Stochastic Differential Equations

The SDE section writes a state process as a drift plus a diffusion term,

$$dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dW_t.$$

The coefficients describe deterministic motion and random shocks. Unlike an ordinary differential equation, an SDE is interpreted through an integral equation and a chosen filtration. The notes compare continuous diffusions with jump processes and use Itô's formula to transform solutions.

For pricing, this notation is valuable because a model can be specified by its local characteristics even when there is no closed-form path. Existence, integrability, and the chosen measure determine whether the process is usable as a financial model.