Definition

Let $(\Omega,\mathcal{F},P)$ be a probability space. A market model is a pair $$((S_{t})_{t\ge 0},(A_{t})_{t\ge 0})$$of processes, with $(S_{t})_{t\ge 0}$ representing the “risky” asset and $(A_{t})_{t\ge 0}$ representing the “risk-free” asset.

We denote by $(\mathcal{F}_{t})_{t\ge 0}$ the filtration generated by $(S_{t})_{t\ge 0},(A_{t})_{t\ge 0}$. $(\mathcal{F}_{t})_{t\ge 0}$ is called the filtration of the market model.

We assume $(S_{t})_{t\ge 0}$ is governed by an Itô diffusion, i.e. SDE of the form $$\begin{align*} dS_{t}&=f(t,S_{t})dt+g(t,S_{t})dB_{t}\\ dA_{t}&=r_{t}A_{t}dt \end{align*}$$