The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf

User’s guide to viscosity solutions of second order partial differential equations

This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

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Controlled Markov processes and viscosity solutions

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Stochastic differential equations and diffusion processes

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

This paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and $n$ risky assets whose prices are log-normal diffusions. We suppose that transactions between the assets incur a cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic programming leads to a variational inequality for the value function. Existence and uniqueness of a viscosity solution are proved. The variational inequality is solved by using a numerical algorithm based on policies, iterations, and multigrid methods. Numerical results are displayed for $n=1$ and $n=2$.

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On an Investment-Consumption Model With Transaction Costs

In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier-Stokes martingale problem where the probability space is also obtained as a part of the solution.

/pdf/stochastic-2-d-navier-stokes-equation-1f59t88wms.pdf

Stochastic 2-D Navier-Stokes Equation ∗

PREFACE GLOSSARY OF BASIC NOTATIONS ELLIPTIC EQUATIONS Background Problems Not in Divergence Form Problems in Divergence Form Markov-Feller Processes INTEGRO-DIFFERENTIAL OPERATORS Discussion The Whole Space Bounded Domains Adjoint Operators Unbounded Functions and Commutator Relation with Jump Processes INTEGRO-DIFFERENTIAL EQUATIONS Problems Not in Divergence Form Problems in Divergence Form GREEN FUNCTION ESTIMATES Discussion Basic Properties Green Spaces Dirichlet Boundary Conditions INVARIANT DENSITY MEASURE Discussion Ergodicity Asymptotic Behavior Boundary Singularity STOPPING TIME PROBLEMS Discussion Setting of the Problem Variational Inequality Asymptotic Behavior ERGODIC CONTROL PROBLEMS Stochastic Control Hamilton-Jacobi-Bellman Equation BIBLIOGRAPHY INDEX

Second Order Elliptic Integro-Differential Problems

Hybrid control and dynamic programming

We consider a dynamic system whose state is governed by a linear stochastic differential equation with time-dependent coefficients. The control acts additively on the state of the system. Our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process. It is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space-time domain. By using some decomposition arguments, the problems of a two-sided control, i.e. optimal corrections, and the case with constraints on the resources, i.e. finite fuel, can be reduced to a simpler case of only one-sided control, i.e. a monotone follower. These results are applied to solving some examples by the so-called method of similarity solutions.

/pdf/additive-control-of-stochastic-linear-systems-with-finite-el56c2alkw.pdf

Jose Luis Menaldi

Papers

On an Investment-Consumption Model With Transaction Costs

Stochastic 2-D Navier-Stokes Equation ∗

Second Order Elliptic Integro-Differential Problems

Hybrid control and dynamic programming

Additive control of stochastic linear systems with finite horizon