E[N_t|\mathcal{F}_0]&=&E[N_t1_{\{S=\infty\}}+N_t1_{\{S<\infty\}}|\mathcal{F}_0]\\
&=&E[M_t1_{\{S=\infty\}}|\mathcal{F}_0]+E[M_{S\wedge t}1_{\{S<\infty\}}|\mathcal{F}_0]
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  1. http://formula.s21g.com/?E[N_t%7C%5Cmathcal%7BF%7D_0]%26%...
  2. https://mkprob.hatenablog.com/entry/20120726/1343262108
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