some little improvement

Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex

@@ -58,9 +58,12 @@
     Recall that $s\wedge t:=\min(s,t)$ and $s\vee t:=\max(s,t)$.
   \end{remark}
   \subsubsection{Martingales}
+  \begin{definition}
+    Let ${(X_t)}_{t\geq 0}$ be a stochastic process. We define the \emph{natural filtration} of $X$ as $\mathcal{F}^X:={(\mathcal{F}_t^X)}_{t\geq 0}$, where $\mathcal{F}_t^X:=\sigma(X_s:s\leq t)$.
+  \end{definition}
   From now on, we will assume that we work in a filtered probability space $(\Omega,\mathcal{F},\Prob,{(\mathcal{F}_t)}_{t\geq 0})$.
   \begin{proposition}
-    Let ${(B_t)}_{t\geq 0}$ be a Brownian motion. Then, the following processes are martingales ${(M_t)}_{t\geq 0}$ with respect to the filtration induced by ${(B_t)}_{t\geq 0}$:
+    Let $B={(B_t)}_{t\geq 0}$ be a Brownian motion. Then, the following processes are martingales ${(M_t)}_{t\geq 0}$ with respect to the natural filtration induced by $B$:
     \begin{itemize}
       \item $M_t=B_t$
       \item $M_t=B_t^2-t$