I am working on the following problem in a course using "Stochastic Differential Equations" by Oksendal.
Consider the two coupled It SDEs.
$dXt=−λX_tdt+σdB_t$
$dYt=−\sin Y_tdt+sX_t\cos Ytdt$
Where $λ,σ,s$ are constant parameters. $B_t$ is standard brownian motion.
Now I am asked to show whether existence and uniquenes of a strong solution can be guaranteed from Lipschitz continuity of and linear bounds on the functions. My question is twofold. How do I write the above as a system and how to show the bounds and the Lipschitz condition. Does one write it like this?
$ \begin{pmatrix} dX_t\\dY_t \end{pmatrix} =\begin{pmatrix} -\lambda X_t & 0\\ -\sin Y_t & s X_t\cos Y_t \end{pmatrix}dt + \begin{pmatrix} \sigma \\ 0 \end{pmatrix} dB_t $
And then check the conditions elementwise, for each term in the matrix?
Thanks for looking!