Existence and Uniqueness for Stochastic Differential Equations

A stochastic differential equation (SDE) is an equation of the form $dX_t = b(t, X_t) dt + \sigma(t, X_t) dB_t$ with initial condition $X_0 = x_0$ . The fundamental question: under what conditions does a unique solution exist?

Statement

Theorem7.1Existence and uniqueness

Suppose $b$ and $\sigma$ satisfy:

Lipschitz continuity: $|b(t,x) - b(t,y)| + |\sigma(t,x) - \sigma(t,y)| \leq K|x-y|$ .
Linear growth: $|b(t,x)| + |\sigma(t,x)| \leq K(1 + |x|)$ .

Then the SDE $dX_t = b(t, X_t) dt + \sigma(t, X_t) dB_t$ with $X_0 = x_0$ has a unique strong solution on $[0,T]$ for any $T > 0$ .

Strong solution: A process $(X_t)$ adapted to the filtration of $B_t$ satisfying the SDE.

ExampleOrnstein-Uhlenbeck process

$dX_t = -\theta X_t dt + \sigma dB_t$ , with $b(x) = -\theta x$ and $\sigma(x) = \sigma$ (constant). Both are Lipschitz and have linear growth, so a unique solution exists. The solution is:

$X_t = e^{-\theta t} X_0 + \sigma \int_0^t e^{-\theta(t-s)} dB_s,$

a Gaussian process with mean $e^{-\theta t} X_0$ and stationary variance $\sigma^2/(2\theta)$ .

Proof technique: Picard iteration

Step 1: Define $X_t^{(0)} = x_0$ and

$X_t^{(n+1)} = x_0 + \int_0^t b(s, X_s^{(n)}) ds + \int_0^t \sigma(s, X_s^{(n)}) dB_s.$

Step 2: Show $\{X_t^{(n)}\}$ is Cauchy in the space of continuous processes with $\mathbb{E}[\sup_{0 \leq s \leq T} |X_s|^2] < \infty$ using Gronwall's inequality.

Step 3: The limit $X_t = \lim_{n \to \infty} X_t^{(n)}$ exists and solves the SDE.

Weak solutions

Definition7.1Weak solution

A weak solution is a pair $(X_t, B_t)$ on some probability space such that $X_t$ satisfies the SDE, where $B_t$ may not be a given Brownian motion.

Weak solutions exist under weaker conditions (continuity instead of Lipschitz), but may not be unique. Pathwise uniqueness implies uniqueness in law, but not conversely (Yamada-Watanabe theorem).

Summary

Under Lipschitz and linear growth conditions, SDEs have unique strong solutions constructed via Picard iteration. Weaker conditions lead to weak solutions with potential non-uniqueness.