11 research outputs found
Some non monotone schemes for Hamilton-Jacobi-Bellman equations
We extend the theory of Barles Jakobsen to develop numerical schemes for
Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes
can be relaxed still leading to the convergence to the viscosity solution of
the equation. We give some examples of such numerical schemes and show that the
bounds obtained by the framework developed are not tight. At last we test some
numerical schemes.Comment: 24 page
Numerical approximation of BSDEs using local polynomial drivers and branching processes
We propose a new numerical scheme for Backward Stochastic Differential
Equations based on branching processes. We approximate an arbitrary (Lipschitz)
driver by local polynomials and then use a Picard iteration scheme. Each step
of the Picard iteration can be solved by using a representation in terms of
branching diffusion systems, thus avoiding the need for a fine time
discretization. In contrast to the previous literature on the numerical
resolution of BSDEs based on branching processes, we prove the convergence of
our numerical scheme without limitation on the time horizon. Numerical
simulations are provided to illustrate the performance of the algorithm.Comment: 28 page
Volatility Uncertainty Quantification in a Stochastic Control Problem Applied to Energy
This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solutions to a system of second order parabolic non-linear PDEs. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We provide an example of the methodology in the context of a swing contract (energy contract with flexibility in purchasing energy power), this allows us to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model.This research is part of the Chair Financial Risks of the Risk Foundation, the Finance for Energy Market Research Centre (FiME) and the ANR project CAESARS (ANR-15-CE05-0024)
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature
Numerical approximation of general Lipschitz BSDEs with branching processes
We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Convergence of our algorithm is proved without any limitation on the time horizon. We also provide numerical simulations to illustrate the performance of the algorithm
Some Non-monotone Schemes for Time Dependent Hamilton–Jacobi–Bellman Equations in Stochastic Control
Incentives, lockdown, and testing: from Thucydides's analysis to the COVID-19 pandemic
We consider the control of the COVID-19 pandemic via incentives, through
either stochastic SIS or SIR compartmental models. When the epidemic is
ongoing, the population can reduce interactions between individuals in order to
decrease the rate of transmission of the disease, and thus limit the epidemic.
However, this effort comes at a cost for the population. Therefore, the
government can put into place incentive policies to encourage the lockdown of
the population. In addition, the government may also implement a testing policy
in order to know more precisely the spread of the epidemic within the country,
and to isolate infected individuals. We provide numerical examples, as well as
an extension to a stochastic SEIR compartmental model to account for the
relatively long latency period of the COVID-19 disease. The numerical results
confirm the relevance of a tax and testing policy to improve the control of an
epidemic. More precisely, if a tax policy is put into place, even in the
absence of a specific testing policy, the population is encouraged to
significantly reduce its interactions, thus limiting the spread of the disease.
If the government also adjusts its testing policy, less effort is required on
the population side, so individuals can interact almost as usual, and the
epidemic is largely contained by the targeted isolation of positively-tested
individuals