \(L^2\) norm imply, for \(t\le T\),

and the same bound holds with \(\sigma\) in place of \(\mu\). Cauchy–Schwarz makes the drift integrals converge in \(L^2\); the Itô isometry makes the stochastic integrals converge in \(L^2\) at each fixed \(t\). Passing to the limit in the Picard equation proves the integral equation at each fixed \(t\). The same estimates for suprema of the drift and stochastic integral differences give convergence uniformly in time in \(L^2\); hence the equality holds simultaneously for all \(t\le T\) after choosing continuous versions. The integrand \(\sigma(s,X_s)\) is in \(\mathcal{L}^2\), since \(X\) has finite second moment of its supremum and the coefficients have linear growth. This proves strong existence.

Why the Uniqueness Estimate Applies to Every Solution

A subtle point in pathwise uniqueness is that one cannot apply a supremum second-moment estimate to an arbitrary solution before knowing that the supremum has finite second moment. The definition of solution already requires \(\sigma(s,X_s)\in\mathcal{L}^2\), but this directly controls the stochastic integral, not yet the supremum of \(X\). The following estimate fills that gap.

Proof. Let \(M_t=\int_0^t\sigma(s,X_s)\,dB_s\). Because \(\sigma(s,X_s)\in\mathcal{L}^2\), \(M\) is a square-integrable martingale, with conditional expectation \(\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s\) and \(\mathbb{E}|M_T|^2=\mathbb{E}\int_0^T\sigma(s,X_s)^2\,ds<\infty\). Doob’s \(L^2\) inequality gives \(\mathbb{E}\sup_{s\le T}|M_s|^2\le4\mathbb{E}\int_0^T\sigma(s,X_s)^2\,ds<\infty\). The linear-growth assumption implies \(|\mu(t,x)|\le K(1+|x|)\). For each path, write \(S_t=\sup_{u\le t}|X_u|\). The integral equation gives \(S_t\le |x_0|+KT+\sup_{u\le T}|M_u|+K\int_0^tS_s\,ds\). Gronwall’s inequality, applied pathwise with the finite random constant \( |x_0|+KT+\sup_{u\le T}|M_u|\), yields \(S_T\le (|x_0|+KT+\sup_{u\le T}|M_u|)e^{KT}\). Squaring and taking expectations gives a finite result, since \(\mathbb{E}\sup_{u\le T}|M_u|^2<\infty\). \(\square\)

Now let \(X\) and \(Y\) be any two solutions with the same initial value and Brownian motion. The proposition applies to both, so \(u(t)=\mathbb{E}\sup_{s\le t}|X_s-Y_s|^2\) is finite. Subtract their integral equations. Cauchy–Schwarz bounds the squared supremum of the drift difference by \(t\) times the integral of its squared integrand; the Doob inequality and Itô isometry bound the stochastic difference by four times the integral of its squared integrand. Using the Lipschitz bound and \(t\le T\) gives

Gronwall’s inequality with zero initial term implies \(u(t)=0\) for every \(t\le T\). In particular, \(\mathbb{E}\sup_{s\le T}|X_s-Y_s|^2=0\). A nonnegative random variable with expectation zero is zero almost surely, so the paths agree for all \(s\le T\) outside one null event. This proves pathwise uniqueness and completes the theorem.

Worked Examples

What the Theorem Does—and Does Not—Say

The drift coefficient and diffusion coefficient have different roles in the SDE: the drift is integrated against time, while the diffusion is integrated with respect to Brownian motion. In this theorem, both are subject to the same global Lipschitz control, and their combined size is controlled by linear growth. These assumptions are sufficient conditions; they are not asserted to be necessary for existence or uniqueness.

The Picard construction is a strong-existence argument because every iterate uses the given \(B_t\) and is adapted to the given filtration. Pathwise uniqueness compares solutions on that same space with that same Brownian motion. Neither statement should be confused with uniqueness in law, which compares probability distributions of solutions and is a distinct property.

Common mistake. It is not justified to begin the uniqueness proof by writing \(u(t)=\mathbb{E}\sup_{s\le t}|X_s-Y_s|^2\) and applying the stochastic maximal inequality without first establishing that this expectation is finite. The gap is real: being a solution initially gives square integrability of \(\sigma(s,X_s)\), not by definition the supremum moment of \(X\). The correction is to first use the square-integrable martingale \(M_t=\int_0^t\sigma(s,X_s)\,dB_s\), the maximal inequality, linear growth and pathwise Gronwall to prove \(\mathbb{E}\sup_{s\le T}|X_s|^2<\infty\) for each solution. Only then are the difference estimates and Gronwall applicable.