European Institute for Statistics, Probability, Stochastic Operations Research
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   Mini workshop CWI-EUR

BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
(BSDE's)

January 18, 2012

BACKGROUND

Backward stochastic differential equations (BSDE's) are used for describing stochastic processes with a fixed terminal value at some future time T. They were first introduced by J.M. Bismut in 1973 and generalized to their current form by Pardoux and Peng in 1990. Since then, they have been widely studied and used in connection with partial differential equations, stochastic control theory, and in particular mathematical finance, since many problems regarding the pricing and hedging of contingent claims can be expressed in terms of BSDEs. The analytical and numerical treatment of BSDE's is more complicated than that of classical (forward) SDE's, and there remain many interesting and challenging open problems in the study of BSDE's. We would like to bring together leading senior as well as junior experts working on BSDE's at several different institutions across Europe so they can exchange ideas, provide feedback on each other’s work and possibly form collaborations.

Organisers

Roger Laeven
Eurandom and University of Amsterdam
R.J.A.Laeven@uva.nl

Kees Oosterlee
CWI Amsterdam and TU Delft
c.w.oosterlee@cwi.nl

Wim Schoutens
Eurandom and KU Leuven
wim.schoutens@wis.kuleuven.be

REGISTRATION

Please use the online registration form to submit your details for registration.
 

SPEAKERS

PROGRAMME

ABSTRACTS

Shashi Jain (CWI)

Stochastic Grid Method and Application for Optimal Modular Construction in Nuclear Power Plants

Stochastic grid method is a Monte Carlo based method to price Bermudan options. The method follows the dynamic programming principle to recursively price option moving backwards in time. An application of stochastic grid method is to find the optimal time to start the construction of a module in a nuclear power plant and thus determining the real option value of sequential modular reactors. In the talk we discuss the stochastic grid method and its application to value real options in nuclear power plants.

Antoon Pelsser (University of Maastricht)

Numerical Approximation of BSDE's using Hermite Martingales

When solving BSDE’s numerically, one of the crucial steps in the algorithm at each time-step is to calculate conditional expectations for the solution Y_t and the “gradient” Z_t.
We propose a new algorithm that uses orthogonal polynomials of the Brownian Motion to approximate Y_t in each time-step. In particular, we advocate the use of Hermite polynomials.
These have the advantage that Hermite polynomials of the Brownian Motion are martingales for any polynomial order. This property is very useful for calculating the conditional expectations in an exact way during each time-step, thereby eliminating a potential source of error in our algorithm. Furthermore, the gradient Z_t can be calculated directly from the polynomial approximation of Y_t, thus eliminating the need to estimate the gradient with regression methods.

Marjon Ruijter (CWI)

On the Fourier cosine series expansion (COS) method for stochastic control problems in finance and (climate) economics

We develop a method for solving stochastic control problems under one-dimensional Lévy processes. The method is based on the dynamic programming principle and a Fourier cosine expansion method (F. Fang, C.W. Oosterlee, 2009). Local errors in the vicinity of boundaries of the domain may disrupt the algorithm. An extensive error analysis provides new insights based on which we develop an extrapolation method to deal with the propagation of local errors. An exponentially converging error in N, the number of terms in the series expansions, is found for a sufficiently accurate extrapolation method and a probability density function in the class C∞([a,b]).
We test the method by solving two stochastic control problems of practical interest. The first is the valuation of an option under uncertain volatility. The second problem we discuss is a consumption-portfolio problem from economics. The model used is a simplified version of the well-known Merton’s portfolio selection problem.

John Schoenmakers (WIAS Berlin)

In this talk we propose a novel approach to reduce the computational complexity of the dual method for pricing American options.
We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at a multilevel type dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example. (joint work with Denis Belomestny).
In the last part of the talk we will elaborate shortly on dual representations for optimal stopping under general dynamic utility functionals, i.e. pricing of American options under ambiguity.

Mitja Stadje (University of Tilburg)

Existence, minimality and approximation of solutions to BSDEs with convex drivers

We study the existence of solutions to backward stochastic differential equations with drivers ƒ(t, W, y, z) that are convex in z. We assume ƒ to be Lipschitz in y and W but do not make growth assumptions with respect to z. We first show the existence of a unique solution (Y, Z ) with
bounded Z if the terminal condition is Lipschitz in W and that it can be approximated by the solutions to properly discretized equations. If the terminal condition is bounded and uniformly continuous in W we show the existence of a minimal continuous supersolution by uniformly
approximating the terminal condition with Lipschitz terminal conditions. Finally, we prove existence of a minimal RCLL supersolution for bounded lower semicontinuous terminal conditions by approximating the terminal condition pointwise from below with Lipschitz terminal conditions.

PRACTICAL INFORMATION

Conference Location
The workshop location is EURANDOM,  Den Dolech 2, 5612 AZ Eindhoven, Laplace Building, 1st floor, LG 1.105.

EURANDOM is located on the campus of Eindhoven University of Technology, in the 'Laplacegebouw' building' (LG on the map). The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm

CONTACT
For more information please contact Patty Koorn,
Workshop officer Eurandom

SPONSORS

Last updated 13-01-12,
by PK