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Michael Hardy
Michael Hardy
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We consider the following SDE: $$dX_t = 1(X_t = 0) dt + 1(X_t >0) dB_t, \quad X_0= x > 0,$$$$dX_t = 1(X_t = 0) \, dt + 1(X_t >0) \, dB_t, \quad X_0= x > 0,$$ where $(B_t, \, t \ge 0)$ is linear Brownian motion. Let $\tau: = \inf\{t >0: X_t = 0\}$ be the first time at which $X$ hits $0$. It is obvious that $(X_t, \, 0 \le t \le \tau)$ is Brownian motion up to the first hitting to $0$.

Question: Can we say something for $X$ after $\tau$? Is it well-defined?

It is clear that $X$ is not reflected Brownian motion but is also supported on $[0,\infty)$. It seems that there is accumulation of zeros after $\tau$ which would lead to local times.

We consider the following SDE: $$dX_t = 1(X_t = 0) dt + 1(X_t >0) dB_t, \quad X_0= x > 0,$$ where $(B_t, \, t \ge 0)$ is linear Brownian motion. Let $\tau: = \inf\{t >0: X_t = 0\}$ be the first time at which $X$ hits $0$. It is obvious that $(X_t, \, 0 \le t \le \tau)$ is Brownian motion up to the first hitting to $0$.

Question: Can we say something for $X$ after $\tau$? Is it well-defined?

It is clear that $X$ is not reflected Brownian motion but is also supported on $[0,\infty)$. It seems that there is accumulation of zeros after $\tau$ which would lead to local times.

We consider the following SDE: $$dX_t = 1(X_t = 0) \, dt + 1(X_t >0) \, dB_t, \quad X_0= x > 0,$$ where $(B_t, \, t \ge 0)$ is linear Brownian motion. Let $\tau: = \inf\{t >0: X_t = 0\}$ be the first time at which $X$ hits $0$. It is obvious that $(X_t, \, 0 \le t \le \tau)$ is Brownian motion up to the first hitting to $0$.

Question: Can we say something for $X$ after $\tau$? Is it well-defined?

It is clear that $X$ is not reflected Brownian motion but is also supported on $[0,\infty)$. It seems that there is accumulation of zeros after $\tau$ which would lead to local times.

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KDD
KDD
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A singular stochastic differential equation

We consider the following SDE: $$dX_t = 1(X_t = 0) dt + 1(X_t >0) dB_t, \quad X_0= x > 0,$$ where $(B_t, \, t \ge 0)$ is linear Brownian motion. Let $\tau: = \inf\{t >0: X_t = 0\}$ be the first time at which $X$ hits $0$. It is obvious that $(X_t, \, 0 \le t \le \tau)$ is Brownian motion up to the first hitting to $0$.

Question: Can we say something for $X$ after $\tau$? Is it well-defined?

It is clear that $X$ is not reflected Brownian motion but is also supported on $[0,\infty)$. It seems that there is accumulation of zeros after $\tau$ which would lead to local times.