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ito_integral [2016/07/06 13:01] nikolaj |
ito_integral [2016/07/06 13:14] nikolaj |
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| One writes | One writes | ||
| - | ${\mathrm d}X_t = \mu_t(X_s, s) \, {\mathrm d}t + \sigma_t(X_s, s) \, {\mathrm d}W_t$ | + | ${\mathrm d}X_t = \mu_t(X_t, t) \, {\mathrm d}t + \sigma_t(X_t, t) \, {\mathrm d}W_t$ |
| If $X_t$ isn't known, this is called a stochastic differential equation in $X_t$. | If $X_t$ isn't known, this is called a stochastic differential equation in $X_t$. | ||
| Line 159: | Line 159: | ||
| $[x,p]_{\Delta t}:=x(t+\delta^2{\Delta t})\,p_{\Delta t}(t)-x(t)\,p_{\Delta t}(t)$ | $[x,p]_{\Delta t}:=x(t+\delta^2{\Delta t})\,p_{\Delta t}(t)-x(t)\,p_{\Delta t}(t)$ | ||
| - | is | + | equals $m\,\kappa^2=m\frac{i\hbar}{2m}=\frac{i\hbar}{2}$. |
| - | + | ||
| - | $m\,\kappa^2=m\frac{i\hbar}{2m}=\frac{i\hbar}{2}$. | + | |
| == Fractional quantum mechanics == | == Fractional quantum mechanics == | ||
| Line 172: | Line 170: | ||
| $P(\Delta x) \propto \exp\left(c\frac{(\Delta x)^2}{\Delta t}\right)$ | $P(\Delta x) \propto \exp\left(c\frac{(\Delta x)^2}{\Delta t}\right)$ | ||
| - | as next-step distribution, and then | + | as next-step distribution, and then $\langle |x|\rangle\propto t^{1/2}$ gives the non-smooth curve. |
| - | + | ||
| - | $\langle |x|\rangle\propto t^{1/2}$ | + | |
| - | + | ||
| - | gives the non-smooth curve. | + | |
| You may want to look at other next-step distributions, | You may want to look at other next-step distributions, | ||
