&&cP[R_c<\infty]\\ &=&E[1_{\{R_c<\infty\}}Z_{R_c}]\quad (Z_t\stackrel{\rm def}{=}\exp(\nu W_t-\frac{1}{2}\nu^2 t))\\ &=&\lim_{t\to\infty}E[1_{\{R_c\leq t\}}Z_{R_c}]\quad ({\rm by\ the\ monotone\ convergence\ thoerem})\\ &=&\lim_{t\to\infty}E[1_{\{R_c\leq t\}}Z_{R_c\wedge t}]\\ &=&\lim_{t\to\infty}E[1_{\{R_c\leq t\}}E[Z_t|\mathcal{F}_{t\wedge R_c}]]\quad ({\rm by\ the\ optional\ sampling\ theorem})\\ &=&\lim_{t\to\infty}E[E[1_{\{R_c\leq t\}}Z_t|\mathcal{F}_{t\wedge R_c}]]\\ &=&\lim_{t\to\infty}E[1_{\{R_c\leq t\}}Z_t]\\ &=&\lim_{t\to\infty}(EZ_t-E[1_{\{R_c> t\}}Z_t])\\ &=&1\quad ({\rm by\ the\ Lebesgue\ convergence\ theorem})
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Recent Referers:
  1. http://formula.s21g.com/?%26%26cP[R_c%3C%5Cinfty]%5C%5C%0...
  2. https://mkprob.hatenablog.com/entry/20120920/1348137802
  3. http://formula.s21g.com/
  4. http://mkprob.hatenablog.com/entry/20130220/1361363246
  5. http://d.hatena.ne.jp/mkprob/20120920/1348137802