In this paper we introduce a new technique to construct unique strong solutions of SDEs with sing... more In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
В этой статье мы преследуем двоякую цель. Во-первых, мы распространяем хорошо известное соотношен... more В этой статье мы преследуем двоякую цель. Во-первых, мы распространяем хорошо известное соотношение между оптимальной остановкой и рандомизированной остановкой заданного случайного процесса на ситуацию, когда доступный поток информации - это фильтрация, которая априори не предполагается как-либо связанной с фильтрацией случайного процесса. В этом случае мы говорим об оптимальной остановке и рандомизированной остановке при общем потоке информации. Во-вторых, следуя идее Н. В. Крылова (1977), мы вводим специальную задачу сингулярного стохастического управления с общей информацией и показываем, что она эквивалентна задачам оптимального управления и рандомизированного управления с частичной информацией. Далее мы показываем, что решение указанной задачи сингулярного управления может быть выражено в терминах вариационных неравенств для частичной информации.
We prove maximum principles for the problem of optimal control for a jump diffusion with infinite... more We prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information. The results are applied to partial information optimal consumption and portfolio problems in infinite horizon.
In this paper we introduce a new technique to construct unique strong solutions of SDEs with sing... more In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
We prove a maximum principle for the problem of optimal control for a fractional diffusion with i... more We prove a maximum principle for the problem of optimal control for a fractional diffusion with infinite horizon. Further, we show existence of fractional backward stochastic differential equations on infinite horizon. We illustrate our findings with an example.
We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon... more We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results by an application to the optimal consumption rate from an economic quantity.
In this paper we introduce a new technique to construct unique strong solutions of SDE's with sin... more In this paper we introduce a new technique to construct unique strong solutions of SDE's with singular coefficients driven by certain Lévy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
Stochastics An International Journal of Probability and Stochastic Processes, 2012
We study stochastic differential games of jump diffusions driven by Brownian motions and compensa... more We study stochastic differential games of jump diffusions driven by Brownian motions and compensated Poisson random measures, where one of the players can choose the stochastic control and the other player can decide when to stop the system. We prove a verification theorem for such games in terms of a Hamilton–Jacobi–Bellman variational inequality. The results are applied to study some specific examples, including optimal resource extraction in a worst-case scenario, and risk minimizing optimal portfolio and stopping.
In this paper we introduce a new technique to construct unique strong solutions of SDEs with sing... more In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
В этой статье мы преследуем двоякую цель. Во-первых, мы распространяем хорошо известное соотношен... more В этой статье мы преследуем двоякую цель. Во-первых, мы распространяем хорошо известное соотношение между оптимальной остановкой и рандомизированной остановкой заданного случайного процесса на ситуацию, когда доступный поток информации - это фильтрация, которая априори не предполагается как-либо связанной с фильтрацией случайного процесса. В этом случае мы говорим об оптимальной остановке и рандомизированной остановке при общем потоке информации. Во-вторых, следуя идее Н. В. Крылова (1977), мы вводим специальную задачу сингулярного стохастического управления с общей информацией и показываем, что она эквивалентна задачам оптимального управления и рандомизированного управления с частичной информацией. Далее мы показываем, что решение указанной задачи сингулярного управления может быть выражено в терминах вариационных неравенств для частичной информации.
We prove maximum principles for the problem of optimal control for a jump diffusion with infinite... more We prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information. The results are applied to partial information optimal consumption and portfolio problems in infinite horizon.
In this paper we introduce a new technique to construct unique strong solutions of SDEs with sing... more In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
We prove a maximum principle for the problem of optimal control for a fractional diffusion with i... more We prove a maximum principle for the problem of optimal control for a fractional diffusion with infinite horizon. Further, we show existence of fractional backward stochastic differential equations on infinite horizon. We illustrate our findings with an example.
We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon... more We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results by an application to the optimal consumption rate from an economic quantity.
In this paper we introduce a new technique to construct unique strong solutions of SDE's with sin... more In this paper we introduce a new technique to construct unique strong solutions of SDE's with singular coefficients driven by certain Lévy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
Stochastics An International Journal of Probability and Stochastic Processes, 2012
We study stochastic differential games of jump diffusions driven by Brownian motions and compensa... more We study stochastic differential games of jump diffusions driven by Brownian motions and compensated Poisson random measures, where one of the players can choose the stochastic control and the other player can decide when to stop the system. We prove a verification theorem for such games in terms of a Hamilton–Jacobi–Bellman variational inequality. The results are applied to study some specific examples, including optimal resource extraction in a worst-case scenario, and risk minimizing optimal portfolio and stopping.
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Papers by Sven Haadem