Risk-sensitive control of continuous time Markov chains

We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterize the value function via Hamilton Jacobi Bellman equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.

Markov chains, R-trivial monoids and representation theory

We develop a general theory of Markov chains realizable as random walks on R-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Mobius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

Structural characterization of folded and extended conformations in peptides containing gamma amino acids with proteinogenic side chains: crystal structures of gamma(n), (alpha gamma)(n) and gamma gamma delta gamma sequences

The crystal structures of nine peptides containing gamma(4)Val and gamma(4)Leu are described. The short sequences Boc-gamma(4)(R)Val](2)-OMe 1, Boc-gamma(4)(R)Val](3)-NHMe 2 and Boc-gamma(4)(S)Val-gamma(4)(R)Val-OMe 3 adopt extended apolar, sheet like structures. The tetrapeptide Boc-gamma(4)(R)Val](4)-OMe 4 adopts an extended conformation, in contrast to the folded C-14 helical structure determined previously for Boc-gamma(4)(R)Leu](4)-OMe. The hybrid alpha gamma sequence Boc-Ala-gamma(4)(R)Leu](2)-OMe 5 adopts an S-shaped structure devoid of intramolecular hydrogen bonds, with both alpha residues adopting local helical conformations. In sharp contrast, the tetrapeptides Boc-Aib-gamma(4)(S)Leu](2)-OMe 6 and Boc-Leu-gamma(4)(R)Leu](2)-OMe 7 adopt folded structures stabilized by two successive C-12 hydrogen bonds. gamma(4)Val residues have also been incorporated into the strand segments of a crystalline octapeptide, Boc-Leu-gamma(4)(R)Val-Val-(D)Pro-Gly-Leu-gamma(4)(R)Val-Val-OMe 8. The gamma gamma delta gamma tetrapeptide containing gamma(4)Val and delta(5)Leu residues adopts an extended sheet like structure. The hydrogen bonding pattern at gamma residues corresponds to an apolar sheet, while a polar sheet is observed at the lone delta residue. The transition between folded and extended structures at gamma residues involves a change of the torsion angle from the gauche to the trans conformation about the C-beta-C-alpha bond.

A mean-reverting stochastic model for the political business cycle

In this article, we look at the political business cycle problem through the lens of uncertainty. The feedback control used by us is the famous NKPC with stochasticity and wage rigidities. We extend the New Keynesian Phillips Curve model to the continuous time stochastic set up with an Ornstein-Uhlenbeck process. We minimize relevant expected quadratic cost by solving the corresponding Hamilton-Jacobi-Bellman equation. The basic intuition of the classical model is qualitatively carried forward in our set up but uncertainty also plays an important role in determining the optimal trajectory of the voter support function. The internal variability of the system acts as a base shifter for the support function in the risk neutral case. The role of uncertainty is even more prominent in the risk averse case where all the shape parameters are directly dependent on variability. Thus, in this case variability controls both the rates of change as well as the base shift parameters. To gain more insight we have also studied the model when the coefficients are time invariant and studied numerical solutions. The close relationship between the unemployment rate and the support function for the incumbent party is highlighted. The role of uncertainty in creating sampling fluctuation in this set up, possibly towards apparently anomalous results, is also explored.