User bernard - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:44:12Z http://mathoverflow.net/feeds/user/22620 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118684/bounded-time-integral-of-ornstein-uhlenbeck-process bounded time integral of Ornstein Uhlenbeck process Bernard 2013-01-11T23:01:38Z 2013-06-07T19:22:00Z <p>With $dF_t=-kF_tdt+\sigma dW_t$ an Ornstein-Uhlenbeck process, I define $Q_t$ as: $Q_t=\int_0^\infty F_t \mathbb{1}_{-L\leq Q_t\leq L} dt$.</p> <p>Intuitively, $F_t$ is a flow and $Q_t$ is a quantity, that we do not authorize to grow to more than $L$.</p> <p>I am interested in the stationary distribution of Q. I can write the Fokker-Planck equation as:</p> <p>$0=\frac{\partial}{\partial F}[k F p + \frac{\sigma^2}{2} \frac{\partial p}{\partial F}] - \frac{\partial}{\partial Q}[F p]$ with $p(F,Q)$</p> <p>$0 = \frac{\partial}{\partial F}[k F b + \frac{\sigma^2}{2} \frac{\partial b}{\partial F}] + f P|_{Q=L}$ and the same equation for $Q=-L$</p> <p>where $p$ is the density on $(F,Q)\in \mathbb{R} \times [-L,L]$ and b is a density on $F\in \mathbb{R}^+$</p> <p>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.</p> http://mathoverflow.net/questions/95195/ode-in-symmetric-definite-positive-matrices ODE in symmetric definite positive matrices Bernard 2012-04-25T20:21:44Z 2012-04-25T21:51:21Z <p>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.</p> <p>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.</p> <p>Would you know of any techniques I can use to solve such a problem?</p> http://mathoverflow.net/questions/93017/median-of-matrices median of matrices Bernard 2012-04-03T17:32:02Z 2012-04-19T20:57:06Z <p>I have $n$ positive definite Hermitian matrices $M_n$ and I want to define and compute their median.</p> <p>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).</p> <p>I could:</p> <p>(1) look for a positive definite Hermitian matrix that minimizes $d(M,(M_k)) = \sum \|M-M_k\|_1$.</p> <p>(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 $d(Q,(Q_k))$.</p> <p>(3) or simply look for the closest (for norm $\|.\|_2$) matrix to the matrix of the element-wise medians.</p> <p>Not sure if anything smart can be said about this... thanks for any help.</p> http://mathoverflow.net/questions/93096/buffon-needle-experiment/93157#93157 Answer by Bernard for buffon needle experiment Bernard 2012-04-04T19:10:19Z 2012-04-05T14:42:14Z <p>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 $l=d$.</p> http://mathoverflow.net/questions/93161/stochastic-control-geometric-mean stochastic control / geometric mean Bernard 2012-04-04T20:02:23Z 2012-04-04T20:02:23Z <p>Consider the following problem:</p> <p>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:</p> <p>$dx_t = -K x_t + \Sigma dW_t$.</p> <p>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 $U$.</p> <p>The solution of this problem is $K^* = U^{-1}(U \sharp \Omega )$ with $U \sharp \Omega$ the geometric mean of $U$ and $\Omega$.</p> <p>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.</p> <p>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..."</p> http://mathoverflow.net/questions/93137/solving-axb-where-a-is-an-unknown-toeplitz-matrix-x-and-b-are-known/93150#93150 Answer by Bernard for Solving Ax=b, where A is an unknown Toeplitz matrix, x and b are known. Bernard 2012-04-04T17:55:16Z 2012-04-04T17:55:16Z <p>You have more unknowns ($2n-1$) than equations ($n$)</p> http://mathoverflow.net/questions/95195/ode-in-symmetric-definite-positive-matrices Comment by Bernard Bernard 2012-04-25T21:10:01Z 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. http://mathoverflow.net/questions/93017/median-of-matrices/93044#93044 Comment by Bernard Bernard 2012-04-04T15:43:01Z 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?