The damped Crank–Nicolson time-marching scheme for the adaptive solution of the Black–Scholes equation

Goll, Christian; Rannacher, Wolf; Woolner, Winnifried

doi:10.21314/jcf.2015.301

Cited by 12 publications

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“…holds for all w σ ∈ W σ . This allows us to formulate the variational discrete problem (15). We refrain from giving a detailed derivation for an optimality system and the sparsity structure for this problem as this would closely follow the procedure in the continuous setting (see Lemma 7 and Remark 8).…”

Section: Sparsity Structurementioning

confidence: 99%

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Maximal discrete sparsity in parabolic optimal control with measures

Herberg¹,

Hinze²,

Schumacher³

2020

Mathematical Control &Amp; Related Fields

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We consider variational discretization [18] of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in L q and weakly- * in M, respectively, to their smooth counterparts, where q ∈ (1, min{2, 1 + 2/d}] is the spatial dimension. Furthermore, we compare our approach to [8], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.

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Section: Sparsity Structurementioning

confidence: 99%

“…In order to solve (P * σ ), we want to represent L σ : W σ → Y * σ by a matrix, as done in [7]. From [12,Section 4] and [15] we know that the matrix representation of L * σ : Y σ → W * σ yields a Crank-Nicolson scheme with a smoothing step. We will derive this first.…”

Section: Sparsity Structurementioning

confidence: 99%

Maximal discrete sparsity in parabolic optimal control with measures

Herberg¹,

Hinze²,

Schumacher³

2020

Mathematical Control &Amp; Related Fields

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“…This setting yields a Crank-Nicholson scheme with smoothing step (see [3] and [4]). We define the discrete state space consisting of piecewise linear and continuous finite elements in space and piecewise constant functions with respect to time and the discrete test space consisting of continuous and piecewise linear functions in space and time.…”

Section: Variational Discretizationmentioning

confidence: 99%

Sparse discretization of sparse control problems

Herberg

Hinze

Schumacher

2019

Proc Appl Math and Mech

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We consider optimal control problems that inherit a sparsity structure, especially we look at problems governed by measure controls. Our goal is to achieve maximal sparsity on the discrete level. We use variational discretization of the control problems utilizing a Petrov-Galerkin approximation of the state, which induces controls that are composed of Dirac measures. In the parabolic case this allows us to achieve sparsity on the discrete level in space and time. Numerical experiments show the differences of this approach to a full discretization approach.Inspired by [2] we consider the continuous minimization problemwhere the state y ∈ L q (Q) solves a heat equation with right hand side u, initial data u 0 and zero boundary values in a very weak sense, i.e.For more details we refer to [2, Definition 2.1.]. Here, Ω is an open bounded domain in IR d , d ∈ {1, 2, 3} with Lipschitz boundary Γ := ∂Ω, and Ω c ⊂⊂ Ω. For given T > 0 we consider the time interval (0, T ) and the space-time domain Q c ⊂⊂ Q := Ω × (0, T ). Let α > 0, β > 0 be suitable parameters and q ∈ (1, min{2, (d + 2)/d}). As control spaces we consider the real and regular Borel measures M(Ω c ) := C(Ω c ) * and M(Q c ) := C(Q c ) * , respectively. In this setting, (P ) has a unique solution (ū 0 ,ū) with associated stateȳ, which can be proven similar to [2, Theorem 2.7.]. For the controls we observe the following sparsity (see [2, Corollary 3.2.]):wherew ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ C(Q) is the adjoint state (see [2, Theorem 3.1.]) andū 0 =ū + 0 −ū − 0 andū =ū + −ū − are the respective Jordan decompositions. Considering it the generic case thatw is not constant on sets of measure greater than zero, this implies that the support sets of the controls are of measure zero. This motivates us to suggest a discretization strategy, which preserves this sparsity structure on the discrete level in space and time. Variational discretizationWe want to achieve the desired maximal discrete sparsity, i.e. Dirac-measures in space-time, by choosing the Petrov-Galerkinansatz and -test spaces that will induce this structure in combination with the variational discretization concept introduced in [5]. This concept, via the discretization of the test space and the optimality conditions, induces an implicit discretization of the controls (u 0 , u) ∈ M(Ω c ) × M(Q c ). This is how we control the discrete structure of the controls.In the following we will indicate discretization in space-time by index σ := (τ, h). We define the discrete state space consisting of piecewise linear and continuous finite elements in space and piecewise constant functions with respect to time

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“…It was proved that for initial data from L 2 that two damping steps are sufficient. For more details we refer [10,11].…”

Section: Black-scholes Equationmentioning

confidence: 99%

Fourth order scheme for wavelet based solution of Black-Scholes equation

Finěk¹

2017

AIP Conference Proceedings

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Abstract. The present paper is devoted to the numerical solution of the Black-Scholes equation for pricing European options. We apply the Crank-Nicolson scheme with Richardson extrapolation for time discretization and Hermite cubic spline wavelets with four vanishing moments for space discretization. This scheme is the fourth order accurate both in time and in space. Computational results indicate that the Crank-Nicolson scheme with Richardson extrapolation significantly decreases the amount of computational work. We also numerically show that optimal convergence rate for the used scheme is obtained without using startup procedure despite the data irregularities in the model.

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The damped Crank–Nicolson time-marching scheme for the adaptive solution of the Black–Scholes equation

Cited by 12 publications

References 0 publications

Maximal discrete sparsity in parabolic optimal control with measures

Maximal discrete sparsity in parabolic optimal control with measures

Sparse discretization of sparse control problems

Fourth order scheme for wavelet based solution of Black-Scholes equation

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