Tanaka's Formula
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High Quality Content by WIKIPEDIA articles! In the stochastic calculus, Tanaka's formula states that B_t = int_0^t sgn(B_s), dB_s + L_t where Bt is the standard Brownian motion, sgn denotes the sign function sgn (x) = begin{cases} +1, & x geq 0; -1, & x 0. end{cases} and Lt 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} frac1{2 varepsilon} { s in [0, t] B_{s} in (- varepsilon, + varepsilon) } . Tanaka's formula is the explicit Doob Meyer decomposition of the submartingale Bt into the martingale part (the integral...
High Quality Content by WIKIPEDIA articles! In the stochastic calculus, Tanaka's formula states that B_t = int_0^t sgn(B_s), dB_s + L_t where Bt is the standard Brownian motion, sgn denotes the sign function sgn (x) = begin{cases} +1, & x geq 0; -1, & x 0. end{cases} and Lt 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} frac1{2 varepsilon} { s in [0, t] B_{s} in (- varepsilon, + varepsilon) } . Tanaka's formula is the explicit Doob Meyer decomposition of the submartingale Bt 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 Ito's Lemma for the (nonsmooth) absolute value function f(x) = x , with f'(x) = sgn(x) and f''(x) = 2 (x); see local time for a formal explanation of the Ito term.
