fixed error
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2
Makefile
2
Makefile
@ -1,5 +1,5 @@
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pdf: init
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pdf: init
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latexmk < /dev/null
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latexmk -halt-on-error < /dev/null
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clean:
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clean:
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latexmk -c
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latexmk -c
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@ -208,7 +208,7 @@ we need the following theorem, which we won't prove here:
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then $X^T$ is also a supermartingale,
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then $X^T$ is also a supermartingale,
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and we have $\bE[X_{T \wedge n}] \le \bE[X_0]$ for all $n$.
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and we have $\bE[X_{T \wedge n}] \le \bE[X_0]$ for all $n$.
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If $(X_n)_n$ is a martingale, then so is $X^T$
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If $(X_n)_n$ is a martingale, then so is $X^T$
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and $\bE[X_{T \wedge n}] \le \bE[X_0]$.
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and $\bE[X_{T \wedge n}] = \bE[X_0]$.
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\end{theorem}
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\end{theorem}
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\begin{proof}
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\begin{proof}
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First, we need to show that $X^T$ is adapted.
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First, we need to show that $X^T$ is adapted.
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