Shuenn-Jyi Sheu

Risk‐Sensitive Control and an Optimal Investment Model

We consider an optimal investment model in which the goal is to maximize the long-term growth rate of expected utility of wealth. In the model, the mean returns of the securities are explicitly affected by the underlying economic factors. The utility function is HARA. The problem is reformulated as an infinite time horizon risk-sensitive control problem. We study the dynamic programming equation associated with this control problem and derive some consequences of the investment problem.

Diffusion processes on graphs: stochastic differential equations, large deviation principle

Abstract. Itos rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices.

ON THE STRUCTURE OF SOLUTIONS OF ERGODIC TYPE BELLMAN EQUATION RELATED TO RISK-SENSITIVE CONTROL

Bellman equations of ergodic type related to risk-sensitive control are considered. We treat the case that the nonlinear term is positive quadratic form on first-order partial derivatives of solution, which includes linear exponential quadratic Gaussian control problem. In this paper we prove that the equation in general has multiple solutions. We shall specify the set of all the classical solutions and classify the solutions by a global behavior of the diffusion process associated with the given solution. The solution associated with ergodic diffusion process plays particular role. We shall also prove the uniqueness of such solution. Furthermore, the solution which gives us ergodicity is stable under perturbation of coefficients. Finally, we have a representation result for the solution corresponding to the ergodic diffusion.

Risk Sensitive Portfolio Management with Cox--Ingersoll--Ross Interest Rates: The HJB Equation

This paper presents an application of risk sensitive control theory in financial decision making. The investor has an infinite horizon objective that can be interpreted as maximizing the portfolios risk adjusted exponential growth rate. There are two assets, a stock and a bank account, and two underlying Brownian motions, so this model is incomplete. The novel feature here is that the interest rate for the bank account is governed by Cox--Ingersoll--Ross dynamics. This is significant for risk sensitive portfolio management because the factor process, unlike in the Gaussian and all other cases treated in the literature, cannot be negative (under appropriate parameterization).

ASYMPTOTICS OF THE PROBABILITY MINIMIZING A "DOWN-SIDE" RISK

We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussian model we obtain explicit solutions for the primal problem.

Large-time behavior of perturbed diffusion markov processes with applications to the second eigenvalue problem for fokker-planck operators and simulated annealing

AbstractWe study the large-time behavior and rate of convergence to the invariant measures of the processes dXε(t)=b(X)ε(t)) dt + εσ(Xε(t)) dB(t). A crucial constant Λ appears naturally in our study. Heuristically, when the time is of the order exp(Λ − α)/ε2 , the transition density has a good lower bound and when the process has run for about exp(Λ − α)/ε2, it is very close to the invariant measure. LetLε=(ε2/2)Λ − ∇U · ∇ be a second-order differential operator on ℝd. Under suitable conditions,Lz has the discrete spectrum

The behavior of the spectral gap under growing drift

\begin{gathered} 0 = \lambda _1^\varepsilon > - \lambda _2^\varepsilon ...and lim \varepsilon ^2 log \lambda _2^\varepsilon = - \Lambda \hfill \\ \varepsilon \to 0 \hfill \\ \end{gathered}

On the Hamilton--Jacobi--Bellman Equation for an Optimal Consumption Problem: I. Existence of Solution

LetU be a function from ℝd to [0,∞) with suitable conditions. A nonhomogeneous Markov processY(·) governed by