# All Questions

Tagged with stochastic-processes linear-algebra

16 questions

**1**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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 [...