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# All Questions

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### 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}(\...
473 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 $$B^\top - T B^\top = B + B T$$ ...
255 views

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 ...
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-...
946 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'' ...
211 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 ...
305 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 ...
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 ...
199 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! ...
161 views

1k views

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 $\rho^W_0(s)=\... 0answers 194 views ### Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix Hello, Denote$\log_\star$as the entrywise logarithm operation, and let$A$be some row-stochastic matrix such that$\lim_{p\rightarrow\infty}A^p$exists and all its entries are non-zero. As a part ... 0answers 236 views ### stochastic control / geometric mean Consider the following problem: Given$\Omega$and$U$two symmetric definite positive matrices, choose a matrix$K$to minimize the expectation$x' \Omega x + x'K'UKx$when$x$follows the invariant ... 2answers 622 views ### Spectral gap of a product of Markov processes For$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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