An implicit method for the finite time horizon Hamilton–Jacobi–Bellman quasi-variational inequalities

Ieda, Masashi

doi:10.1016/j.amc.2015.04.031

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…We formulate our execution problem as a combined stochastic control problem over a finite time horizon. The corresponding HJBQVI is solved numerically using a scheme similar to that proposed by Ieda [Ied13]. The performance criterion is used to maximize the terminal wealth including the terminal execution.…”

Section: Discussionmentioning

confidence: 99%

“…The goal of this section is to derive the corresponding Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). In Section 3, we present a numerical method for solving the HJBQVI, following a similar procedure to that found in [Ied13]. The HJBQVI is discretized by a finite difference scheme and is converted to an equivalent fixed point problem.…”

Section: Introductionmentioning

confidence: 99%

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A dynamic optimal execution strategy under stochastic price recovery

Ieda

2015

J. Finan. Eng.

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In the present paper, we study the optimal execution problem under stochastic price recovery based on limit order book dynamics. We model price recovery after execution of a large order by accelerating the arrival of the refilling order, which is defined as a Cox process whose intensity increases by the degree of the market impact. We include not only the market order but also the limit order in our strategy in a restricted fashion. We formulate the problem as a combined stochastic control problem over a finite time horizon. The corresponding Hamilton-Jacobi-Bellman quasivariational inequality is solved numerically. The optimal strategy obtained consists of three components: (i) the initial large trade; (ii) the unscheduled small trades during the period; (iii) the terminal large trade. The size and timing of the trade is governed by the tolerance for market impact depending on the state at each time step, and hence the strategy behaves dynamically. We also provide competitive results due to inclusion of the limit order, even though a limit order is allowed under conservative evaluation of the execution price.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A dynamic optimal execution strategy under stochastic price recovery

Ieda

2015

J. Finan. Eng.

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“…In water resources management, Unami and Mohawesh [25] proved unique existence of a viscosity solution to the Hamilton-Jacobi-Bellman (HJB) equation appearing in a stochastic DP problem for reservoir operation. Ieda [18] dealt with the HJB quasi-variational inequality (QVI) associated with the combined impulse and stochastic optimal control problem over a finite time horizon, applying it to an optimal forest harvesting problem. Some types of control problems for discrete-time decision processes including the one discussed here have continuous-time counterparts which can be regarded as impulse control problems [4].…”

Section: Introductionmentioning

confidence: 99%

Time periodic optimal policy for operation of a water storage tank using the dynamic programming approach

Unami

Mohawesh

Fadhil

2019

Applied Mathematics and Computation

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Operation of a water storage tank in a specific environment motivates mathematical studies on a discrete-time deterministic dynamic programming problem. The operator decides whether or not to open the valve releasing the water in the tank to a drip irrigation system, based on the information on the storage volume of the tank. Two cases of functional regularity, which are Lipschitz continuous and of bounded variations, are considered for the reward defining the performance index to be maximized. Firstly, it is shown that the value function inherits the Lipschitz continuity of the reward in the infinite time horizon problem with discounting. Then, time periodic value functions are discussed in terms of the fixedpoint theorem. Discrete approximation of value functions is discussed as well, to conduct numerical experiments with a-posteriori error estimation applied to the real-world problem where the discount rate approaches to unity. It is found that a Skiba point appears as a threshold of valve opening for each day in an optimal policy for operation. Practically, setting a constant threshold throughout the period is quite reasonable and acceptable for the operator of the water storage tank to irrigate the farmland.

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“…We refer the detail of the method to solve the fixed point problem to Ieda [6] and references therein. Solving the fixed point problem (10) by this method, we obtain the optimal strategy as a Markov strategy, i.e., l = l(t, x) and ζ = ζ(t, x).…”

Section: Optimal Execution Strategymentioning

confidence: 99%

Optimal execution problem: a combined stochastic optimal control approach

Ieda

2014

Stochastic Systems Theory and its Applications (SSS)

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We propose the optimal execution strategy including the limit order. The arrival of the counterpart market order of our limit order is modelled by the Poisson process reduced from the Cox process. We formulate the limit order and market order by the regular stochastic control and impulse control respectively. The performance criterion consists of the terminal wealth and the inventory penalty function which copes with the uncertainty of the limit order. We solve the corresponding Hamilton-Jacobi-Bellman quasi variational inequality numerically using the implicit method. The result of the numerical simulation with the obtained optimal strategy gives a quantitative evaluation of the advantage to include the limit order into the execution strategy.

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An implicit method for the finite time horizon Hamilton–Jacobi–Bellman quasi-variational inequalities

Cited by 4 publications

References 12 publications

A dynamic optimal execution strategy under stochastic price recovery

A dynamic optimal execution strategy under stochastic price recovery

Time periodic optimal policy for operation of a water storage tank using the dynamic programming approach

Optimal execution problem: a combined stochastic optimal control approach

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