Optimal consumption and investment with bounded downside risk for power utility functions

We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.Comment: 36 page

Non-equilibrium Statistical Mechanics of Anharmonic Crystals with Self-consistent Stochastic Reservoirs

We consider a d-dimensional crystal with an arbitrary harmonic interaction and an anharmonic on-site potential, with stochastic Langevin heat bath at each site. We develop an integral formalism for the correlation functions that is suitable for the study of their relaxation (time decay) as well as their behavior in space. Furthermore, in a perturbative analysis, for the one-dimensional system with weak coupling between the sites and small quartic anharmonicity, we investigate the steady state and show that the Fourier's law holds. We also obtain an expression for the thermal conductivity (for arbitrary next-neighbor interactions) and give the temperature profile in the steady state

A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics

This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation ("the cell problem"), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature.Comment: 32 page

Normal forms approach to diffusion near hyperbolic equilibria

We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighborhood of a hyperbolic critical point. We show that if the distribution of the initial condition has a scaling limit then the exit distribution and exit time also have a joint scaling limit as the noise intensity goes to zero. The limiting law is computed explicitly. The result completes the theory of noisy heteroclinic networks in two dimensions. The analysis is based on normal forms theory.Comment: 21 page

Models of Passive and Reactive Tracer Motion: an Application of Ito Calculus

By means of Ito calculus it is possible to find, in a straight-forward way, the analytical solution to some equations related to the passive tracer transport problem in a velocity field that obeys the multidimensional Burgers equation and to a simple model of reactive tracer motion.Comment: revised version 7 pages, Latex, to appear as a letter to J. of Physics

The Tychonoff uniqueness theorem for the G-heat equation

In this paper, we obtain the Tychonoff uniqueness theorem for the G-heat equation

Soft and hard wall in a stochastic reaction diffusion equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra diffusive scaling.Comment: 33 page

Synchronization of coupled stochastic limit cycle oscillators

For a class of coupled limit cycle oscillators, we give a condition on a linear coupling operator that is necessary and sufficient for exponential stability of the synchronous solution. We show that with certain modifications our method of analysis applies to networks with partial, time-dependent, and nonlinear coupling schemes, as well as to ensembles of local systems with nonperiodic attractors. We also study robustness of synchrony to noise. To this end, we analytically estimate the degree of coherence of the network oscillations in the presence of noise. Our estimate of coherence highlights the main ingredients of stochastic stability of the synchronous regime. In particular, it quantifies the contribution of the network topology. The estimate of coherence for the randomly perturbed network can be used as means for analytic inference of degree of stability of the synchronous solution of the unperturbed deterministic network. Furthermore, we show that in large networks, the effects of noise on the dynamics of each oscillator can be effectively controlled by varying the strength of coupling, which provides a powerful mechanism of denoising. This suggests that the organization of oscillators in a coupled network may play an important role in maintaining robust oscillations in random environment. The analysis is complemented with the results of numerical simulations of a neuronal network. PACS: 05.45.Xt, 05.40.Ca Keywords: synchronization, coupled oscillators, denoising, robustness to noise, compartmental modelComment: major revisions; two new section

Diffusive behavior for randomly kicked Newtonian particles in a spatially periodic medium

We prove a central limit theorem for the momentum distribution of a particle undergoing an unbiased spatially periodic random forcing at exponentially distributed times without friction. The start is a linear Boltzmann equation for the phase space density, where the average energy of the particle grows linearly in time. Rescaling time, the momentum converges to a Brownian motion, and the position is its time-integral showing superdiffusive scaling with time $t^{3/2}$. The analysis has two parts: (1) to show that the particle spends most of its time at high energy, where the spatial environment is practically invisible; (2) to treat the low energy incursions where the motion is dominated by the deterministic force, with potential drift but where symmetry arguments cancel the ballistic behavior.Comment: 55 pages. Some typos corrected from previous versio

Hedging American contingent claims with constrained portfolios

The valuation theory for American Contingent Claims, due to Bensoussan (1984) and Karatzas (1988), is extended to deal with constraints on portfolio choice , including incomplete markets and borrowing/short-selling constraints, or with different interest rates for borrowing and lending. In the unconstrained case, the classical theory provides a single arbitrage-free price ; this is expressed as the supremum, over all stopping times, of the claim's expected discounted value under the equivalent martingale measure. In the presence of constraints, is replaced by an entire interval of arbitrage-free prices, with endpoints characterized as . Here is the analogue of , the arbitrage-free price with unconstrained portfolios, in an auxiliary market model ; and the family is suitably chosen, to contain the original model and to reflect the constraints on portfolios. For several such constraints, explicit computations of the endpoints are carried out in the case of the American call-option. The analysis involves novel results in martingale theory (including simultaneous Doob-Meyer decompositions), optimal stopping and stochastic control problems, stochastic games, and uses tools from convex analysis.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42331/1/780-2-3-215_80020215.pd