⧼exchistory⧽
1 exercise(s) shown, 0 hidden
BBy Bot
May 08'24
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Let [math](\Omega,\F,(\F_n)_{n\geq0},\p)[/math] be a filtered probability space. Let [math]X=(X_n)_{n\geq 0}[/math] be a supermartingale and let [math]T[/math] be a stopping time. Then
[[math]]
X_T\in L^1(\Omega,\F,(\F_n)_{n\geq 0},\p)
[[/math]]
and
[[math]]
\E[X_T]\leq \E[X_0]
[[/math]]
in each case of the following situations.
- [math]T[/math] is bounded.
- [math]X[/math] is bounded and [math]T[/math] is finite.
- [math]\E[T] \lt \infty[/math] and for some [math]k\geq 0[/math], we have
[[math]] \vert X_n(\omega)-X_{n-1}(\omega)\vert\leq k, [[/math]]for all [math]\omega\in\Omega[/math].