I'm struggling to understand if I have this idea of an orthogonal transform (to simplify a set of SDE's in Ito form) understood. Suppose I have the orthogonal transform \begin{align} \begin{bmatrix} W_3^\star \\ W_4^\star \end{bmatrix} = \int_{0}^{z}\begin{bmatrix} D_2^s & D_2^c \\ -D_2^c & D_2^s \end{bmatrix} d\begin{bmatrix} W_3 \\ W_4 \end{bmatrix}, \end{align} where \begin{align} (D_2^s)^2+(D_2^c)^2 = 1, \quad \begin{bmatrix} D_2^s,D_2^c \end{bmatrix}.\begin{bmatrix} -D_2^c,D_2^s \end{bmatrix} =0, \end{align}
I believe I need to show that the variance and correlation are zero for the transform to hold? So if we consider the stochastic integral \begin{align}\nonumber \mathbb{E}\bigg[\bigg(\int_{a}^{b}X(z)dW\bigg)\bigg(\int_{a}^{b}Y(z)dW\bigg)\bigg] &= \mathbb{E}\bigg[\bigg(\sum_{i}X(z_i)\Delta W_i\bigg)\bigg(\sum_j Y(z_j)\Delta W_j\bigg)\bigg] \\ \nonumber &=\mathbb{E}\bigg[X(z_1)\Delta W_1\bigg(Y(z_1)\Delta W_1 + Y(z_2)\Delta W_2 + ...\bigg) \\ \nonumber &+X(z_2)\Delta W_2\bigg(...\bigg) +...\bigg] \\ \nonumber &=\mathbb{E}\bigg[\sum_i X(z_i)Y(z_i)(\Delta W_i)^2\bigg] \\ \nonumber &=\sum_i\mathbb{E}\bigg[X(z_i)Y(z_i)(\Delta W_i)^2\bigg] \\ \nonumber &= \sum_i\mathbb{E}\bigg[X(z_i)Y(z_i)\bigg](z_{i+1}-z_i) \\ \nonumber &= \mathbb{E}\bigg[\sum_i X(z_i)Y(z_i)(z_{i+1}-z_i)\bigg] \\ &=\mathbb{E}\bigg[\int_{a}^{b}X(z)Y(z)dz\bigg]. \end{align} For two independent Wiener processes \begin{align} \mathbb{E}[W_1W_2] = 0. \end{align} This implies that \begin{align} \mathbb{E}\bigg[\bigg(\int_{a}^{b}X(z)dW_1\bigg)\bigg(\int_{a}^{b}Y(z)dW_2\bigg)\bigg] = 0. \end{align} Furthermore... \begin{align}\nonumber \mathbb{E}[W_3^\star W_4^\star] &= \mathbb{E}\bigg[\bigg(\int_{0}^{x_3}D_{2}^{s}dW_3 + D_2^cdW_4\bigg)\bigg(\int_{0}^{x_3}D_2^sdW_4 - D_2^cdW_3\bigg)\bigg] \\ \nonumber &=\mathbb{E}\bigg[\int_{0}^{x_3}D_{2}^{s}dW_3\int_{0}^{x_3}D_{2}^{c}dW_4 - \int_{0}^{x_3}D_{2}^{c}dW_4\int_{0}^{x_3}D_{2}^{c}dW_3 \\ \nonumber &+ \int_{0}^{x_3}D_{2}^{c}dW_4\int_{0}^{x_3}D_{2}^{s}dW_4 - \int_{0}^{x_3}D_{2}^{c}dW_3\int_{0}^{x_3}D_{2}^{s}dW_3\bigg] \\ \nonumber &=\mathbb{E}\bigg[\int_{0}^{x_3}D_{2}^{c}dW_4\int_{0}^{x_3}D_{2}^{s}dW_4 - \int_{0}^{x_3}D_{2}^{c}dW_3\int_{0}^{x_3}D_{2}^{s}dW_3\bigg] \\ \nonumber &=\mathbb{E}\bigg[\int_{0}^{z}D_{2}^{c}D_{2}^{s}dz - \int_{0}^{z}D_{2}^{c}D_{2}^{s}dz\bigg] \\ &=0. \end{align}
Now my question is, would the transform work on a set of Stratonovich SDE's? I.e. would the transform
\begin{align} \begin{bmatrix} W_3^\star \\ W_4^\star \end{bmatrix} = \int_{0}^{x_3}\begin{bmatrix} D_2^s & D_2^c \\ -D_2^c & D_2^s \end{bmatrix} \circ d\begin{bmatrix} W_3 \\ W_4 \end{bmatrix}, \end{align} be justified? Could someone give me some help if I can apply the same tests, but change to Stratonovich integral? Thanks if you made it this far...