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

Event-triggered recursive state estimation for dynamical networks under randomly switching topologies and multiple missing measurements

A general maximum principle for optimal control of forward–backward stochastic systems

This technical note is concerned with a partially observed optimal control problem, whose novel feature is that the cost functional is of mean-field type. Hence determining the optimal control is time inconsistent in the sense that Bellman's dynamic programming principle does not hold. A maximum principle is established using Girsanov's theorem and convex variation. Some nonlinear filtering results for backward stochastic differential equations (BSDEs) are developed by expressing the solutions of the BSDEs as some Ito's processes. An illustrative example is demonstrated in terms of the maximum principle and the filtering.

Stochastic Maximum Principle for Mean-Field Type Optimal Control Under Partial Information

Leader-follower stochastic differential game with asymmetric information and applications

This paper studies the partial information control problems of backward stochastic systems. There are three major contributions made in this paper: (i) First, we obtain a new stochastic maximum principle for partial information control problems. Our method relies on a direct calculation of the derivative of the cost functional. (ii) Second, we introduce two classes of partial information linear-quadratic backward control problems for the first time and then investigate them using the maximum principle. Complete and explicit solutions are obtained in terms of some forward and backward stochastic differential filtering equations. (iii) Last but not least, we study a class of full information stochastic pension fund optimization problems which can be viewed as a special case of our general partial information ones. Applying the aforementioned maximum principle, we derive the optimal contribution policy in closed-form and present some related economic remarks.

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A Maximum Principle for Partial Information Backward Stochastic Control Problems with Applications

In this paper, we study a partial information optimal control problem derived by forward-backward stochastic systems with correlated noises between the system and the observation. Utilizing a direct method, an approximation method, and a Malliavin derivative method, we establish three versions of maximum principle (i.e., necessary condition) for optimal control. To show their applications, we work out two illustrative examples within the frameworks of linear-quadratic control and recursive utility and then solve them via the maximum principles and stochastic filtering.

Maximum Principles for Forward-Backward Stochastic Control Systems with Correlated State and Observation Noises

This paper studies a linear-quadratic optimal control problem derived by forward-backward stochastic differential equations, where the drift coefficient of the observation equation is linear with respect to the state $x$ , and the observation noise is correlated with the state noise, in the sense that the cross-variation of the state and the observation is nonzero. A backward separation approach is introduced. Combining it with variational method and stochastic filtering, two optimality conditions and a feedback representation of optimal control are derived. Closed-form optimal solutions are obtained in some particular cases. As an application of the optimality conditions, a generalized recursive utility problem from financial markets is solved explicitly.

Guangchen Wang

Papers

Stochastic Maximum Principle for Mean-Field Type Optimal Control Under Partial Information

A Maximum Principle for Partial Information Backward Stochastic Control Problems with Applications

Maximum Principles for Forward-Backward Stochastic Control Systems with Correlated State and Observation Noises

Leader-follower stochastic differential game with asymmetric information and applications

A Linear-Quadratic Optimal Control Problem of Forward-Backward Stochastic Differential Equations With Partial Information