Publisher's Synopsis
Stochastic analysis is analysis based on Ito's calculus. This calculus was developed to cope with questions arising in probability theory in which processes are modelled by motion along paths which typically are not differentiable. The development of this calculus now rests on linear analysis and measure theory. Stochastic analysis is a basic tool in much of modern probability theory and is used in many applied areas from biology to physics, especially statistical mechanics. It has become particularly well known via the Black-Scholes formula as a way of modelling financial markets and strategies. Stochastic control plays an important role in many scientific and applied disciplines including communications, engineering, medicine, finance and many others. It is one of the effective methods being used to find optimal decision-making strategies in applications. Possible control variables include reinsurance, investment, volume control, portfolio selection, and combinations of all these actions. The optimal strategy is determined dynamically, selected and changed at each point in time depending on the risk position of the business. The general stochastic control problem is intractable to solve and requires an exponential amount of memory and computation time. The reason is that the state space needs to be discretized and thus becomes exponentially large in the number of dimensions. Computing the expectation values means that all states need to be visited and requires the summation of exponentially large sums. The same intractabilities are encountered in reinforcement learning. Non-linear stochastic control problems display features not shared by deterministic control problems nor by linear stochastic control. In deterministic control, only one globally optimal solution exists. In stochastic control, the optimal solution is a weighted mixture of suboptimal solutions. Methods and Tools for Stochastic Analysis and Control is a compilation of some of the latest research in this important area. In particular, this book gives an overview of some of the theoretical methods and tools for stochastic analysis, and it presents the applications of these methods to problems in systems theory, science, and economics. The book provides a self-contained treatment on practical aspects of stochastic modeling and calculus including applications drawn from engineering, statistics, and computer science. Readers should be familiar with basic probability theory and have a working knowledge of stochastic calculus. PhD students and researchers in stochastic control will find this book useful. Stochastic analysis is analysis based on Ito's calculus. This calculus was developed to cope with questions arising in probability theory in which processes are modelled by motion along paths which typically are not differentiable. The development of this calculus now rests on linear analysis and measure theory. Stochastic analysis is a basic tool in much of modern probability theory and is used in many applied areas from biology to physics, especially statistical mechanics. It has become particularly well known via the Black-Scholes formula as a way of modelling financial markets and strategies. Stochastic control plays an important role in many scientific and applied disciplines including communications, engineering, medicine, finance and many others. It is one of the effective methods being used to find optimal decision-making strategies in applications. Possible control variables include reinsurance, investment, volume control, portfolio selection, and combinations of all these actions. The optimal strategy is determined dynamically, selected and changed at each point in time depending on the risk position of the business. The general stochastic control problem is intractable to solve and requires an exponential amount of memory and computation time. The reason is that the state space needs to be discretized and thus becomes exponentially large in the number of dimensions. Computing the expectation values means that all states need to be visited and requires the summation of exponentially large sums. The same intractabilities are encountered in reinforcement learning. Non-linear stochastic control problems display features not shared by deterministic control problems nor by linear stochastic control. In deterministic control, only one globally optimal solution exists. In stochastic control, the optimal solution is a weighted mixture of suboptimal solutions. Methods and Tools for Stochastic Analysis and Control is a compilation of some of the latest research in this important area. In particular, this book gives an overview of some of the theoretical methods and tools for stochastic analysis, and it presents the applications of these methods to problems in systems theory, science, and economics. The book provides a self-contained treatment on practical aspects of stochastic modeling and calculus including applications drawn from engineering, statistics, and computer science. Readers should be familiar with basic probability theory and have a working knowledge of stochastic calculus. PhD students and researchers in stochastic control will find this book useful.
