A stochastic partially reversible investment problem on a finite time-horizon: Free-boundary analysis

We study a continuous-time, finite horizon, stochastic partially reversible investment problem for a firm producing a single good in a market with frictions. The production capacity is modeled as a one-dimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment–disinvestment strategy. We associate to the investment–disinvestment problem a zero-sum optimal stopping game and characterize its value function through a free-boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment–disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.     

Asymptotic analysis for a downside risk minimization problem under partial information

We give an analytic characterization of a large-time “downside risk” probability associated with an investor’s wealth. We assume that risky securities in our market model are affected by “hidden” economic factors, which evolve as a finite-state Markov chain. We formalize and prove a duality relation between downside risk minimization and the related risk-sensitive optimization. The proof is based on an analysis of an ergodic-type Hamilton–Jacobi–Bellman equation with large (exponentially growing) drift.     

Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter

In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

Analysis of a Degenerate Obstacle Problem on an Unbounded Set Arising in the Environment

We study a class of optimization dynamics problems related to investment under uncertainty. The general model problem is reformulated in terms of an obstacle problem associated to a second-order elliptic operator which is not in divergence form. The spatial domain is unbounded and no boundary conditions are a priori specified. By using the special structure of the differential operator and the spatial domain, and some approximating arguments, we show the existence and uniqueness of a solution of the problem. We also study the regularity of the solution and give some estimates on the location of the coincidence set.     

Analysis of a Degenerate Obstacle Problem on an Unbounded Set Arising in the Environment

We study a class of optimization dynamics problems related to investment under uncertainty. The general model problem is reformulated in terms of an obstacle problem associated to a second-order elliptic operator which is not in divergence form. The spatial domain is unbounded and no boundary conditions are a priori specified. By using the special structure of the differential operator and the spatial domain, and some approximating arguments, we show the existence and uniqueness of a solution of the problem. We also study the regularity of the solution and give some estimates on the location of the coincidence set.     

Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs

We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods.     

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton–Jacobi–Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear–quadratic model.     

Controlling Inflation: The Infinite Horizon Case

This paper studies the two-dimensional singular stochastic control problem over an infinite time-interval arising when the Central Bank tries to contain the inflation by acting on the nominal interest rate. It is shown that this problem admits a variational formulation which can be differentiated (in some sense) to lead to a stochastic differential game with stopping times between the conservative and the expansionist tendencies of the Bank. Substantial regularity of the free boundary associated to the differential game is obtained. Existence of an optimal policy is established when the regularity of the free boundary is strengthened slightly, and it is shown that the optimal process is a diffusion reflected at the boundary. Accepted 22 May 1998     

Viscosity Solutions of an Infinite-Dimensional Black—Scholes—Barenblatt Equation

We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman equations.     

Two‐parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. A functional analytic approach

We consider a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ, and the level of anisotropy of the cell is determined by a diagonal matrix γ with positive diagonal entries. The relative size of each periodic perforation is instead determined by a positive parameter ε. For a given value of γ, we analyze the behavior of the unique solution of the problem as tends to by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory.     

Anticipative portfolio optimization under constraints and a higher interest rate for borrowing

We study a stochastic control problem to maximize expected utility from terminal and/or consumption. The novel feature of our work is that the portfolio is allowed to anticipate the future with constraints and a higher interest rate for borrowing. The investor possesses information about the terminal values of the components of the Brownian motion, possibly distorted by ‘noise’. We use the technique from the so-called enlargement of filtrations, to model our problem. General existence results are established for optimal portfolio and consumption strategies. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients.