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INNOVATIONS IN STOCHASTIC ANALYSIS AND APPLICATIONS with emphasis on STOCHASTIC CONTROL AND INFORMATION

Final Report Summary - INNOSTOCH (INNOVATIONS IN STOCHASTIC ANALYSIS AND APPLICATIONS with emphasis on STOCHASTIC CONTROL AND INFORMATION)

The main objectives of this project have been to introduce novel ideas, new approaches and methods in stochastic analysis, with the purpose of being able to solve problems that have been beyond reach by standard theories. Particular emphasis has been on problems related to optimal control and its interplay with information and applications, particularly to finance. In my opinion these objectives have been reached in several research areas. More specifically, the achievements can be summarised as follows:

(i) Insider equilibrium models
My coauthors and I have introduced new ideas and methods in insider trading, including the classical insider equilibrium models of Kyle and Back. By using a filter theory approach we have been able to generalise results of earlier papers in the literature. For example, we have proved a new anticipative linear filtering equation and we extended the Kyle-Back model to include memory in the system and to a situation with partially informed noise traders.

(ii) Optimal control with model uncertainty (worst case scenario models)
With my coauthor Agnès Sulem I have studied optimal control problems under model uncertainty, also called robust control. With model uncertainty we mean that the underlying probability measure is not assumed to be known, but we seek a control process which performs optimally in the least favourable case (“worst case scenario”).This type of problems has had a renewed attention in the aftermath of the financial crisis. We have introduced a general machinery for dealing with such problems in terms of games of forward-backward stochastic differential equations (FBSDEs).

(iii) Optimal control of stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) with delay.
With my coauthors Agnès Sulem and Tusheng Zhang I have obtained efficient solution methods for SDEs with delay. The solution involves time-advanced BSDEs, which is a topic of independent interest. We have also obtained extensions to SPDEs. This allows for more realistic models in applications, e.g. to descriptions of population growth that combines effects from individual growth in time and spreading (diffusion) in space, often called stochastic reaction-diffusion equations.

(iv) Singular control and optimal stopping
With my coauthor Agnès Sulem I have developed a maximum principle for singular control. This makes it possible to study singular control under partial information. Since singular control is related to optimal stopping, this also gives results on partial information optimal stopping. The results have applications to problems of optimal consumption in finance and optimal harvesting strategies of populations. This part relates both to Group (iii) above and to Group (vi) below.

(v) Stochastic control with infinite horizon
The classical maximum principle for optimal control has the disadvantage that it only works for fine horizon control. With coauthors I have shown how to obtain a infinite horizon versions of this powerful method. This is useful for many applications. For example, it makes it possible to study sustainable harvesting policies.

(vi) Optimal control of mean-field stochastic differential equations
Mean-field stochastic differential equations appear as models of stochastic systems with many interacting particles. It is also of interest as models of systemic risk in finance. Such systems are not Markovian, and neither the dynamic programming method nor the classical maximum principle method apply to them. With coauthors I have introduced other solution methods, and extended the situation to include mean-field stochastic differential games and singular control.