Mfrhtyj

.mw-parser-output table.ambox{margin:0 10%;border-collapse:collapse;background:#fbfbfb;border:1px solid #aaa;border-left:10px solid #608ec2}.mw-parser-output table.ambox th.ambox-text,.mw-parser-output table.ambox td.ambox-text{padding:.25em .5em;width:100%}.mw-parser-output table.ambox td.ambox-image{padding:2px 0 2px .5em;text-align:center;vertical-align:middle}.mw-parser-output table.ambox td.ambox-imageright{padding:2px 4px 2px 0;text-align:center;vertical-align:middle}.mw-parser-output table.ambox-notice{border-left:10px solid #608ec2}.mw-parser-output table.ambox-delete,.mw-parser-output table.ambox-serious{border-left:10px solid #b22222}.mw-parser-output table.ambox-content{border-left:10px solid #f28500}.mw-parser-output table.ambox-style{border-left:10px solid #f4c430}.mw-parser-output table.ambox-merge{border-left:10px solid #9932cc}.mw-parser-output table.ambox-protection{border-left:10px solid #bba}.mw-parser-output .ambox+.ambox,.mw-parser-output .topbox+.ambox,.mw-pars...

0 ÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_beta$. $$P{T_beta leq t}=P{max_{0leq sleq t} B(s) geq beta}=$$ $$=2P{B(t) geq beta} = sqrt{frac{2}{pi}} int_{frac{beta}{sqrt{t}}}^{infty} exp{-frac{w^2}{2}}dw$$ Because the book used the last result : $$P{T_beta leq t}=P{max_{0leq sleq t} B(s) geq beta}= 2P{B(t) geq beta}$$ Now I don't undersatnd why $T_beta$ has the inverse gaussian density $$f_{beta}(t)= frac{|beta|}{sqrt{2pi t^3}}expleft{-frac{beta^2}{2t}right}$$ for $t >0$. Where did it find? inverse brownian-motion stopping-times share | cite | improve this question ...