Hidehiro Kaise

Idempotent Expansions for Continuous-Time Stochastic Control

Max-plus methods have previously been used to solve deterministic control problems. The methods are based on max-plus (or min-plus) expansions and can yield curse-of-dimensionality-free numerical methods. In this paper, we explore min-plus methods for continuous-time stochastic control on a finite-time horizon. We first approximate the original value function via time-discretization. By generalizing the min-plus distributive property to continuum spaces, we obtain an algorithm for recursive computation of the time-discretized values, which we refer to as the idempotent distributed dynamic programming principle (IDDPP). Under the IDDPP, the value function at each step can be represented as an infimum of functions in a certain class. This is a min-plus expansion for the value function. For the specific class of problems considered here, we see that the class can be taken as that consisting of the quadratic functions. A means for reducing the numbers of constituent quadratic functions is discussed.

Idempotent Method for Continuous-Time Stochastic Control and Complexity Attenuation

Abstract We consider a min-plus based numerical method for solution of finite time-horizon control of nonlinear diffusion processes. The approach belongs to the class of curse-of-dimensionality-free methods. The min-plus distributive property is required. The price to pay is a very heavy curse-of-complexity. These methods perform well due to the complexity-attenuation step. This projects the solution down onto a near-optimal min-plus subspace.

Idempotent expansions for continuous-time stochastic control: compact control space

It is now well-known that many classes of deterministic control problems may be solved by max-plus or minplus (more generally, idempotent) numerical methods. These methods include max-plus basis-expansion approaches [1], [2], [7], [10], as well as the more recently developed curse-of-dimensionality-free methods [10], [15]. It has recently been discovered that idempotent methods are applicable to stochastic control and games. The methods are related to the above curse-of-dimensionality-free methods for deterministic control. In particular, a min-plus based method was found for stochastic control problems [11], [16], and a min-max method was discovered for games [12].

Path-dependent differential games of inf-sup type and Isaacs partial differential equations

We consider two-person zero-sum differential games under path-dependent dynamics and costs fixing the order of the inf and the sup for the payoffs. Using dynamic programming methods, the inf-sup type value functions are defined on sets of past trajectories of states which are infinite-dimensional spaces. Under a notion of co-invariant derivatives for the infinite-dimensional spaces, we obtain infinitesimal generators of the dynamic programming operators related to the value functions. Then, we associate the inf-sup type value functions with path-dependent Isaacs partial differential equations in the sense of viscosity solutions proposed by N. Lukoyanov.

Optimal portfolio for a highly risk-averse investor: A differential game interpretation

Risk-sensitive portfolio optimization is treated with a linear-Gaussian-factor model. The main interest is how a highly risk-averse investor controls his/her interest rate risk. A simple risk-averse limit with increasing risk-averse parameter γ ↑ ∞ is not appropriate: the associated maximized risk-sensitized expected growth rate goes to −∞ as γ ↑ ∞, and a “breakdown” occurs in the limit. Instead, a small-noise and large-risk-aversion limit is considered, assuming the factor-noise has a small parameter e 1, taking e-dependent risk-averse parameter γ(e) = O(e−2) 1, and letting e ↓ 0. The limit value is characterized as the value of a linear-quadratic differential game. A sequence (π(e))e>0 of e-dependent dynamic investment strategies is constructed from a saddle point of the game, and its asymptotic optimality is shown as e ↓ 0.

Structure on Solutions of Ergodic Type Bellman Equations of First and Second Orders: Some Observations through the Singular Limits

AbstractThe following sections are included:IntroductionStructure of Solutions in Second Order CaseSingular LimitsSome Observation on Structure of Viscosity Solutions in First Order CaseReferences

Deduction of some Bellman-Isaacs equations of ergodic type and their singular limits-relationships to eigenvalue problems of Schrodinger operators

The Bellman-Isaacs equation of ergodic type is studied in a particular case. A solution of the equation related to the principal eigenfunction of the Schrodinger operator is obtained as the limit of the solution of a Bellman-Isaacs equation of discounted type. Furthermore a singular limit of the equation is studied in relation to semiclassical analysis.