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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!

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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 $\langle W^{(i)},W^{(j)}\rangle_t=\delta_{ij}t$.

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  • $\begingroup$ Many thanks for your input! Could you tell me what the scalar (like d<H,Y>) is supposed to mean? $\endgroup$ – tobias Mar 29 '17 at 12:31
  • $\begingroup$ Also, when n=1, do the equations come down to the ones like in math.stackexchange.com/questions/1788853/… or en.wikipedia.org/wiki/… ? $\endgroup$ – tobias Mar 29 '17 at 13:31
  • $\begingroup$ @tobias Thank you for your replication. I already add the definition of W, <W,W>, and <H,Y>. $\endgroup$ – JGWang Mar 30 '17 at 3:09
  • $\begingroup$ Okay, that looks much better. Could you write down a bit more on \sigma_{ij} then as well? That would be great! Many thanks in advance! $\endgroup$ – tobias Mar 30 '17 at 6:41
  • $\begingroup$ You are doing a great job! But with the current definition of \sigma_{ij} couldn't we dramatically simplify the solution by considerung in the sums ranging over i and j simultaneaously only those terms with i=j? $\endgroup$ – tobias Mar 30 '17 at 7:07

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