Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Stochastic differential equations and diffusion processes

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

Incentive or performance fees for money managers are frequently accompanied by high-water mark provisions which condition the payment of the performance fee upon exceeding the maximum achieved share value. In this paper, we show that hedge fund performance fees are valuable to money managers, and conversely represent a claim on a significant proportion of investor wealth. The high-water mark provisions in these contracts limit the value of the performance fees. We provide a closed-form solution to the high-water mark

High-Water Marks and Hedge Fund Management Contracts

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is nondominated. We propose studying this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power, and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally, several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models.

Robust utility maximization in nondominated models with 2bsde: the uncertain volatility model

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimization, over a set of possibly nondominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalizations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semimartingale characterization of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149–190]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of Nutz [Stochastic Process. Appl. 125 (2015) 4543–4555], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path–dependent partial differential equation (PPDE).

/pdf/stochastic-control-for-a-class-of-nonlinear-kernels-and-kdt9xnkd7s.pdf

Stochastic control for a class of nonlinear kernels and applications

SelectNet: Self-paced learning for high-dimensional partial differential equations

We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. As an example of application, we prove that reflected BSDEs are well-posed in a general framework.

/pdf/a-unified-approach-to-a-priori-estimates-for-supersolutions-2wi9hdcxuf.pdf

A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose studying this problem in the framework of second order backward stochastic dierential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2].

http://www.cmapx.polytechnique.fr/~possamai/1201.0769v4.pdf

Chao Zhou

Papers

Robust utility maximization in nondominated models with 2bsde: the uncertain volatility model

Stochastic control for a class of nonlinear kernels and applications

SelectNet: Self-paced learning for high-dimensional partial differential equations

A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations

Robust Utility Maximization in Non-dominated Models with 2BSDEs