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      1
      vote
      0answers
      56 views

      Non-diagonalizable matrix in a discretized Ornstein-Uhlenbeck process

      I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\...
      14
      votes
      1answer
      461 views

      For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$

      Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy $$B^\top - T B^\top = B + B T$$ ...
      2
      votes
      1answer
      213 views

      Steady state Kalman filter

      My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from. Kalman filter allows us to estimate state at time $t$ as ...
      4
      votes
      0answers
      48 views

      Is there an equivalent line time-invariant system for a linear time-varying system with specific properties? [closed]

      Given a discrete-time linear time-varying system (LTV) $$x(k+1) = A(k) x(k) + B(k) u(k)$$ where $A(k)$ and $B(k)$ are generated by a stationary random process. Is there an equivalent linear time-...
      1
      vote
      1answer
      838 views

      Correlation for a discrete time markov chain

      Question Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible Discrete Time Markov Chain (DTMC) with finite state space $S$, transition matrix $P$ and steady state $\pi$. Assume that we are ''far enough'' ...
      0
      votes
      0answers
      210 views

      Hadamard product (Schur product) in $L^2[0,1]$

      Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
      0
      votes
      0answers
      300 views

      Comparison of Parameter estimation using maximum likelihood and Maximum entropy

      I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
      7
      votes
      1answer
      186 views

      approximate stationary distributions of a doubly stochastic matrix and its supports

      Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...
      5
      votes
      0answers
      196 views

      Existence or construction of a sequence of orthogonal matrices with three properties

      This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help .... Any pointers or suggestions are appreicated! ...
      3
      votes
      1answer
      159 views

      Sum of two parts of a continuous stochastic process

      Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all $...
      4
      votes
      0answers
      390 views

      Sum of the entries of the inverse covariance matrix

      Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = \left[sinc\left(\frac{T\left(r-s\right)}{n}\right)\right]^n_{r,s=...
      2
      votes
      0answers
      997 views

      Random matrices whose limit gives exact Wigner surmise

      Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write $\rho^W_0(s)=\...
      1
      vote
      0answers
      194 views

      Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix

      Hello, Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero. As a part ...
      5
      votes
      0answers
      235 views

      stochastic control / geometric mean

      Consider the following problem: Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
      2
      votes
      2answers
      617 views

      Spectral gap of a product of Markov processes

      For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n_m]$. Write $J \equiv (j_1,\dots,j_N) \in \prod_m [...

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