Semi-Lagrangian schemes for linear and fully non-linear diffusion equations

Debrabant, Kristian; Jakobsen, Espen R.

doi:10.1090/s0025-5718-2012-02632-9

Cited by 93 publications

(198 citation statements)

References 31 publications

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“…Typical operator S ρ can be obtained by using an explicit or implicit finite difference method on (3.5a) (see [16,15,33]), or a semi-Lagrangian (SL) scheme ( [35,19,24]). In (4.1), the value of W n−1 i,j may depend on the whole function W ρ (i.e., all the values W k i,j for k = 0, · · · , N ).…”

Section: Numerical Approximationmentioning

confidence: 99%

“…The operator T corresponds to a discretisation of the equation (3.5a) by a semi-Lagrangian scheme (see [35,19,24]). Now, define an approximation method for the system (3.5) as follows:…”

Section: 2mentioning

confidence: 99%

“…The error estimates theory is based on some technique of "shaking coefficients" and regularization introduced by Krylov in [30,31] and studied later by many authors [6,7,8,13,14,24]. The main idea consists of regularizing the exact solution ϑ in order to obtain a smooth sub-solution ϑ ε (for an approximation parameter ε > 0) to the equation (3.5), then the upper-bound error estimate can be obtained by using the consistency estimate (4.16).…”

Section: Error Bounds For the Semi-lagrangian Schemementioning

confidence: 99%

“…In the framework of viscosity solutions, semi-Lagrangian schemes for first order HJB equations have been studied in [20]. Extensions to the second order case can be found in [35,19,23,24]. In our context, the numerical scheme that will be studied couples the classical semi-Lagrangian scheme with an additional projection step on the boundary.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

2014

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Abstract. This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order HamiltonJacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.

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Section: Numerical Approximationmentioning

confidence: 99%

“…The operator T corresponds to a discretisation of the equation (3.5a) by a semi-Lagrangian scheme (see [35,19,24]). Now, define an approximation method for the system (3.5) as follows:…”

Section: 2mentioning

confidence: 99%

Section: Error Bounds For the Semi-lagrangian Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

2014

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“…Then, we introduce a SL scheme and prove its convergence. Recall that SL schemes have been introduced in [16] for first order Hamilton-Jacobi equations and then extended to the second order case in [15,18,19,32,33]. For time-dependent equations with an oblique boundary condition (that appears in the case of finite horizon control problems), the SL method has been investigated in [11].…”

Section: Introductionmentioning

confidence: 99%

Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost

Kröner¹,

Picarelli²,

Zidani³

2018

SIAM J. Control Optim.

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Abstract. An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order Hamilton-Jacobi-Bellman (HJB) equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity and stability of the scheme. These properties are verified for a specific semi-Lagrangian scheme.

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A semi‐Lagrangian ε$$ \varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate

Lu,

Dang

2023

Numerical Methods Partial

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We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.

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Semi-Lagrangian schemes for linear and fully non-linear diffusion equations

Cited by 93 publications

References 31 publications

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost

A semi‐Lagrangian ε$$ \varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate

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