4,072 research outputs found
Two semi-Lagrangian fast methods for Hamilton-Jacobi-Bellman equations
In this paper we apply the Fast Iterative Method (FIM) for solving general
Hamilton-Jacobi-Bellman (HJB) equations and we compare the results with an
accelerated version of the Fast Sweeping Method (FSM). We find that FIM can be
indeed used to solve HJB equations with no relevant modifications with respect
to the original algorithm proposed for the eikonal equation, and that it
overcomes FSM in many cases. Observing the evolution of the active list of
nodes for FIM, we recover another numerical validation of the arguments
recently discussed in [Cacace et al., SISC 36 (2014), A570-A587] about the
impossibility of creating local single-pass methods for HJB equations
Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?
The use of local single-pass methods (like, e.g., the Fast Marching method)
has become popular in the solution of some Hamilton-Jacobi equations. The
prototype of these equations is the eikonal equation, for which the methods can
be applied saving CPU time and possibly memory allocation. Then, some natural
questions arise: can local single-pass methods solve any Hamilton-Jacobi
equation? If not, where the limit should be set? This paper tries to answer
these questions. In order to give a complete picture, we present an overview of
some fast methods available in literature and we briefly analyze their main
features. We also introduce some numerical tools and provide several numerical
tests which are intended to exhibit the limitations of the methods. We show
that the construction of a local single-pass method for general Hamilton-Jacobi
equations is very hard, if not impossible. Nevertheless, some special classes
of problems can be actually solved, making local single-pass methods very
useful from the practical point of view.Comment: 19 page
A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance
We present a simple and easy to implement method for the numerical solution
of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many
cases, the considered problems have only a viscosity solution, to which,
fortunately, many intuitive (e.g. finite difference based) discretisations can
be shown to converge. However, especially when using fully implicit time
stepping schemes with their desirable stability properties, one is still faced
with the considerable task of solving the resulting nonlinear discrete system.
In this paper, we introduce a penalty method which approximates the nonlinear
discrete system to first order in the penalty parameter, and we show that an
iterative scheme can be used to solve the penalised discrete problem in
finitely many steps. We include a number of examples from mathematical finance
for which the described approach yields a rigorous numerical scheme and present
numerical results.Comment: 18 Pages, 4 Figures. This updated version has a slightly more
detailed introduction. In the current form, the paper will appear in SIAM
Journal on Numerical Analysi
Nonlinear Parabolic Equations arising in Mathematical Finance
This survey paper is focused on qualitative and numerical analyses of fully
nonlinear partial differential equations of parabolic type arising in financial
mathematics. The main purpose is to review various non-linear extensions of the
classical Black-Scholes theory for pricing financial instruments, as well as
models of stochastic dynamic portfolio optimization leading to the
Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both
problems can be represented by solutions to nonlinear parabolic equations.
Qualitative analysis will be focused on issues concerning the existence and
uniqueness of solutions. In the numerical part we discuss a stable
finite-volume and finite difference schemes for solving fully nonlinear
parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387
An Efficient Policy Iteration Algorithm for Dynamic Programming Equations
We present an accelerated algorithm for the solution of static
Hamilton-Jacobi-Bellman equations related to optimal control problems. Our
scheme is based on a classic policy iteration procedure, which is known to have
superlinear convergence in many relevant cases provided the initial guess is
sufficiently close to the solution. In many cases, this limitation degenerates
into a behavior similar to a value iteration method, with an increased
computation time. The new scheme circumvents this problem by combining the
advantages of both algorithms with an efficient coupling. The method starts
with a value iteration phase and then switches to a policy iteration procedure
when a certain error threshold is reached. A delicate point is to determine
this threshold in order to avoid cumbersome computation with the value
iteration and, at the same time, to be reasonably sure that the policy
iteration method will finally converge to the optimal solution. We analyze the
methods and efficient coupling in a number of examples in dimension two, three
and four illustrating its properties
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