the existence-and-uniqueness theorem under Lipschitz conditions
This is the foundational well-posedness theorem for SDEs: it gives clean, checkable conditions on the coefficients b and sigma under which the equation dX = b(X) dt + sigma(X) dB has a unique strong solution for every initial condition. It is the stochastic analogue of the Picard-Lindelof theorem for ODEs, and it is the result you invoke first whenever you write down a new SDE and want to know it actually defines a process.
The hypotheses are a global Lipschitz condition and a linear-growth condition. Lipschitz: there is a constant K with |b(x) - b(y)| + |sigma(x) - sigma(y)| <= K |x - y| for all x, y. Linear growth: |b(x)| + |sigma(x)| <= K(1 + |x|). Under these (with a square-integrable, independent initial condition), there exists a unique continuous adapted process X_t solving the integral equation, it is a strong solution, and it has finite moments: E[sup_{s <= T} |X_s|^2] < infinity. The proof is Picard iteration in the stochastic setting: define X^(n+1)_t = X_0 + integral_0^t b(X^(n)_s) ds + integral_0^t sigma(X^(n)_s) dB_s, estimate the L^2 distance between successive iterates using the Cauchy-Schwarz bound on the drift term and the Ito isometry E[(integral sigma dB)^2] = E[integral sigma^2 ds] on the diffusion term, then apply Gronwall's inequality to get a contraction; the limit solves the SDE and uniqueness follows from the same estimate. Linear growth is what prevents the solution from blowing up to infinity in finite time (no explosion).
Why it matters and where the conditions bite: global Lipschitz buys you both existence and pathwise uniqueness (hence, by Yamada-Watanabe, a unique strong solution), and linear growth buys you non-explosion and finite moments. The honest caveats are real. Drop Lipschitz and you can lose uniqueness (the deterministic dx/dt = sqrt(|x|) with x_0 = 0 already has many solutions). Drop linear growth and the solution can explode in finite time (dX = X^2 dt blows up). Many important SDEs — geometric Brownian motion (globally Lipschitz, fine), but CIR/Bessel (square-root diffusion, only locally Holder), or SDEs with superlinear drift — fall outside the theorem's exact hypotheses and need refined tools: local Lipschitz + a Lyapunov function for non-explosion, or the Yamada-Watanabe square-root condition for pathwise uniqueness.
Geometric Brownian motion dX = mu X dt + sigma X dB has b(x) = mu x and sigma(x) = sigma x, both globally Lipschitz with constant max(|mu|, |sigma|) and of linear growth, so the theorem applies directly and gives a unique strong solution with all moments finite. By contrast dX = X^2 dt violates linear growth and its solution explodes in finite time.
Lipschitz gives uniqueness, linear growth prevents explosion — and dropping either breaks the conclusion.
Linear growth is not optional: without it the solution can explode in finite time (dX = X^2 dt). And Lipschitz is global here; merely local Lipschitz needs an extra non-explosion (Lyapunov) argument to extend the solution for all time.