Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

Biswas, Imran H.; Jakobsen, Espen R.; Karlsen, Kenneth H.

doi:10.1007/s00245-009-9095-8

Cited by 55 publications

(73 citation statements)

References 38 publications

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“…On 31 The combined switching and impulse in Tang and Yong's paper is slightly different from ours as the switching and impulse cannot happen at the time in their setting. the other hand, Biswas et al [35] prove that the value function of a optimal switching of a Levy process is the uniqueness viscosity solution of a system of nonlocal variational inequalities.…”

Section: Solving the Optimal Control Problemmentioning

confidence: 99%

“…The figures here are meant for illustration only. 35 It is well-known that market has different behavior during the opening and closing period [22], so we only use the data between 12-3pm. 36 https://www.nasdaqtrader.com/trader.aspx?ID=marketsharedaily 37 NYSE TAQ contains consolidated trades and quotes from all US exchanges, but the timestamps of the trades and quotes are not synchronized [39] and subjected to significant delay [40].…”

Section: An Order Book Examplementioning

confidence: 99%

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Market making under a weakly consistent limit order book model

Law¹,

Viens

2019

High Frequency

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We develop a new market‐making model, from the ground up, which is tailored toward high‐frequency trading under a limit order book (LOB), based on the well‐known classification of order types in market microstructure. Our flexible framework allows arbitrary order volume, price jump, and bid‐ask spread distributions as well as the use of market orders. It also honors the consistency of price movements upon arrivals of different order types. For example, it is apparent that prices should never go down on buy market orders. In addition, it respects the price‐time priority of LOB. In contrast to the approach of regular control on diffusion as in the classical Avellaneda and Stoikov (Quantitative Finance, 8, 217, 2008) market‐making framework, we exploit the techniques of optimal switching and impulse control on marked point processes, which have proven to be very effective in modeling the order book features. The Hamilton‐Jacobi‐Bellman quasi‐variational inequality (HJBQVI) associated with the control problem can be solved numerically via finite‐difference method. We illustrate our optimal trading strategy with a full numerical analysis, calibrated to the order book statistics of a popular exchanged‐traded fund (ETF). Our simulation shows that the profit of market‐making can be severely overstated under LOBs with inconsistent price movements.

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Section: Solving the Optimal Control Problemmentioning

confidence: 99%

Section: An Order Book Examplementioning

confidence: 99%

Market making under a weakly consistent limit order book model

Law¹,

Viens

2019

High Frequency

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“…We mention that the system of IPDEs (1.1) and related to optimal switching problems of jump-diffusion process have been studied in [5], [16], [13], [14] or [17].…”

Section: Introductionmentioning

confidence: 99%

Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition

Hamadène

Neffati

2019

ASY

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In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution which is unique in the class of functions with polynomial growth. In our study, the main tool is the notion of reflected backward stochastic differential equations with jumps with interconnected obstacles for which we show the existence of a solution.

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“…In particular, let us recall that Cockburn et al [15] tackled up the continuous dependence estimate for quasi-linear second-order equations with Neumann boundary conditions, while Grinpenberg [23] addressed the case of the Dirichlet boundary data for the same equations. Afterwards, Jakobsen and Karlsen [27] extended their results to more general classes of equations (see also [28] for elliptic problems; we refer the reader to the papers [12,29] for integro-differential HJB equations). Furthermore, Jakobsen and Georgelin [26] extended the previous results to problems with more general boundary conditions and domains.…”

Section: Introductionmentioning

confidence: 99%

Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem

Marchi¹

2012

ESAIM: COCV

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Abstract. This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space [0, +∞)× R n and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.

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Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

Cited by 55 publications

References 38 publications

Market making under a weakly consistent limit order book model

Market making under a weakly consistent limit order book model

Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition

Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem

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