Existence and Uniqueness Theorem for SDEs

Theorem7.1Strong existence and uniqueness

If $b(t,x)$ and $\sigma(t,x)$ are Lipschitz in $x$ uniformly in $t$ and satisfy linear growth, 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 any finite interval $[0,T]$ .

The proof uses Picard iteration and Gronwall's inequality. Extensions to non-Lipschitz coefficients (e.g., Yamada-Watanabe conditions) allow weak uniqueness.