. PR ] 1 6 N ov 1 99 9 Splitting : Tanaka ’ s SDE revisited
نویسنده
چکیده
1 What follows is my attempt to understand a set of ideas being developed by Boris Tsirelson. I do this by studying a specific, and I hope interesting, example. Tanaka's SDE is one of the easiest examples of a stochastic differential equation with no strong solution. Suppose X t ; t ≥ 0 is a real-valued Brownian motion starting from zero and we put B t = t 0 sgn(X s)dX s then B is also a Brownian motion and Tanaka's SDE X t = t 0 sgn(X s)dB s , (1) is satisfied. But the trajectory of B does not determine that of X. Recall Tanaka's formula |X t | = t 0 sgn(X s)dX s + L t , (2) where L t ; t ≥ 0 is the local time process of X at zero. We find |X t | = B t + sup s≤t (−B s) (3) but B does not tell us the signs of the excursions from zero made by X.
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