Singularities of Invariant Densities for Random Switching between Two Linear ODEs in 2D

In the presented paper the response of nonlinear random oscillators is calculated by the numerical solution of the associated Fokker-Planck-Equation. This approach can be used to approximate the invariant probability density of chaotic nonlinear oscillators or to calculate the response of a nonlinear oscillator, driven by additive white noise. A weighted residual approach is proposed to solve the Fokker-Planck-Equations numerically. By this method the partial differential equation is transformed to a coupled set of linear ordinary differential equations. These equations can be derived easily, because the integrals associated to the Galerkin procedure can be solved analytically. Considering stationary densities the solution is given by a generalized Eigenproblem. Utilizing sparse matrix techniques the solution can be calculated very efficiently.

Regularity of Invariant Densities for Random Switching between Two Linear ODEs in \(\mathbb{R}^{{d}}\)

Deals with a continuous-time version of the Markowitz mean-variance portfolio selection problem. The proposed model is concerned with a basket of securities consisting of one bank account and multiple stocks. One distinct feature of the model is that the market parameters, including the bank interest rate and the appreciation and volatility rates of the stocks, depend on the market mode that switches between a finite number of states. It is assumed that the random regime switching is independent of the random sources that drive the stock prices. Although the model is more realistic and takes into account possible random volatility, it essentially renders the underlying market incomplete. We use a Markov-chain modulated diffusion formulation to model the problem. By using techniques of stochastic linear-quadratic control, we obtain mean-variance efficient portfolios and efficient frontiers in explicitly closed forms, based on solutions of two systems of linear ordinary differential equations. We also address related issues such as minimum variance portfolio and mutual fund theorem. The results are notably different from those for the case when there is no regime switching. Nonetheless, if the interest rate is independent of the Markov chain and is deterministic, then the results exhibit (rather unexpected) similarity to their no-regime-switching counterparts, even if the stocks appreciation and volatility rates are Markov-modulated.

In this paper, a class of stochastic hybrid systems comprising of multiple operation modes is studied. In each mode, the state evolves according to a linear stochastic differential equation. We allow for stochastic switching between operational modes with switching times controlled by an underlying renewal process such that the time spent in each mode is a random variable with an arbitrary given probability distribution. We present a novel method to derive exact analytical solutions for the statistical moments, and illustrate the applicability of the method on an example drawn from systems biology. More specifically, we study how random switching of a gene between transcriptionally active and inactive states drives stochastic variation in the level of the expressed protein. Our results show that while randomness in gene switching times has no effect on the mean protein level, it critically impacts the magnitude of fluctuations in the protein level. This effect is further amplified for proteins with high decay rate. We finally discuss how noise in protein levels can be used to infer the underlying gene expression mechanisms.