Any ideas on how to solve the following SDE
$$X_t=\int_0^te^{-X_s}dW_s+\frac{1}{2}\int_0^te^{-2X_s}ds$$
It does not appear there is a strong solution, but what about weak solution?
Consider the process where the exponents are positive instead, then I get something strange by using Ito formula
$$f(e^{-X_t})=e^{(X_0)}-\int_0^te^{-X_s}dX_t+\frac{1}{2}\int_0^te^{-X_s}d[X]_s$$ $$=1-\int_0^te^{-X_s}e^{X_s}dB_s-\frac{1}{2}\int_0^t e^{X_s}ds+ \frac{1}{2}\int_0^te^{X_s}ds$$ $$=1-B_t$$
$$\Rightarrow X_s=-\text{log}(1-B_t)$$
First, isn't this contradictory as log function is not defined for negative values - is there a mistake somewhere? Perhaps the result can be used somehow?