An $n \times n$ matrix polynomial $L(\lambda)$ (with real or complex coefficients) is called self-adjoint if $L(\lambda) = (L(\bar{\lambda}))*$ and symmetric if $L(\lambda) = (L(\pm\lambda))^T$. Factorizations of selfadjoint and symmetric matrix polynomials of the form $L(\lambda) = (M(\bar{\lambda}))*DM(\lambda)$ or $L(\lambda) = (M(\pm\lambda))^{T}DM(\lambda)$ are studied, where $D$ is a constant matrix and $M(\lambda)$ is a matrix polynomial. In particular, the minimal possible size of $D$ is described in terms of the elementary divisors of $L(\lambda)$ and (sometimes) signature of the Hermitian values of $L(\lambda)$.

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Perturbation theory of eigenvalue problems

Factorization of Matrix Polynomials with Symmetries

If $A(t)$ is a $C^{1,\al}$-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized $C^1$ in $t$. If $A$ is $C^\infty$ then the eigenvalues can be parameterized twice differentiable.

Differentiable perturbation of unbounded operators

In lectures given in 1953 at New York University, Franz Rellich proved that for all f∈C0∞(Rn \{0}) and n≠2

background

On the Rellich inequality with magnetic potentials

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

https://www.math.kit.edu/stoch/~baeuerle/seite/finance_nb/media/bli02.pdf

Optimal portfolios for financial markets with wishart volatility

We prove the nonexistence of smooth stable solution to the biharmonic problem $\Delta^2 u= u^p$, $u>0$ in $\R^N$ for $1 5$ when $p > 1$, we obtain the nonexistence of smooth stable solution for any $N \leq 12$ and $p > 1$. Moreover, we consider also the corresponding problem in the half space $\R^N_+$, or the elliptic problem $\Delta^2 u= l(u+1)^p$ on a bounded smooth domain $O$ with the Navier boundary conditions. We will prove the regularity of the extremal solution in lower dimensions. Our results improve the previous works.

Perturbation theory of eigenvalue problems

Citations

Factorization of Matrix Polynomials with Symmetries

Differentiable perturbation of unbounded operators

On the Rellich inequality with magnetic potentials

Optimal portfolios for financial markets with wishart volatility

On stable solutions of biharmonic problem with polynomial growth

Related Papers (5)

Perturbation theory for linear operators

Methods of Modern Mathematical Physics

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