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I'm trying to write a fokker planck equation for a particular SDE, but I'm caught up on an aside by the author I'm following.

He has a SDE with drift [tex]b \in \mathbb{R}^n[/tex], a dispersion matrix [tex]\sigma \in \mathbb{R}^{n\times k}[/tex], and k-dimensional brownian motion [tex]W_t[/tex], resulting in something like this

[tex]

dX_t &=& b(X_t)dt + \sigma(X_t)dW_t

[/tex]

My confusion comes from this k-dimensional brownian motion. What is this k-th dimension? I'm guessing that X_t will take a value in n-d,and W_t is k-by-1, so that would make sigma like a covariance matrix. But what do these k components actually mean physically?

Thanks

He has a SDE with drift [tex]b \in \mathbb{R}^n[/tex], a dispersion matrix [tex]\sigma \in \mathbb{R}^{n\times k}[/tex], and k-dimensional brownian motion [tex]W_t[/tex], resulting in something like this

[tex]

dX_t &=& b(X_t)dt + \sigma(X_t)dW_t

[/tex]

My confusion comes from this k-dimensional brownian motion. What is this k-th dimension? I'm guessing that X_t will take a value in n-d,and W_t is k-by-1, so that would make sigma like a covariance matrix. But what do these k components actually mean physically?

Thanks

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