<div dir="ltr">Hi Lucas,<div><br></div><div>I figured it out after you left office hours -- by definition of conditional expectation we have that <img alt="\int_A X dP = \int_A E[X|\mathcal{F}] dP \ \forall \ A \in \mathcal{F}" title="\int_A X dP = \int_A E[X|\mathcal{F}] dP \ \forall \ A \in \mathcal{F}" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09%5Cint%5FA%09X%09dP%09=%09%5Cint%5FA%09E[X|%5Cmathcal{F}]%09dP%09%5C%09%5Cforall%09%5C%09A%09%5Cin%09%5Cmathcal{F}" id="gmail-l0.8087272519338846" height="19" width="245" style="display: inline; vertical-align: -5.501px;"> so we get the equality by just applying this definition to the random variable <img alt="Z(T)\tilde{E}(X|\mathcal{F}_s)" title="Z(T)\tilde{E}(X|\mathcal{F}_s)" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09Z(T)%5Ctilde{E}(X|%5Cmathcal{F}%5Fs)" id="gmail-l0.5946236877249782" height="19" width="100" style="display: inline; vertical-align: -3.667px;">. </div><div><br></div><div>Hope this helps,</div><div><br></div><div>David</div></div><div class="gmail_extra"><br><div class="gmail_quote">On Fri, Dec 15, 2017 at 1:19 PM, Lucas Duarte Bahia <span dir="ltr"><<a href="mailto:lduarteb@andrew.cmu.edu" target="_blank">lduarteb@andrew.cmu.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">Professor,<div><br></div><div>Could you please explain the following step in Lemma 1.11 in the notes?</div><div><br></div><div><img src="cid:ii_1605b666475908eb" alt="Inline image 1" width="182" height="64"><br clear="all"><div><div class="m_-3619069669695387705gmail_signature" data-smartmail="gmail_signature"><div dir="ltr"><p style="font-size:12.8px"><br></p><p style="font-size:12.8px">Thanks,</p><div style="font-size:12.8px"><b>Lucas Duarte Bahia</b></div><div style="font-size:12.8px">MS. Computational Finance Student</div><div style="font-size:12.8px">Carnegie Mellon University</div><div style="font-size:12.8px">Telephone: <a href="tel:(412)%20378-1892" value="+14123781892" style="color:rgb(17,85,204)" target="_blank">(412) 378-1892</a></div></div></div></div>
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