Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Bokanowski, Olivier; Debrabant, Kristian

doi:10.1093/imanum/draa014

Cited by 6 publications

(13 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this scheme, under the assumption of uniform parabolicity, we establish new stability results in the H 1 -norm for fully nonlinear HJB and Isaacs equations, and in the L 2 -norm for the semilinear case (see Theorems 4 and 5, respectively). These generalize some results of [4,5,10] to more general non-linear situations. From this analysis we deduce error bounds for classical smooth and piecewise smooth solutions in the semilinear uniformly parabolic case (see Theorems 7 and 19).…”

Section: Introductionsupporting

confidence: 82%

“…The previous arguments can also be used to derive error estimates for piecewise smooth solutions. In this case, we will need to limit the number of non-regular points that may appear in the exact solution (assumption (A6)(i) is similar to [5]).…”

Section: Proposition 16mentioning

confidence: 99%

“…This result is extended to a semi-linear example in [10]; the application to incompressible Navier-Stokes equations has been analyzed in [14]. In [5], a closely related BDF scheme is studied for a diffusion problem with an obstacle term (which includes the American option problem in mathematical finance).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

2021

Self Cite

View full text Add to dashboard Cite

We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the $$L^2$$ L 2 norm for linear and semi-linear equations, and in the $$H^1$$ H 1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in $$L^2$$ L 2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.

show abstract

Section: Introductionsupporting

confidence: 82%

Section: Proposition 16mentioning

confidence: 99%

See 1 more Smart Citation

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…The second order version of the method is not theoretically guaranteed to converge to the viscosity solution in the degenerate case, however, recent results in Bokanowski and Debrabant [5] and Bokanowski et al [6] show stability of BDF schemes in more regular cases and we will demonstrate excellent empirical properties of the scheme below. 6.1.1.…”

Section: Numerical Scheme and Convergence Testsmentioning

confidence: 55%

“…The monotonicity is violated for the BDF2 scheme due to the alternating signs in the approximations to the first time and space derivatives. It is shown in Bokanowski and Debrabant [5] that such schemes still have good stability properties for American options under Black-Scholes. Although this analysis is not applicable here due to the degeneracy of the diffusion operator, we observe no stability issues in the numerical tests.…”

Section: Numerical Scheme and Convergence Testsmentioning

confidence: 99%

Executive Stock Option Exercise with Full and Partial Information on a Drift Change Point

Henderson¹,

Kladívko²,

Monoyios³

et al. 2020

SIAM J. Finan. Math.

View full text Add to dashboard Cite

We analyse the optimal exercise of an American call executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point, and seek to optimally exercise the option by maximising its discounted payoff under the physical measure. The first agent has full information, and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration F in which the stock drift becomes a diffusion driven by the innovations process, an F-Brownian motion also driving the stock under F, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterise the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.

show abstract

High-order filtered schemes for time-dependent second order HJB equations

2018

Self Cite

View full text Add to dashboard Cite

In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal., 51(1): 2013, and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations. For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by "filtering" them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes. 1 arXiv:1611.04939v1 [math.NA] 15 Nov 2016 b, σ, f, are bounded, Lipschitz and Hölder continuous respectively in space and time, i.e. there is a constant K such that, for any ϕ ∈ {b, σ, f, }, sup a∈A sup (t,x) =(s,y)Moreover, the solution is then also Lipschitz continuous in space and locally 1/2-Hölder continuous in time [14]. The boundedness assumption can be removed, see also [14]. Moreover, in the paper we consider the equation on bounded numerical domains so that the boundedness assumption is automatically satisfied.In this article, we propose approximation schemes for (1.1) for which convergence is guaranteed in a general setting, and which exhibit high order convergence under sufficient regularity of the solution. As we will explain in the following, these are in a sense two conflicting goals, and we will meet them by application of a so-called "filter", an idea introduced in [18].The seminal work by Barles and Souganidis [6] establishes that a consistent and stable scheme converges to the viscosity solution of (1.1) if it is also monotone. That this is not simply a requirement of the proof, but can be crucial in practice, is demonstrated, e.g., by [30] (and also in Section 4.2 here). It is shown there empirically that for the uncertain volatility model from [25], perhaps the simplest non-trivial second order HJB equation there is, the (consistent and stable but) non-monotone Crank-Nicolson scheme fails to converge to the correct viscosity solution in the presence of Lipschitz initial data without higher regularity. This is in contrast to classical solutions where there is no such monotonicity requirement. As the setting of solutions above (i.e., Lipschitz in space and 1/2-Hölder in time) is the natural setting for HJB equations and often no higher global regularity is observed, a wide literature on monotone schemes has developed.By Godunov's theorem [19], in the case of explicit linear schemes for the approximation of the line...

show abstract

Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Cited by 6 publications

References 25 publications

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

Executive Stock Option Exercise with Full and Partial Information on a Drift Change Point

High-order filtered schemes for time-dependent second order HJB equations

Contact Info

Product

Resources

About