Lipschitz existence and pathwise uniqueness

For each finite horizon, require a common Lipschitz constant for drift and diffusion as functions of the state, uniformly over time up to that horizon. Use Euclidean norm for the drift and Hilbert–Schmidt norm for the diffusion matrix. Also require at most linear growth in the state with a finite horizon-dependent constant. The latter follows from global Lipschitz control if the coefficient values at zero are uniformly bounded on that horizon, but should not be silently omitted when those values have not been controlled.

Under these hypotheses, the equation has a unique continuous adapted strong solution with a finite expected squared supremum on the horizon. Applying the statement consistently on longer horizons gives a nonexplosive global solution. The constants may depend on the horizon; no claim of a time-uniform moment bound or an invariant distribution follows. A process can be globally defined while its moments grow exponentially, as happens for a linear equation with positive drift.

Picard iteration starts with the initial value as a constant process and repeatedly substitutes the preceding approximation into both integrals. Cauchy–Schwarz controls the drift integral, while Itô isometry and a martingale maximal inequality control the stochastic integral. Lipschitz continuity turns the resulting difference estimate into an integral of the preceding difference. Repeating this estimate gives factorial decay on a bounded horizon, producing a limit in expected squared supremum and allowing the equation to pass to that limit.

For two solutions driven by the same Brownian motion, the identical estimates bound their expected squared difference by a constant times its own time integral, plus the initial difference. Gronwall's inequality forces equality when the initial variables agree and gives quantitative continuous dependence otherwise. The noise terms cannot simply be canceled for a general state-dependent diffusion coefficient: they have the same Brownian driver but different integrands. The Lipschitz estimate, rather than literal cancellation, is what closes the argument.

Locally Lipschitz coefficients permit truncation outside expanding balls. The resulting global solutions agree before their exit times by uniqueness, giving a maximal local solution. Its lifetime is the limit of those exit times. Local Lipschitz regularity alone does not force that lifetime to be infinite: the deterministic equation with drift equal to the square of the state already explodes from a positive initial value. Smoothness does not compensate for uncontrolled growth.

Linear growth, or an appropriate Lyapunov estimate, supplies a separate nonexplosion argument. Stopped moment bounds make the probability of leaving a large ball before a fixed time tend to zero, after which the stopping levels can be removed. Conversely, failure of a sufficient Lipschitz condition does not prove nonuniqueness. Some less regular diffusions remain well posed through sharper one-dimensional arguments. A theorem's hypotheses are a usable sufficient criterion, rather than a complete classification of every solvable coefficient.