# Marginal conditional mean of two dimensional Brownian motion, using more than one time point.

EDIT: I found the error! I do not think this question is relevant for anyone, but I cannot find out how to delete it - please feel free to if you have the read this and have the option. I have indicated in the question where the error is.

Consider a standard two-dimensional Wiener process $(W^1,W^2)$, but with correlation $\rho$ between them. What I am trying to do is find the mean and variance for one of them at some deterministic time $T$, conditional on the other at both time $T$ and $T-1$. I get non-intuitive results and hoped you guys could help clairify. My standing theory is that I've made some mistake, but I can't find it.

We can derive that (this is standard/using independent increments) $$\rho_{11} := Corr(W^1_T,W^1_{T-1}) = \sqrt{\frac{T-1}{T}}$$ and $$\rho_{21}:=Corr(W^2_T,W^1_{T-1}) = \rho \sqrt{\frac{T-1}{T}}.$$ Note the relation $\rho_{21}=\rho \cdot \rho_{11}$.

Now let us derive the conditional distribution of $W^2(T)$ using results from wikipedia. Summing up we found $$(W^2(T),W^1(T),W^1(T-1)) \sim N(0,\Sigma)$$ where \begin{equation*} \Sigma = \begin{pmatrix} 1 & \rho & \rho_{21} \\ \rho & 1 & \rho_{11} \\ \rho_{21} & \rho_{11} & 1 \\ \end{pmatrix} \end{equation*} as argued above. Then set \begin{align*} \Sigma_{11} = {} & 1 \\ \Sigma_{12} = {} & (\rho_{RS} ,\rho_{SS} ) \\ \Sigma_{21} = {} & (\rho_{RS} , \rho_{SS} )' \\ \Sigma_{22} = {} & \begin{pmatrix} 1 & \rho \\ \rho & 1 \\ \end{pmatrix} \end{align*} The error is here! $\Sigma_{22}$ is not what it should be.

We get \begin{equation*} \det ( \Sigma_{22}) = 1- \rho^2 \end{equation*} so \begin{equation*} \Sigma_{22}^{-1} = \frac{1}{1-\rho^2} \begin{pmatrix} 1 & -\rho \\ -\rho & 1 \\ \end{pmatrix} \end{equation*} and using the observation $\rho_{21} = \rho \cdot \rho_{11}$ we get \begin{align*} \Sigma_{12} \Sigma_{22}^{-1} ={} & \frac{1}{1-\rho^2} \bigl ( \rho_{21} -\rho \cdot \rho_{11}, \rho_{11} -\rho\cdot \rho_{21} \bigr ) \\ = {} & \frac{1}{1-\rho^2} \bigl ( 0, (1-\rho^2) \rho_{11 } \bigr ) \\ = {} & \bigl ( 0, \rho_{11 } \bigr ) \end{align*} and \begin{align*} \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21}= {} & \frac{\rho_{21}}{1- \rho^2 } \cdot ( \rho_{21}- \rho \cdot \rho_{11}) + \frac{\rho_{11} }{1- \rho^2} \cdot (\rho_{11} - \rho\cdot \rho_{21} ) \\ = {} & \frac{\rho_{11} }{1- \rho^2} \cdot ( (1-\rho^2) \rho_{11} ) \\ = {} & \rho_{11}. \end{align*}

This means that $$E[W^2_T \lvert W^1_{T}=w_2, W^1(T-1) =w_1 ] = \rho_{11} w_1$$ and $$Var(W^2_T \lvert W^1_{T}=w_2, W^1(T-1) =w_1 ) = \rho_{11},$$ by the formula for finding the marginal distribution.

As mentioned I'm finding this result really strange, so, first of all: Are there anything wrong with the derivation? If not, what is the intuition behind that all information from $W^1_T$ say seems to magically dissapear ? Even worse the parameter measuring correlation (which would be my guess would measure the information, were it $0$ I would say there were none) is not even in the formulas!