> [!definition] > > Let $\bracs{X_n}_0^\infty$ be a discrete-time, $\bracs{\cf_n}_0^\infty$-[[Filtration|adapted]] [[Stochastic Process|stochastic process]], and let $\bracs{C_n}_1^\infty$ be the *investment* into the process, such that in each unit of time, the profit/loss is $C_{n + 1}(X_{n + 1} - X_n)$. > > The process $\bracs{C_n}_1^\infty$ is $\bracs{\cf_n}_0^\infty$-**previsible** if $C_{n + 1} \in L^1(\Omega, \cf_n, \bp)$ for all $n \ge 0$. In other words, future investments are chosen based on past observations, in which case > $ > \begin{align*} > \Delta_{n + 1} &= \ev\braks{C_{n + 1}(X_{n + 1} - X_{n})} \\ > &= \int \ev\bracs{C_{n + 1}(X_{n + 1} - X_{n})|\cf_n} \\ > &= \ev(C_{n + 1}\ev\bracs{X_{n + 1}|\cf_n}) - \ev(C_{n + 1}X_n) > \end{align*} > $