Continuous-time Stochastic Control and Optimization with Financial Applications

verfasst von: Huyên Pham

Verlag: Springer Berlin Heidelberg

Buchreihe : Stochastic Modelling and Applied Probability

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Some elements of stochastic analysis

Abstract

In this chapter, we present some useful concepts and results of stochastic analysis. There are many books focusing on the classical theory presented in this chapter. We mention among others Dellacherie and Meyer [DM80], Jacod [Jac79], Karatzas and Shreve [KaSh88], Protter [Pro90] or Revuz and Yor [ReY91], from which are quoted most of the results recalled here without proof. The reader is supposed to be familiar with the elementary notion of the theory of integration and probabilities (see e.g. Revuz [Rev94], [Rev97]). In the sequel, (Ω,ℱ,P) denotes a probability space. For p ∈ [1,∞), we denote by L ^p = L ^p(Ω,ℱ,P) the set of random variables ξ (valued in ℝ^d) such that |ξ|^p is integrable, i.e. E|ξ|^p < +∞.

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2. Stochastic optimization problems. Examples in finance

Abstract

In this chapter, we outline the basic structure of a stochastic optimization problem in continuous time, and we illustrate it through several examples from mathematical finance. The solution to these problems will be detailed later.

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3. The classical PDE approach to dynamic programming

Abstract

In this chapter, we use the dynamic programming method for solving stochastic control problems. We consider in Section 3.2 the framework of controlled diffusion and the problem is formulated on finite or infinite horizon. The basic idea of the approach is to consider a family of control problems by varying the initial state values, and to derive some relations between the associated value functions. This principle, called the dynamic programming principle and initiated in the 1950s by Bellman, is stated precisely in Section 3.3. This approach yields a certain partial differential equation (PDE), of second order and nonlinear, called Hamilton-Jacobi-Bellman (HJB), and formally derived in Section 3.4. When this PDE can be solved by the explicit or theoretical achievement of a smooth solution, the verification theorem proved in Section 3.5, validates the optimality of the candidate solution to the HJB equation. This classical approach to the dynamic programming is called the verification step. We illustrate this method in Section 3.6 by solving three examples in finance. The main drawback of this approach is to suppose the existence of a regular solution to the HJB equation. This is not the case in general, and we give in Section 3.7 a simple example inspired by finance pointing out this feature.

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4. The viscosity solutions approach to stochastic control problems

Abstract

As outlined in the previous chapter, the dynamic programming method is a powerful tool to study stochastic control problems by means of the Hamilton-Jacobi-Bellman equation. However, in the classical approach, the method is used only when it is assumed a priori that the value function is smooth enough. This is not necessarily true even in very simple cases.

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5. Optimal switching and free boundary problems

Abstract

The theory of optimal stopping and its generalization as optimal switching is an important and classical field of stochastic control, which knows a renewed increasing interest due to its numerous and various applications in economy and finance, in particular for real options. Actually, it provides a suitable modeling framework for the evaluation of optimal investment decisions in capital for firms. Hence, it permits to capture the value of managerial flexibility to adapt decisions in response to unexpected markets developments, which is a key element in the modern theory of real options.

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6. Backward stochastic differential equations and optimal control

Abstract

The theory of backward stochastic differential equations (BSDEs) was pioneered by Pardoux and Peng [PaPe90]. It became now very popular, and is an important field of research due to its connections with stochastic control, mathematical finance, and partial differential equations. BSDEs provide a probabilistic representation of nonlinear PDEs, which extends the famous Feynman-Kac formula for linear PDEs. As a consequence, BSDEs can be used for designing numerical algorithms to nonlinear PDEs.

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7. Martingale and convex duality methods

Abstract

In the optimization methods by dynamic programming or BSDEs studied in the previous chapters, the optimization carried essentially on the control process α influencing the state process. The basic idea of martingale methods is to reduce the initial problem to an optimization problem on the state variable by means of a linear representation under an expectation formula weighted by a variable, called a dual variable.

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Backmatter

Titel: Continuous-time Stochastic Control and Optimization with Financial Applications
verfasst von: Huyên Pham
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-89500-8
Print ISBN: 978-3-540-89499-5
DOI: https://doi.org/10.1007/978-3-540-89500-8