Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks

We aim at characterizing viability, invariance and some reachability properties of controlled piecewise deterministic Markov processes (PDMPs). Using analytical methods from the theory of viscosity solutions, we establish criteria for viability and invariance in terms of the first order normal cone. We also investigate reachability of arbitrary open sets. The method is based on viscosity techniques and duality for some associated linearized problem. The theoretical results are applied to general On/Off systems, Cook's model for haploinssuficiency, and a stochastic model for bacteriophage lambda.Comment: submitte

Large deviations for Markov jump processes with mean-field interaction via the comparison principle for an associated Hamilton-Jacobi equation

We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation

On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.Comment: 28 pages. Added DOI https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final publication, corrected typo (missing gamma) in example 4.1

Piecewise deterministic Markov processes in biological models

We present a short introduction into the framework of piecewise deterministic Markov processes. We illustrate the abstract mathematical setting with a series of examples related to dispersal of biological systems, cell cycle models, gene expression, physiologically structured populations, as well as neural activity. General results concerning asymptotic properties of stochastic semigroups induced by such Markov processes are applied to specific examples.Comment: in: Semigroup of Operators - Theory and Applications, J. Banasiak et al. (eds.), Springer Proceedings in Mathematics & Statistics 113, (2015), pp. 235-25

Optimal dividend policies with random profitability

We study an optimal dividend problem under a bankruptcy constraint. Firms face a trade-off between potential bankruptcy and extraction of profits. In contrast to previous works, general cash flow drifts, including Ornstein--Uhlenbeck and CIR processes, are considered. We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as comparison between the sub- and supersolutions of the Hamilton--Jacobi--Bellman equation, and we provide an efficient and convergent numerical scheme for finding the solution. The value function is given by a nonlinear PDE with a gradient constraint from below in one dimension. We find that the optimal strategy is both a barrier and a band strategy and that it includes voluntary liquidation in parts of the state space. Finally, we present and numerically study extensions of the model, including equity issuance and credit lines