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Considering a linear difference equation \begin{equation} \underbrace{\begin{bmatrix} -p & 1 & 0 & 0 & 0 & \cdots & 0\\ 1 &-p & 1 & 0 & 0 & \cdots & 0\\ 0 & 1 &-p & 1 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots &\vdots\\ \vdots & \vdots & \vdots &\vdots & \ddots & \vdots &\vdots\\ 0 & \cdots & 0 & 0 & 1 & -p & 1 \\ 0 & \cdots & 0 & 0 & 0 & 1 & -p \end{bmatrix}}_P \underbrace{\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ \vdots \\ x_{N-2} \\ x_{N-1} \end{bmatrix}}_x = -\underbrace{\begin{bmatrix} x_0 \\ 0 \\ 0 \\ \vdots \\ \vdots \\ 0 \\ 0 \end{bmatrix}}_C, \end{equation} where $p:=2+\tilde{\kappa}^2 \tau^2$ and $\tau:=T/N$ are given; sequence $\{x_j\}_{j=1}^{N-1}$ needs to be estimated by solving the linear difference equation.

Normally, the solution of a well-defined linear difference equation can be estimated using linear least squares (LLS), such that $x=-P^{-1}C$; however, according to the paper "Optimal Execution of Portfolio Transactions":

"a linear difference equation whose solution may be written as a combination of the exponentials $e^{\pm \kappa(j\tau)}$, where $\kappa$ satisfies \begin{equation} \frac{2}{\tau^2}(\cosh(\kappa\tau)-1) = \tilde{\kappa}^2 \end{equation} and the solution can be written as \begin{equation} x_j = \frac{\sinh(\kappa(T-j\tau))}{\sinh(\kappa T)} x_0. \end{equation}"

Can anyone explain to me how $P^{-1}$ is related to the hyperbolic function $\sinh$?

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  • $\begingroup$ I’m voting to close this question because it is more suited for math SE but to old to migrate $\endgroup$ – kjetil b halvorsen Jan 30 at 21:43