User bernard - MathOverflow

bounded time integral of Ornstein Uhlenbeck process

2013-01-11T23:01:38Z

With $dF_t=-kF_tdt+\sigma dW_t$ an Ornstein-Uhlenbeck process, I define $Q_t$ as: $Q_t=\int_0^\infty F_t \mathbb{1}_{-L\leq Q_t\leq L} dt$.

Intuitively, $F_t$ is a flow and $Q_t$ is a quantity, that we do not authorize to grow to more than $L$.

I am interested in the stationary distribution of Q. I can write the Fokker-Planck equation as:

$0=\frac{\partial}{\partial F}[k F p + \frac{\sigma^2}{2} \frac{\partial p}{\partial F}] - \frac{\partial}{\partial Q}[F p]$ with $p(F,Q)$

$0 = \frac{\partial}{\partial F}[k F b + \frac{\sigma^2}{2} \frac{\partial b}{\partial F}] + f P|_{Q=L}$ and the same equation for $Q=-L$

where $p$ is the density on $(F,Q)\in \mathbb{R} \times [-L,L]$ and b is a density on $F\in \mathbb{R}^+$

But unfortunately I am stuck there and not sure how to get -even approximate- ways to look at this. I am only interested in the marginal on $Q$ (the marginal on $F$ is trivial since $F$ does not depend on $Q$, so you just get the pdf of the OU process), but not sure how to approach this. Any help would be useful.

ODE in symmetric definite positive matrices

2012-04-25T20:21:44Z

It is easy to solve the ODE: $\frac {dx}{dt} = a - b x^2$ with $x(0)=0$, $a>0$, and $b>0$, indeed all one has to do is write $dt = \frac {dx}{a-bx^2} = \frac 1 {2\sqrt a}(\frac {dx}{\sqrt a-\sqrt bx} + \frac {dx}{\sqrt a+\sqrt bx})$ and integrate both parts.

I am interested in the same equation with $x$ taking its values in symmetric positive matrices, ie $\frac {dx}{dt} = a - x b x$, with $a$ and $b$ symmetric positive. In that case, because the involved matrices might not commute, I cannot use the same trick to get a closed form solution.

Would you know of any techniques I can use to solve such a problem?

median of matrices

2012-04-03T17:32:02Z

I have $n$ positive definite Hermitian matrices $M_n$ and I want to define and compute their median.

These matrices correspond to independent estimations of a covariance matrix in the presence of some noise I cannot quantify, hence the desire to use a median as opposed to a mean (non gaussian residuals / outliers).

I could:

(1) look for a positive definite Hermitian matrix that minimizes $d(M,(M_k)) = \sum \|M-M_k\|_1$.

(2) or I could look a the eigen decompositions $M_k = Q_k \Lambda_k Q_k^T$ (with $\Lambda_k$ sorted) and define $\Lambda = med\ \Lambda_k$ and $Q$ as the orthogonal matrix that minimizes $d(Q,(Q_k))$.

(3) or simply look for the closest (for norm $\|.\|_2$) matrix to the matrix of the element-wise medians.

Not sure if anything smart can be said about this... thanks for any help.

Answer by Bernard for buffon needle experiment

2012-04-04T19:10:19Z

The question is well defined for estimating $\frac 1 \pi$, but not for estimating $\pi$. If $l>d$, you need to evaluate $cos^{-1}$ which requires knowledge of $\pi$, otherwise the variance of the estimator decreases with$ \frac l d$, so you'd "practically" settle for $l=d$.

stochastic control / geometric mean

2012-04-04T20:02:23Z

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 distribution of the Ornstein-Uhlenbeck process:

$dx_t = -K x_t + \Sigma dW_t$.

The intuition of the problem is straightforward: there is a penalty on $x$ for the metric associated to $\Omega$ and a penalty on $Kx$ for the metric associated to $U$.

The solution of this problem is $K^* = U^{-1}(U \sharp \Omega )$ with $U \sharp \Omega$ the geometric mean of $U$ and $\Omega$.

I can get this result by a convoluted way: expressing the variance of the invariant distribution of $x$ as a function of $K$ and $\Sigma$ (it gives the Algebraic Riccati equation as a constraint) and then solving the optimization problem under this constraint.

Given that the solution involves the mean of my two matrices $U$ and $\Omega$ for the Riemannian metric on positive definite matrices, I am looking for more elegant proofs of this result: something along the lines of "the total cost is the sum of two costs which correspond to some distances in a well defined space and by minimizing the sum of these distances, we end up with a geometric mean..."

Answer by Bernard for Solving Ax=b, where A is an unknown Toeplitz matrix, x and b are known.

2012-04-04T17:55:16Z

You have more unknowns ($2n-1$) than equations ($n$)

Comment by Bernard

2012-04-25T21:10:01Z

sorry, it is indeed (2). To avoid confusion, t is a scalar and x is a positive matrix. When dimension is 1, everything goes well, when dimension is higher I'm not sure what can be done.

Comment by Bernard

2012-04-04T15:43:01Z

I realize my question is ill-defined, but is there a strong reason to prefer the Riemannian distance on positive definite matrices (which is just the Euclidian distance on their logs if I understand well) to any other distance? If I were to look at the case of dimension 1, assuming that I have estimated the variance of a random variable using 2 different samples, producing estimators $\sigma_1^2$ and $\sigma_2^2$, I would combine them by taking the arithmetic mean, not the geometric mean. Am I missing something?