The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

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 equations in infinite dimensions

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

1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.

Stochastic flows and stochastic differential equations

-1We consider the problem of optimal multiple switching in a finite horizon when the state of the system, including the switching costs, is a general adapted stochastic process. The problem is formulated as an extended impulse control problem and solved using probabilistic tools such as the Snell envelope of processes and reflected backward stochastic differential equations. Finally, when the state of the system is a Markov process, we show that the associated vector of value functions provides a viscosity solution to a system of variational inequalities with interconnected obstacles.

/pdf/a-finite-horizon-optimal-multiple-switching-problem-1ee6yc1nel.pdf

A Finite Horizon Optimal Multiple Switching Problem

We analyze multidimensional BSDEs in a filtration that supports a Brownian motion and a Poisson random measure. Under a monotonicity assumption on the driver, the paper extends several results from the literature. We establish existence and uniqueness of solutions in provided that the generator and the terminal condition satisfy appropriate integrability conditions. The analysis is first carried out under a deterministic time horizon, and then generalized to random time horizons given by a stopping time with respect to the underlying filtration. Moreover, we provide a comparison principle in dimension one.

/pdf/bsdes-with-monotone-generator-driven-by-brownian-and-poisson-4uqs61mkk6.pdf

BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting

http://perso.univ-lemans.fr/~apopier/documents/Research/P06.pdf

Backward stochastic differential equations with singular terminal condition

This paper deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation (BSDE for short) with one reflecting barrier in the case when the terminal value, the generator and the obstacle process are Lp-integrable with p ∈ ]1, 2[. To construct the solution we use two methods: penalization and Snell envelope. As an application we broaden the class of functions for which the related obstacle partial differential equation problem has a unique viscosity solution.

Alexandre Popier

Papers

A Finite Horizon Optimal Multiple Switching Problem

BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting

Backward stochastic differential equations with singular terminal condition

Lp-SOLUTIONS FOR REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS