<div dir="ltr"><div>Hi,</div><div><br></div><div>I am struggling here with some processes that have integral with W. Could you please give me some guidance if the way I am thinking is correct?</div><div><br></div><div><br></div><div>2015 midterm ex 1)</div><div><br></div><div><br></div><div><div class="va_le" align="center" style="display:block"><img alt="\frac{\partial}{\partial W(t)}\int_{0}^{t} (W(u)^3 -1) du " title="\frac{\partial}{\partial W(t)}\int_{0}^{t} (W(u)^3 -1) du " class="va_ld" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cdisplaystyle%09%5Cfrac{%5Cpartial}{%5Cpartial%09W(t)}%5Cint%5F{0}%5E{t}%09(W(u)%5E3%09-1)%09du%09" id="l0.9740872621577457" style="display: block;" height="44" width="193"></div><br><br></div><div><div>We get that this is 0. My intuition (after seeing the solution) is that this is zero because every increment in of W is independent of W(t).</div><div><br></div><div>Assignment 4 -- ex 1 d)</div><div><br></div><div><div class="gmail-va_le" align="center" style="display:block"><br></div><div class="gmail-va_le" align="center" style="display:block"><img alt="\frac{\partial}{\partial t} \int_{0}^{W(t)} e^{-ts^2} ds" title="\frac{\partial}{\partial t} \int_{0}^{W(t)} e^{-ts^2} ds" class="gmail-va_ld" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cdisplaystyle%09%5Cfrac{%5Cpartial}{%5Cpartial%09t}%09%5Cint%5F{0}%5E{W(t)}%09e%5E{-ts%5E2}%09ds" id="gmail-l0.4112549885205148" style="display: block;" height="46" width="125"></div><br>Here my understanding is that we should apply the Leibniz Rule and set W(t) as a constant when we do the partial derivative in relation to t. <br></div></div><div><br></div><div><div class="gmail-m_-3660482582878540534gmail_signature"><div dir="ltr"><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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