I am new to stochastic differential equation and ran into a question of solving $$dX_t=\sqrt t(X_t+\sin t)dW_t$$ where $W_t$ is the standard Wiener Process and $X_0 \equiv K\in \mathbb R$.

I know Ito's formula and could solve some easy SDEs using the formula. However, this one, I cannot find the appropriate function to use. Any hint would be appreciated! Thanks and regards.


The SDE is a particular example of a so-called linear SDE

$$dX_t = (\alpha(t)+\beta(t) X_t) \, dt + (\gamma(t)+\delta(t) X_t) \, dW_t \tag{1}$$

where $\alpha, \beta,\gamma,\delta$ are deterministic functions. Such linear SDEs can be solved explicitly, and you can find formula for the solution for instance in the book Brownian motion - An Introduction to Stochastic Processes by Schilling & Partzsch. The idea is to solve first the homogeneous SDE

$$dX_t = \beta(t) X_t \, dt + \delta(t) X_t \, dW_t$$

and then to use a "variation of constants"-approach, see this question. For the particular case that $\alpha=\beta=0$ the solution to $(1)$ is given by

$$X_t = \exp \left( M_t \right) \left[ X_0 + \int_0^t \exp(-M_s) \gamma(s) \, dW_s - \int_0^t \exp(-M_s) \gamma(s) \delta(s) \, ds \right]$$


$$M_t := \int_0^t \delta(s) \, dW_s - \frac{1}{2} \int_0^t \delta(s)^2 \, ds.$$

Plugging in $\delta(t) = \sqrt{t}$ and $\gamma(t) = \sqrt{t} \sin t$ gives the solution to the SDE

$$dX_t = \sqrt{t} (X_t+\sin t) \, dW_t. \tag{2}$$

You can use the approach, which I mentioned above, to "reprove" the formula for the solution, i.e. first solve the SDE

$$dX_t = \sqrt{t} X_t \, dW_t$$

and then use the "variation of constants"-approach to obtain the solution to $(2)$.


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