Existence and Uniqueness for SDEs

Writing down a stochastic differential equation like $dX_t = \mu(X_t, t)\,dt + \sigma(X_t, t)\,dW_t$ doesn't automatically mean it describes one well-behaved process — much like an ordinary differential equation can blow up to infinity in finite time or have several equally valid solutions from the same starting point, an SDE can too, if its drift $\mu$ (the average direction of movement) and diffusion $\sigma$ (the amount of randomness) are badly behaved functions.

The standard fix is the same one used for ordinary differential equations, adapted to the stochastic case: if $\mu$ and $\sigma$ satisfy a Lipschitz condition (they don't change too abruptly as $X_t$ changes) and a linear growth condition (they don't grow faster than linearly in $X_t$, ruling out the kind of runaway feedback that causes finite-time blow-up), then the SDE is guaranteed to have exactly one solution, and that solution stays finite over any finite time horizon.

This matters practically because every everyday model — geometric Brownian motion, the Ornstein-Uhlenbeck process behind mean-reverting rates, the CIR process behind Heston's variance — was specifically built with drift and diffusion terms satisfying these conditions, which is precisely why quants can plug numbers into these models without first re-deriving that a solution exists. The one common exception is CIR-type variance processes, whose square-root diffusion term technically violates the Lipschitz condition at zero, requiring a separate (though well-established) existence argument — and this same technicality is behind the well-known Feller condition that determines whether such a process can actually touch zero or stays strictly positive.