# Solutions to linear SDE with many noise sources

It is well known how to solve the linear stochastic ODEs with one source of noise $$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$ See, for instance, https://math.stackexchange.com/questions/1788853/solution-to-general-linear-sde or https://en.wikipedia.org/wiki/Stochastic_differential_equation#Linear_SDE:_general_case .

I would like to know how solutions to SDEs with many noise sources, i.e., to $$dX_t=(a(t)X_t+c(t))dt+\sum_{i=1}^{n} (b_i(t)X_t+d_i(t))dW^{(i)}_t,$$ do look like.

It wouldd be great if you could help me here! Many thanks for your help in advance!

Write your SDE as following, $$dX_t=X_t\Bigl[a(t)\,dt+\sum_{i=1}^nb_i(t)\,dW^{(i)}_t\Bigr]+\Bigl[c(t)\,dt+\sum_{i=1}^nd_i(t)dW_t^{(i)}\Bigr].$$ Where $W=\{(W^{(1)}_t,\cdots, W^{(n)}_t)^{\top},t\ge 0\}$ is an n-dimensional continuous martingale. Let \begin{align} Y_t&=\int_0^t a(s)\,ds+\sum_{i=1}^n\int_0^tb_i(s)\,dW^{(i)}_s,\\ H_t&=\int_0^t c(s)\,ds+\sum_{i=1}^n\int_0^td_i(s)dW_s^{(i)}. \end{align} Then $$dX_t=X_t\,dY_t+dH_t. \tag{1}$$ Accoding D. Revuz & M. Yor, Continuous martingales & Brownian Motions, 3rd edn(Springer, Berlin 1999), p.378, the solution of (1) is the following $$X_t=\mathscr{E}(Y)_t\Bigl[X_0+\int_0^t\mathscr{E}(Y)^{-1}_s(dH_s-d\langle H,Y \rangle_s)\Bigr].$$ where \begin{gather} \mathscr{E}(Y)_t=\exp\Bigl[\int_0^ta(s)\,ds+\sum_{i=1}^n\int_0^tb_i(s)\,dW^{(i)}_s -\frac12\sum_{i,j=1}^n\int_0^t b_i(s)b_j(s)d\langle W^{(i)},W^{(j)}\rangle_s\Bigr],\\ \langle H,Y\rangle_t=\sum_{i,j=1}^n\int_0^tb_i(s)d_j(s)\,d\langle W^{(i)},W^{(j)}\rangle_s, \end{gather} and $\langle W^{(i)},W^{(j)}\rangle$ is the bracket(or covariation) process of $W^{(i)}$ and $W^{(j)}$(c.f. Revuz & Yor's book, p.125). If $W$ is continuous Gaussian process with $\mathsf{E}[W^{(i)}_t]=0$ and $\mathsf{E}[W^{(i)}_sW^{(j)}_t]=\sigma_{ij}(s\wedge t)$, then $\langle W^{(i)},W^{(j)}\rangle_t=\sigma_{ij}(t)$. If $W$ is components independent n-dimensional Brownian Motion, then $\langle W^{(i)},W^{(j)}\rangle_t=\delta_{ij}t$.