Edit this page (or any of the subsections by clicking "edit" next to the heading). Add, change, correct, modify anything you like. All the changes are logged anyway. If you think you know something (even just a vague general idea), just post it, and we'll see how much we can get out of it.

Phys558 Temp Space

hah. If you thought the 2005 final was bad, go look at the one with carbon nanotube crystals. That's crazy. I can't even find papers on how to do it.

Math580 Temp Space

${\bigg \langle }\int _{0}^{t}G(s)dB(s){\bigg \rangle }=0$

${\bigg \langle }\int _{0}^{t}G(s)dB(s)\cdot \int _{0}^{t}H(s)dB(s){\bigg \rangle }=\int _{0}^{t}\langle G(s)H(s)\rangle ds$

$\sum _{i=0}^{n-1}f(t_{i})\{B_{t_{i+1}}-B_{t_{i}}\}$

${\ddot {\phi }}=-\gamma {\dot {\phi }}+\sigma \eta (t)$

$\phi (t)=\phi _{0}+{\frac {\omega _{0}}{\gamma }}\left(1-e^{-\gamma t}\right)+\sigma \int _{0}^{t}e^{\gamma s}B(s)\,ds$

$d\omega _{t}=-\gamma \,\underbrace {\omega _{t}\,dt} _{d\phi _{t}}+\sigma dB_{t}$

$\omega _{t}={\frac {d\phi (t)}{dt}}=\omega _{0}-\gamma \left(\phi _{t}-\phi _{0}\right)+\sigma B_{t}$

Missing: Show that: $\mathrm {d} \!\left(\int _{0}^{t}{\frac {dB(s)}{1-s}}\right)=\int _{0}^{t}\mathrm {d} \left({\frac {dB(s)}{1-s}}\right)={\frac {dB(t)}{1-t}}$

Attempt: ${\begin{aligned}\mathrm {d} \!\left(\int _{0}^{t}{\frac {dB(s)}{1-s}}\right)&=\mathrm {d} \!\left(\sum _{j=0}^{n-1}{\frac {B((j+1)t/n)-B(jt/n)}{1-jt/n}}\right)\\&=\left(\sum _{j=1}^{n}{\frac {B((j+1)t/n)-B(jt/n)}{1-jt/n}}\right)-\left(\sum _{j=0}^{n-1}{\frac {B((j+1)t/n)-B(jt/n)}{1-jt/n}}\right)\\&={\frac {B((n+1)t/n)-B(t)}{1-t}}-\left[B(t/n)-B(0)\right]\\&\to {\frac {dB(t)}{1-t}}\end{aligned}}$

3

dX(t)=a(X,t)\,dt+b(X,t)\,dB_{t}

where W_t is a Wiener process, and let f(x, t) be a function with continuous second derivatives.

Then $f(x(t),t)$ is also an Itō process, and

df(X(t),t)={\frac {\partial f}{\partial t}}dt+{\frac {\partial f}{\partial X}}\underbrace {\left[a(X,t)dt+b(X,t)\,dB_{t}\right]} _{dX_{t}}+\underbrace {{\frac {1}{2}}b(X,t)^{2}{\frac {\partial ^{2}f}{\partial X^{2}}}} _{\mathrm {Ito} \;\mathrm {correction} }dt

df(x(t),t)={\frac {\partial f}{\partial t}}dt+{\frac {\partial f}{\partial x}}dx

$\langle {\dot {\phi }}\rangle =\omega _{0}e^{-\gamma t}$ $var({\dot {\phi }})=\sigma ^{2}t$ $(dB_{t})^{2}=dt$ $d(B_{T}^{n})$ $d(B_{T}^{n})=nB_{t}^{n-1}dB_{t}+\left[{\frac {n(n-1)}{2}}B_{t}^{n-2}\,dt\right]$