arXiv subjects
Publications and source records attributed to Xavier Warin.
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.
In Electricity markets, illiquidity, transaction costs and market price characteristics prevent managers to replicate exactly contracts. A residual risk is always present and the hedging strategy depends on a risk criterion chosen. We present an algorithm to hedge a position for a mean variance criterion taking into account the transaction cost and the small depth of the market. We show its effectiveness on a typical problem coming from the field of electricity markets.
A new method based on nesting Monte Carlo is developed to solve high-dimensional semi-linear PDEs. Convergence of the method is proved and its convergence rate studied. Results in high dimension for different kind of non-linearities show its efficiency.
We extend a recently developed method to solve semi-linear PDEs to the case of a degenerated diffusion. Being a pure Monte Carlo method it does not suffer from the so called curse of dimensionality and it can be used to solve problems that were out of reach so far. We give some results of convergence and show numerically that it is effective. Besides we numerically show that the new scheme developed can be used to solve some full non linear PDEs. At last we provide an effective algorithm to implement the scheme.
We study an islanded microgrid system designed to supply a small village with the power produced by photovoltaic panels, wind turbines and a diesel generator. A battery storage system device is used to shift power from times of high renewable production to times of high demand. We introduce a methodology to solve microgrid management problem using different variants of Regression Monte Carlo algorithms and use numerical simulations to infer results about the optimal design of the grid.
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.
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.
The branching methods developed are effective methods to solve some semi linear PDEs and are shown numerically to be able to solve some full non linear PDEs. These methods are however restricted to some small coefficients in the PDE and small maturities. This article shows numerically that these methods can be adapted to solve the problems with longer maturities in the semi-linear case by using a new derivation scheme and some nested method. As for the case of full non linear PDEs, we introduce new schemes and we show numerically that they provide an effective alternative to the schemes previously developed.
We numerically study an Asset Liability Management problem linked to the decommissioning of French nuclear power plants. We link the risk aversion of practitioners to an optimization problem. Using different price models we show that the optimal solution is linked to a de-risking management strategy similar to a concave strategy and we propose an effective heuristic to simulate the underlying optimal strategy. Besides we show that the strategy is stable with respect to the main parameters involved in the liability problem.
We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical results, a numerical deterministic approximation of the related HJB variational inequality is provided. We finally show that this numerical approximation converges to the value function. This allows us to describe the investment and dividend optimal policies.
We develop a pure Monte Carlo method to compute $E(g(X_T))$ where $g$ is a bounded and Lipschitz function and $X_t$ an Ito process. This approach extends a previously proposed method to the general multidimensional case with a SDE with varying coefficients. A variance reduction method relying on interacting particle systems is also developped.
We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod (1964), Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in the pair $(u, Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to "small maturity" or "small nonlinearity" of the PDE. Our main ingredient is the automatic differentiation technique as in Henry Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free Central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.
This paper examines the dividend and investment policies of a cash constrained firm that has access to costly external funding. We depart from the literature by allowing the firm to issue collateralized debt to increase its investment in productive assets resulting in a performance sensitive interest rate on debt. We formulate this problem as a bi-dimensional singular control problem and use both a viscosity solution approch and a verification technique to get qualitative properties of the value function. We further solve quasi-explicitly the control problem in two special cases.
We introduce some approximation schemes for linear and fully non-linear diffusion equations of Bellman-Isaacs type. Although they are not monotone one can prove their convergence to the viscosity solution of the problem. Effective implementation of these scheme is discussed and they are extensively tested.
We introduce some sparse grids interpolations used in Semi-Lagrangian schemes for linear and fully non-linear diffusion Hamilton Jacobi Bellman equations arising in stochastic control. We prove that the method introduced converges toward the viscosity solution of the problem and we show that some potentially high order schemes can be efficiently implemented. Numerical test in dimension $2$ to $5$ are achieved and show that deterministic methods can be used efficiently in stochastic control in moderate dimension.
We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integro-differential equations (PIDE), which can be shown to possess unique smooth solutions in our setting. The first equation is non-linear, but does not depend on the pay-off of the option to hedge (the pure investment problem), while the other two equations are linear. We propose convergent finite difference schemes for the numerical solution of these PIDEs and illustrate our results with an application to electricity markets, where time-inhomogeneous pure jump Markov processes appear in a natural manner.
Swing options on the gas market are american style option where daily quantities exercices are constrained and global quantities exerciced each year constrained too. The option holder has to decide each day how much he consumes of the quantities satisfying the constraints and tries to use a strategy in order to maximize its expected profit. The pay off fonction is a spread between the spot gas market and the value of an index composed of the past average of some commodities spot or future prices. We study the valorization and the effectiveness of the dynamic hedging of such a contract.
We introduce a new probabilistic method for solving a class of impulse control problems based on their representations as Backward Stochastic Differential Equations (BSDEs for short) with constrained jumps. As an example, our method is used for pricing Swing options. We deal with the jump constraint by a penalization procedure and apply a discrete-time backward scheme to the resulting penalized BSDE with jumps. We study the convergence of this numerical method, with respect to the main approximation parameters: the jump intensity $λ$, the penalization parameter $p > 0$ and the time step. In particular, we obtain a convergence rate of the error due to penalization of order $(λp)^{α- \frac{1}{2}}, \forall α\in \left(0, \frac{1}{2}\right)$. Combining this approach with Monte Carlo techniques, we then work out the valuation problem of (normalized) Swing options in the Black and Scholes framework. We present numerical tests and compare our results with a classical iteration method.