Existence, uniqueness and comparisons for BSDEs in general spaces

Cohen, Samuel N.; Elliott, Robert J.

doi:10.1214/11-aop679

Cited by 55 publications

(66 citation statements)

References 21 publications

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“…As for sufficient conditions for existence and uniqueness of Markov regime-switching BSDEs, we refer the reader, for example, to [27][28][29] and references therein.…”

Section: Frameworkmentioning

confidence: 99%

Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information

Pamen

2017

J Optim Theory Appl

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This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward-backward stochastic differential equations with jumps. First, a general sufficient maximum principle for optimal control for a system, driven by a Markov regime-switching forward-backward jump-diffusion model, is developed. In the regime-switching case, it might happen that the associated Hamiltonian is not concave and hence the classical maximum principle cannot be applied. Hence, an equivalent type maximum principle is introduced and proved. In view of solving an optimal control problem when the Hamiltonian is not concave, we use a third approach based on Malliavin calculus to derive a general stochastic maximum principle. This approach also enables us to derive an explicit solution of a control problem when the concavity assumption is not satisfied. In addition, the framework we propose allows us to apply our results to solve a recursive utility maximization problem.Keywords Forward-backward stochastic differential equations · Malliavin calculus · Regime switching · Recursive utility maximization · Stochastic maximum principleCommunicated by

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“…As for sufficient conditions for existence and uniqueness of Markov regime-switching BSDEs, we refer the reader, for example, to [27][28][29] and references therein.…”

Section: Frameworkmentioning

confidence: 99%

Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information

Pamen

2017

J Optim Theory Appl

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“…However, it is the natural approach when the driver term can itself jump (see [6]), it ensures that the driver is predictable, and as the integral is with respect to Lebesgue measure and our processes have at most countably many jumps, in this case the equation is unchanged whether the left limits are included or not. However, it is the natural approach when the driver term can itself jump (see [6]), it ensures that the driver is predictable, and as the integral is with respect to Lebesgue measure and our processes have at most countably many jumps, in this case the equation is unchanged whether the left limits are included or not.…”

Section: Remarkmentioning

confidence: 99%

“…In [3] and [6], a general condition under which the comparison theorem holds was presented, and in [5] a condition specific to finite state Markov chain BSDEs was also given. In the case of BSDEs with Markov chain noise, and in general for BSDEs with jumps, a further condition is required to ensure that the result holds.…”

Section: Definitionmentioning

confidence: 99%

Undiscounted Markov Chain BSDEs to Stopping Times

Cohen

2014

J. Appl. Probab.

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We consider backward stochastic differential equations in a setting where noise is generated by a countable state, continuous time Markov chain, and the terminal value is prescribed at a stopping time. We show that, given sufficient integrability of the stopping time and a growth bound on the terminal value and BSDE driver, these equations admit unique solutions satisfying the same growth bound (up to multiplication by a constant). This holds without assuming that the driver is monotone in y, that is, our results do not require that the terminal value be discounted at some uniform rate. We show that the conditions are satisfied for hitting times of states of the chain, and hence present some novel applications of the theory of these BSDEs.

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“…A continuous time theory of our proposed method for dynamically consistent bid ask prices may be possible as an application of solutions of backward stochastic differential equations in more general spaces as studied for example in Cohen and Elliott (2010c). However, one has to investigate how the drivers are to be constructed in such a context.…”

Section: Execution Costs and Efficient Execution Frontiersmentioning

confidence: 99%

Execution Costs and Efficient Execution Frontiers

Madan

2012

Ann. Finan. Econ.

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Single period risks acceptable to the market at zero cost are modeled by a convex set of random variables leading to bid and ask prices that are trade size dependent. The theory of nonlinear expectations is employed to construct dynamically consistent sequences of bid and ask unit size prices that are size and trade date contingent. We then study the optimal design of spot and forward trading to minimize execution costs. Finally, we illustrate the construction of a two period execution cost frontier trading a decrease in execution costs for additional exposure to price risk. Most structured products already have prices that depend on the direction of the trade. Additionally markets already exist for large block trades with their own price structure that takes account of the time allowed for its execution. This paper outlines arbitrage free mechanisms for generating such price structures that could lead to automating quotation systems for such markets. Additionally, we describe the tradeoffs implicit in seeking to commit to trades now as opposed to assuming the price risk implicit in delaying commitments. Such an explicit analysis could lead to the development of optimal execution algorithms that economize on the level of price risk absorbed into the execution strategies.

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Existence, uniqueness and comparisons for BSDEs in general spaces

Cited by 55 publications

References 21 publications

Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information

Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information

Undiscounted Markov Chain BSDEs to Stopping Times

Execution Costs and Efficient Execution Frontiers

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