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Is there a way to solve the following for $y(t)$:

$$ y'(t)a(t)+y(t)b(t) + c(t)X_t=0 $$

where, $X_t$ is a stochastic process satisfying $dX_t=X_t(\mu(t)dt+\sigma(t)dW_t)$? Here $W_t$ is a Brownian Motion.

What is the general concept that deals with such problems? Are there some assumptions that must be satisfied for it to be solvable?

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Since the stochastic process $X_t$ appears only in the source term, you can solve this differential equation with respect to $y(t)$ as usual, hence $$ y(t) = y(0)e^{F(t)} - \int_0^t \frac{c(s)}{a(s)} X_se^{F(t)-F(s)} \mathrm{d}s, $$ where $F(t) = -\displaystyle\int_0^t \frac{b(\tau)}{a(\tau)} \mathrm{d}\tau$.

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  • $\begingroup$ furthermore, if you can sample different implementations of the stochastic input, and thus get sampled solution for the distribution of input noise $\endgroup$ Commented yesterday

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