Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

Continuous-time stochastic control and optimization with financial applications / Huyen Pham

This paper builds a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence provides a novel method for solving high-dimensional singular control problems, and enables us to extend the theory of reversible investment: sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions. Consequently, our regularity result links singular controls and Dynkin games through sequential optimal stopping problems.

Connections Between Singular Control and Optimal Switching

This paper analyzes a class of singular control problems for which value functions are not necessarily smooth. Necessary and su±cient conditions for the well-known smooth ¯t principle, along with the regularity of the value functions, are given. Explicit solutions for the optimal policy and for the value functions are provided. In particular, when payo® functions satisfy the usual Inada conditions, the boundaries between action and no-action regions are smooth and strictly monotonic as postulated and exploited in the existing literature (Dixit and Pindyck (1994); Davis, Dempster, Sethi, and Vermes(1987); Kobila (1993); Abel and Eberly (1997); Âksendal (2000); Scheinkman and Zariphopoulou (2001); Merhi and Zervos (2007); Alvarez (2006)). Illustrative examples for both smooth and non-smooth cases are discussed, to highlight the pitfall of solving singular control problems with a priori smoothness assumptions.

A Class of Singular Control Problems and the Smooth Fit Principle

The objective of this thesis is to develop and analyse two stochastic control problems
arising in the context of investment capacity expansion. In both problems the underlying market ﬂuctuations are modelled by a geometric Brownian motion. The decision
maker’s aim is to determine admissible capacity expansion strategies that maximise
appropriate expected present-value performance criteria.
In the ﬁrst model, capacity expansion has price/demand impact and involves proportional costs. The resulting optimisation problem takes the form of a singular
stochastic control problem. In the second model, capacity expansion has no impact on
price/demand but is associated with ﬁxed as well as proportional costs, thus resulting
in an impulse control problem.
Both problems are completely solved and the optimal strategies are fully characterised. In particular, the value functions are constructed explicitly as suitable classical
solutions to the associated Hamilton-Jacobi-Bellman equations

Hessah Al-Motairi

Papers

Models for investment capacity expansion