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$\text { Consider the } \operatorname{SDE} d X_{t}=\operatorname{sign}\left(X_{t}\right) d t+d B_{t} \text { on } 0 \leq t \leq T, \text { where } \operatorname{sign}(x)=1\\ \text { for } x>0 \text { and } \operatorname{sign}(x)=-1 \text { for } x \leq 0 \text { . Show that it has a weak solution. }$

I am not sure if this is the way I am suppose to answer the question but this is how I did it:

$d x_{t}-\operatorname{sign}\left(x_{t}\right) d t=d B_{t}$

$\text { let } Y_{t}=\int_{0}^{t}-\operatorname{sign}\left(X_{s}\right) d s+\int_{0}^{t} d X_{s}$

We can see here that $\int_{0}^{t}\left|\operatorname{sign}\left(x_{s}\right)\right| d s<\infty$ as sign$(X_s)$ is bounded between -1 and 1.

We can also see that $\int_{0}^{t}\left(d x_{s}\right)^{2}=T<\infty$

Hence $Y_t$ is well defined and is a martingale as $[y, y]_{t}=\int_{0}^{t} \operatorname{sign}^{2}\left(X_{s}\right) d s=T \text { and }$ $\int_{0}^{t} d X_{S}=\int_{0}^{t} d t=T$

Therefore $[Y, Y]_{t}$ is a constant value which is a martingale. And $Y_t$ is Brownian Motion that satisfies the SDE $dX_t$

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Based on Karatzas and Shreve's book, section 5.3.B, Weak Solutions By Means of Girsanov, Proposition 3.6., "the principal method for creating weak solutions is the transformation of drift via the Girsanov theorem". Their proof of the proposition illustrates the approach.

The only condition that needs to be met is for the drift to have bounded growth: if

$$ |b(t,x)| \leq K(1+|x|) $$

for all $t$ and $x$, for some positive constant $K$, then SDE

$$ dX_t = b(t,X_t) dt + dB_t $$

has a weak solution.

Function $\operatorname{sign}$ does meet the condition:

$$ |\operatorname{sign}(x)| < 1 + |x| $$

for all $x$.

Notes:

  1. Contrast with the 'usual' conditions for strong solutions: both drift and diffusion coefficients are to satisfy global linear growth and Lipschitz conditions. Note that $\operatorname{sign}$ fails Lipschitz badly.

  2. The other thing worth mentioning is that, for practical investigations, one can approximate $\operatorname{sign}$ by a sequence of smooth functions that converges pointwise to it (see this article for such research):

$$ f_n(x) = -1_{x<-1/n} + (-n^3x^3/2 + 3nx/2)1_{-1/n\leq x \leq 1/n} + 1_{x>1/n} $$ and study SDE $$ dX_t = f_n(X_t) dt + dB_t. $$

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