Proof of Existence for SDEs

We prove existence of strong solutions via Picard iteration.

Step 1: Define the sequence $X_t^{(0)} = x_0$ 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: Use the Lipschitz condition to show $\mathbb{E}[\sup_{s \leq t} |X_s^{(n+1)} - X_s^{(n)}|^2] \leq K^n t^n / n!$, proving the sequence is Cauchy.

Step 3: The limit $X_t = \lim_{n} X_t^{(n)}$ exists in $L^2$ and satisfies the SDE.

Step 4: Uniqueness follows from showing two solutions must coincide using Gronwall's inequality.