<div dir="ltr">Hi Lucas,<div><br></div><div>Remember when applying Ito's Lemma to a process <img alt="X_t" title="X_t" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09X%5Ft" id="gmail-l0.0856551180636913" height="13" style="display: inline; vertical-align: -2.2px;" width="17"> (in this case <img alt="X_t = \int_0^tW(u)^3-1du" title="X_t = \int_0^tW(u)^3-1du" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09X%5Ft%09=%09%5Cint%5F0%5EtW(u)%5E3-1du" id="gmail-l0.8424815584188432" style="display: inline; vertical-align: -5.867px;" height="22" width="154">) you define a function <img alt="f(t,x)" title="f(t,x)" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09f(t,x)" id="gmail-l0.8041729860346349" style="display: inline; vertical-align: -3.667px;" height="16" width="43"> such that <img alt="f(t,W_t) = X_t" title="f(t,W_t) = X_t" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09f(t,W%5Ft)%09=%09X%5Ft" id="gmail-l0.21856172912871807" height="16" style="display: inline; vertical-align: -3.667px;" width="97">. So here we define <img alt="f(t,x) = \int_0^tW(u)^3-1du" title="f(t,x) = \int_0^tW(u)^3-1du" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09f(t,x)%09=%09%5Cint%5F0%5EtW(u)%5E3-1du" id="gmail-l0.589756704933279" style="display: inline; vertical-align: -5.867px;" height="22" width="181"> which has no x dependence so <img alt="\frac{\partial}{\partial x} f(t,x) = 0" title="\frac{\partial}{\partial x} f(t,x) = 0" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09%5Cfrac{%5Cpartial}{%5Cpartial%09x}%09f(t,x)%09=%090" id="gmail-l0.3703771990974216" height="21" style="display: inline; vertical-align: -5.867px;" width="92">. What may have confused you is that we don't replace <img alt="W(u)" title="W(u)" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09W(u)" id="gmail-l0.46367344606276784" height="16" style="display: inline; vertical-align: -3.667px;" width="38"> -- we would only replace <img alt="W(t)" title="W(t)" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09W(t)" id="gmail-l0.4252192768199212" style="display: inline; vertical-align: -3.667px;" height="16" width="34">'s with x.</div><div><br></div><div>Similarly for Assignment 4 define <img alt="f(t,x) = \int_0^xe^{-ts^2}ds" title="f(t,x) = \int_0^xe^{-ts^2}ds" class="gmail-va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09f(t,x)%09=%09%5Cint%5F0%5Exe%5E{-ts%5E2}ds" id="gmail-l0.07034608286057464" height="22" style="display: inline; vertical-align: -5.867px;" width="140"> and then you can apply regular calculus rules to obtain the derivatives you need. Remember Brownian Motion is not differentiable so writing <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="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09%09%5Cfrac{%5Cpartial}{%5Cpartial%09t}%09%5Cint%5F0%5E{W(t)}e%5E{-ts%5E2}ds" id="l0.2326767734013273" height="24" style="display: inline; vertical-align: -5.867px;" width="111"> does not make sense.</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 Sun, Nov 19, 2017 at 11:56 AM, 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"><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="m_-8836562330807141834va_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="m_-8836562330807141834va_ld" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cdisplaystyle%09%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%09W(t)%7D%5Cint%5F%7B0%7D%5E%7Bt%7D%09(W(u)%5E3%09-1)%09du%09" id="m_-8836562330807141834l0.9740872621577457" 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="m_-8836562330807141834gmail-va_le" align="center" style="display:block"><br></div><div class="m_-8836562330807141834gmail-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="m_-8836562330807141834gmail-va_ld" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cdisplaystyle%09%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%09t%7D%09%5Cint%5F%7B0%7D%5E%7BW(t)%7D%09e%5E%7B-ts%5E2%7D%09ds" id="m_-8836562330807141834gmail-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="m_-8836562330807141834gmail-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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