Inexact arithmetic considerations for direct control and penalty methods: American options under jump diffusion

Huang, Yu; Labahn, George

doi:10.1016/j.apnum.2013.02.007

Cited by 16 publications

(13 citation statements)

References 34 publications

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“…and it has been pointed out in [15,17,16] that a different choice of δ will generally lead to different policy iterations; more precisely, in our context, depending on the choice of δ, we obtain different adaptations of Algorithm B.1, all converging to the same solution. For theoretical considerations on the best choice of δ, we refer to [17]; numerically, we find the following.…”

Section: 23mentioning

confidence: 64%

Penalty Methods for the Solution of Discrete HJB Equations—Continuous Control and Obstacle Problems

Witte¹,

Reisinger²

2012

SIAM J. Numer. Anal.

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In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent iterative solvers for the penalised equations. Furthermore, we discuss under what conditions local quadratic convergence of the iterative solvers can be expected. We include numerical results demonstrating the competitiveness of our methods.

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Section: 23mentioning

confidence: 64%

Penalty Methods for the Solution of Discrete HJB Equations—Continuous Control and Obstacle Problems

Witte¹,

Reisinger²

2012

SIAM J. Numer. Anal.

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“…In a direct control formulation, either the generator (sup w∈W {∂u/∂t + Ł w u − ρu + f w }) or impulse (Mu − u) component is active at any grid point. Since these have different units, comparing them in floating point arithmetic requires a scaling factor δ > 0 to ensure fast convergence [16] (see also Lemma 4.1). Scaling by δ and discretizing (1.1) (ignoring boundary conditions) yields max max…”

Section: Direct Controlmentioning

confidence: 99%

Weakly Chained Matrices, Policy Iteration, and Impulse Control

Azimzadeh¹

2016

SIAM J. Numer. Anal.

100

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This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasivariational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the HJBQVI problem. Scheme (i) takes the form of a Bellman problem involving an operator which is not necessarily contractive. We consider the well-posedness of the Bellman problem and give sufficient conditions for convergence of the corresponding policy iteration. To do so, we use weakly chained diagonally dominant matrices, which give a graph-theoretic characterization of weakly diagonally dominant M-matrices. We compare schemes (i)-(iii) under the following examples: (a) optimal control of the exchange rate, (b) optimal consumption with fixed and proportional transaction costs, and (c) pricing guaranteed minimum withdrawal benefits in variable annuities. We find that one should abstain from using scheme (i). Definition 2.1 (Monotone matrix). A real square matrix A is monotone (in the sense of Collatz) if for all real vectorsWe use the following assumptions:Proposition 2.2 (Convergence of policy iteration). Suppose (H0), (H1), and that A(P ) is a monotone matrix for all P in P.(v ) ≥1 defined by Policy-Iteration is nondecreasing and converges to the unique solution v of (2.1). Moreover, if P is finite, convergence occurs in at most |P| iterations (i.e. v |P| = v |P|+1 = · · · ).The monotone convergence of (v ) ≥1 to the unique solution of (2.1) can be proven similarly to Theorem A.3 of Appendix A. See [7, Theorem 2.1] for a proof of the finite termination when P is finite. In practice, P is often finite, in which case (H0) and (H1) are trivial.Remark 2.3. Theorem A.3 establishes the existence and uniqueness of solutions to (2.1) independent of (H1.ii). Owing to this, results that rely on Proposition 2.2 can be relaxed to exclude (H1.ii), with the caveat that when P is infinite, Policy-Iteration be replaced by -Policy-Iteration (see Appendix A). In this case, the resulting sequence (v ) ≥1 is not necessarily nondecreasing. Weakly chained diagonally dominant matricesWe say row i of a complex matrix A := (a ij ) is strictly diagonally dominant (SDD) if |a ii | > j =i |a ij |. We say A is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with weak inequality instead. Definition 3.1. A complex square matrix A is said to be a weakly chained diagonally dominant (WCDD) if:(i) A is WDD; (ii) for each row r, there exists a path in the graph of A from r to an SDD row p.Recall that the directed graph of an M × M complex matrix A := (a ij ) is given by the vertices {1, . . . , M } and edges defined as follows: there exists an edge from i to j if and only if a ij = 0. Note that if r is itself an SDD row, the trivial path r → r satisfies the requirement of (ii) in the above.The nonsingularity of WCDD matrices is proven in [27]. We provide an elementary proof for the convenience of the reader: Lemma 3.2....

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“…break from the iteration 8: end if 9: end for The scale parameter in line 5 of Algorithm 2.2.1, used throughout the literature by Forsyth, Labahn and coauthors [4,[25][26][27], prevents the enforcement of unrealistic levels of accuracy for points x where v k+1 (x) ≈ 0. Additionally, note that having chosen the initial guess for the payoff v 0 , the initial guess for the strategy is induced by v 0 .…”

Section: Iterative Algorithm For Symmetric Gamesmentioning

confidence: 99%

A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games

Zabaljauregui

2020

Appl Math Optim

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Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author’s best knowledge, the only numerical method in the literature is the heuristic one we put forward in Aïd et al (ESAIM Proc Surv 65:27–45, 2019) to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory.

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Inexact arithmetic considerations for direct control and penalty methods: American options under jump diffusion

Cited by 16 publications

References 34 publications

Penalty Methods for the Solution of Discrete HJB Equations—Continuous Control and Obstacle Problems

Penalty Methods for the Solution of Discrete HJB Equations—Continuous Control and Obstacle Problems

Weakly Chained Matrices, Policy Iteration, and Impulse Control

A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games

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