Radial basis function network

In the stochastic calculus, Tanaka's formula states that

$\left|B_t\right|=\underset{0}{\overset{t}{\int }}\mathrm{sgn}(B_s)d{B}_s+L_t$

where B_t is the standard Brownian motion, sgn denotes the sign function

$\mathrm{sgn}(x)=\{\begin{array}{ll}+1,& x\ge 0;\\ -1,& x<0.\end{array}$

and L_t is its local time at 0 (the local time spent by B at 0 before time t) given by the L²-limit

$L_t=\underset{\epsilon \downarrow 0}{\lim }\frac{1}{2\epsilon }|\{s\in [0,t]|B_s\in (-\epsilon ,+\epsilon )\}|.$

Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale |B_t| into the martingale part (the integral on the right-hand side), and a continuous increasing process (local time). It can also be seen as the analogue of Itō's lemma for the (nonsmooth) absolute value function $f(x)=\left|x\right|$, with $f^{\prime }(x)=\mathrm{sgn}(x)$ and $f^{″}(x)=2\delta (x)$; see local time for a formal explanation of the Itō term.

Outline of proof

The function |x| is not C² in x at x = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero (i.e. in [−ε, ε]) by parabolas

$\frac{x^2}{2\left|\epsilon \right|}+\frac{\left|\epsilon \right|}{2}.$

And using Itō's formula we can then take the limit as ε → 0, leading to Tanaka's formula.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Example 5.3.2)
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534