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Find the expression for stochastic process $X_t$ given that it follows the stochastic differential equation:

$dX_t = \left( \frac{1}{(1+t)^2} - \frac{2}{1+t}X_t\right)dt + \frac{1}{(1+t)^2}dW_t$

This is my first time approaching the topic of SDEs and I'm having difficulties with this problem. I do know we can integrate both sides so that $\int_{t=0}^T dX_t = X_T - X_0$, but I do not know how to approach taking Ito integral of the right side. Can anyone describe the process?

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1 Answer 1

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This SDE falls under the linear sdes,

$\begin{align*} \mathrm{d}X_t = (a(t)X_t+ b(t)) \mathrm{d}t + (g(t)X_t+ h(t))\mathrm{d}B_t \end{align*}$

which are solved here Solution to General Linear SDE and so we get

$$\begin{align*} X_t = & X_0 e^{ \int_0^t \frac{-2}{1+s}\mathrm{d}s}\\ &+ e^{ \int_0^t \frac{-2}{1+s}\mathrm{d}s}\left( \int_0^t e^{ -\int_0^s \frac{-2}{1+r}\mathrm{d}r}\frac{1}{(1+s)^{2}} \mathrm{d}s + \int_0^t e^{ -\int_0^s \frac{-2}{1+r}\mathrm{d}r}\frac{1}{(1+s)^{2}} \mathrm{d}B_s\right). \end{align*}$$

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