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Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE

$dX_t = dW_t + dL_t^0(X_t)$,

where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at $0$. I have two related questions:

  1. Does there exist a (weak) solution to the SDE

$dX_t = dW_t + 2 dL_t^0(X_t)$

where again $L_t^0(X_t)$ is the local time of $X_t$ at $0$?

  1. What about the SDE

$dX_t = \sqrt{X_t}dW_t + 2 dL_t^1(X_t)$,

where $L_t^1(X_t)$ is the local time of $X_t$ at $1$?

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For the first see ch 6 prob 2.24 of Revuz & Yor

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  • $\begingroup$ Thanks, this led me to the paper of Harrison-Shepp which describes the resulting process: On skew Brownian motion. $\endgroup$ – ysys Dec 31 '15 at 6:46

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