Radial basis function network

In the stochastic calculus, Tanaka's formula states that

$|B_{t}|=\int _{0}^{t}\operatorname {sgn}(B_{s})\,dB_{s}+L_{t}$

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

$\operatorname {sgn}(x)={\begin{cases}+1,&x\geq 0;\\-1,&x<0.\end{cases}}$

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

$L_{t}=\lim _{\varepsilon \downarrow 0}{\frac {1}{2\varepsilon }}|\{s\in [0,t]|B_{s}\in (-\varepsilon ,+\varepsilon )\}|.$

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 $f(x)=|x|$, with $f'(x)=\operatorname {sgn}(x)$ 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

${\frac {x^{2}}{2|\varepsilon |}}+{\frac {|\varepsilon |}{2}}.$

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.
  
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• 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