Adriferr

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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

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

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

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

4

$dX_{t}=-\gamma X_{t}dt+\sigma dB_{t}$ $e^{\gamma t}$ $X_{t}=X_{0}e^{-\gamma t}+\sigma \int e^{\gamma (s-t)}dB(s)$

$dN_{t}=rN_{t}\,dt+\alpha N_{t}\,dB_{t}$

$d\left(\ln(N_{t})\right)$

$N_{t}=N_{0}\exp \left[\left(r-\alpha ^{2}/2\right)t+\alpha B_{t}\right]$

$X_{0}e^{-\gamma t}$

${\frac {\sigma ^{2}}{2\gamma }}\left(1-e^{-2\gamma t}\right)$