User: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= $$\bigg\langle\int_0^tG(s)dB(s)\bigg\rangle=0$$

$$\bigg\langle\int_0^tG(s)dB(s)\cdot\int_0^tH(s)dB(s)\bigg\rangle= \int_0^t\langle G(s)H(s)\rangle ds$$

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

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

$$\phi(t)=\phi_0+\frac{\omega_0}{\gamma}\left(1-e^{-\gamma t}\right)+\sigma\int_0^te^{\gamma s}B(s)\,ds$$

$$ d\omega_t=-\gamma \,\underbrace{\omega_t\,dt}_{d\phi_t}+\sigma dB_t$$

$$\omega_t=\frac{d\phi(t)}{dt}=\omega_0-\gamma\left(\phi_t-\phi_0\right)+\sigma B_t$$

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: $$\begin{align}\mathrm{d}\! \left(\int_0^t\frac{dB(s)}{1-s}\right)&=\mathrm{d}\! \left( \sum_{j=0}^{n-1} \frac{B((j+1)t/n)-B(jt/n)}{1-jt/n}\right)\\ &=\left(\sum_{j=1}^{n} \frac{B((j+1)t/n)-B(jt/n)}{1-jt/n}\right)-\left(\sum_{j=0}^{n-1} \frac{B((j+1)t/n)-B(jt/n)}{1-jt/n}\right)\\ &=\frac{B((n+1)t/n)-B(t)}{1-t}-\left[B(t/n)-B(0)\right]\\ &\to\frac{dB(t)}{1-t} \end{align}$$

3

 * $$ dX(t) = a(X,t)\,dt + b(X,t)\,dB_t $$

where Wt 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
$$dX_t=-\gamma X_tdt+\sigma dB_t$$ $$e^{\gamma t}$$ $$X_t=X_0e^{-\gamma t}+\sigma\int e^{\gamma(s-t)}dB(s)$$
 * $$ dN_t = r N_t\,dt + \alpha N_t\,dB_t $$
 * $$d\left(\ln(N_t)\right)$$
 * $$N_t=N_0\exp\left[\left(r-\alpha^2/2\right)t+\alpha B_t\right]$$
 * $$X_0e^{-\gamma t}$$
 * $$\frac{\sigma^2}{2\gamma}\left(1-e^{-2\gamma t}\right)$$

=Phys413/Phyg610 Temp Space=