Let be $n \times n$ system of SDE $$ dX_t^{(i)} = \sum_{k = 1}^n a_{ik}(t) X_t^{(k)} dt + dW_t^{(k)}$$ where $i = 1, 2, \dots, n$ and $a_{ik}(t)$ are continuous function for $t \geq 0$ and $X_0^{(i)} = x_i$. $W^{(1)}, \dots, W^{(n)}$ are independent Brownian motions.
Let define matrix $$ A = \left[ {\begin{array}{cccc} a_{11}(t) & a_{12}(t) & \dotsb & a_{1n}(t) \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1}(t) & a_{n2}(t) & \dotsb & a_{nn}(t) \\ \end{array} } \right]$$ Let be $Y$ fundamental solutions of systems of linear differential equations $\dot{y} = Ay$, for $Y(0) = I$. Let be $C(t) = Y^{-1}(t)$.
I know also that it holds $-(CA)_{ik} = \dot{c}_{ik}$
So i have two questions.
- i need to prove that $$d (\sum_{k = 1}^n c_{ik} X_t^{(k)}) = \sum_{k = 1}^n c_{ik} dW_t^{(k)}.$$
So i did that $$d (\sum_{k = 1}^n c_{ik} X_t^{(k)}) = \sum_{k = 1}^n (\dot{c}_{ik} X_t^{(k)} dt + c_{ik} dX_t^{(k)}) = \sum_{k = 1}^n (-(CA)_{ik} X_t^{(k)} dt + c_{ik} (\sum_{l = 1}^n a_{il}(t) X_t^{(l)} dt + dW_t^{(l)}))$$ And now I don't know how to solve that double summation.
- i need to find a solution of the system of differential equations with the corresponding integrals.