Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
ito_integral [2016/07/06 13:00] nikolaj |
ito_integral [2016/07/06 13:14] nikolaj |
||
---|---|---|---|
Line 119: | Line 119: | ||
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 155: | Line 155: | ||
I.e. the Itō integral is defined with most left of the grid cells as in the explicit Euler-method numerical approximation scheme. The implicit Euler-method simply corresponds to a different notion of integral here and would give a different result. | I.e. the Itō integral is defined with most left of the grid cells as in the explicit Euler-method numerical approximation scheme. The implicit Euler-method simply corresponds to a different notion of integral here and would give a different result. | ||
- | In quantum mechanics, the difference of the product above | + | In quantum mechanics, the difference of the products |
$[x,p]_{\Delta t}:=x(t+\delta^2{\Delta t})\,p_{\Delta t}(t)-x(t)\,p_{\Delta t}(t)$ | $[x,p]_{\Delta t}:=x(t+\delta^2{\Delta t})\,p_{\Delta t}(t)-x(t)\,p_{\Delta t}(t)$ | ||
- | is $m\,\kappa^2=m\frac{i\hbar}{2m}=\frac{i\hbar}{2}$. | + | equals $m\,\kappa^2=m\frac{i\hbar}{2m}=\frac{i\hbar}{2}$. |
== Fractional quantum mechanics == | == Fractional quantum mechanics == | ||
Line 170: | 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, |