41 research outputs found
Discrete-time approximation of multidimensional BSDEs with oblique reflections
In this paper, we study the discrete-time approximation of multidimensional
reflected BSDEs of the type of those presented by Hu and Tang [Probab. Theory
Related Fields 147 (2010) 89-121] and generalized by Hamad\`ene and Zhang
[Stochastic Process. Appl. 120 (2010) 403-426]. In comparison to the penalizing
approach followed by Hamad\`{e}ne and Jeanblanc [Math. Oper. Res. 32 (2007)
182-192] or Elie and Kharroubi [Statist. Probab. Lett. 80 (2010) 1388-1396], we
study a more natural scheme based on oblique projections. We provide a control
on the error of the algorithm by introducing and studying the notion of
multidimensional discretely reflected BSDE. In the particular case where the
driver does not depend on the variable , the error on the grid points is of
order , .Comment: Published in at http://dx.doi.org/10.1214/11-AAP771 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems
This paper deals with existence and uniqueness, in viscosity sense, of a
solution for a system of m variational partial differential inequalities with
inter-connected obstacles. A particular case of this system is the
deterministic version of the Verification Theorem of the Markovian optimal
m-states switching problem. The switching cost functions are arbitrary. This
problem is connected with the valuation of a power plant in the energy market.
The main tool is the notion of systems of reflected BSDEs with oblique
reflection.Comment: 36 page
Stochastic Control Representations for Penalized Backward Stochastic Differential Equations
This paper shows that penalized backward stochastic differential equation
(BSDE), which is often used to approximate and solve the corresponding
reflected BSDE, admits both optimal stopping representation and optimal control
representation. The new feature of the optimal stopping representation is that
the player is allowed to stop at exogenous Poisson arrival times. The
convergence rate of the penalized BSDE then follows from the optimal stopping
representation. The paper then applies to two classes of equations, namely
multidimensional reflected BSDE and reflected BSDE with a constraint on the
hedging part, and gives stochastic control representations for their
corresponding penalized equations.Comment: 24 pages in SIAM Journal on Control and Optimization, 201
Adding constraints to BSDEs with Jumps: an alternative to multidimensional reflections
This paper is dedicated to the analysis of backward stochastic differential
equations (BSDEs) with jumps, subject to an additional global constraint
involving all the components of the solution. We study the existence and
uniqueness of a minimal solution for these so-called constrained BSDEs with
jumps via a penalization procedure. This new type of BSDE offers a nice and
practical unifying framework to the notions of constrained BSDEs presented in
[19] and BSDEs with constrained jumps introduced in [14]. More remarkably, the
solution of a multidimensional Brownian reflected BSDE studied in [11] and [13]
can also be represented via a well chosen one-dimensional constrained BSDE with
jumps.This last result is very promising from a numerical point of view for the
resolution of high dimensional optimal switching problems and more generally
for systems of coupled variational inequalitie
Switching Game of Backward Stochastic Differential Equations and Associated System of Obliquely Reflected Backward Stochastic Differential Equations
This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates
A Full Balance Sheet Two-modes Optimal Switching problem
We formulate and solve a finite horizon full balance sheet two-modes optimal
switching problem related to trade-off strategies between expected profit and
cost yields. Given the current mode, this model allows for either a switch to
the other mode or termination of the project, and this happens for both sides
of the balance sheet. A novelty in this model is that the related obstacles are
nonlinear in the underlying yields, whereas, they are linear in the standard
optimal switching problem. The optimal switching problem is formulated in terms
of a system of Snell envelopes for the profit and cost yields which act as
obstacles to each other. We prove existence of a continuous minimal solution of
this system using an approximation scheme and fully characterize the optimal
switching strategy.Comment: 23 pages. To appear in Stochastic