Donnerstag, 22.10.2015: A primal-dual algorithm for backward SDEs
Prof. Dr. Christian Bender (Universität des Saarlandes)
Backward stochastic differential equations (BSDEs) are terminal value
problems for stochastic differential equations with the additional twist
that
the solution must not anticipate future information. They have numerous
applications, e.g. in stochastic control and mathematical finance, and
provide
stochastic representations for several classes of parabolic partial
differential equations.
Numerical methods for backward stochastic differential equations
(BSDEs) typically consist of two steps. In a first step a time
discretization is performed, which leads to a backward dynamic
programming equation. In a second step this dynamic program has to be
solved numerically. This second step requires to approximate high order
nestings of conditional expectations, which is a challenging problem in
particular when the BSDE is driven by a high-dimensional Brownian
motion.
After a brief survey on BSDEs and their time discretization, we present
a method to construct confidence intervals on the value of the dynamic
program, and hence on the solution of the time-discretized BSDE. This
method generalizes the primal-dual approach, which is popular and
well-studied for Bermudan option pricing problems. In a nutshell, the
idea is to derive a maximization problem and a minimization problem such
that the value process of both problems coincides with the solution of
the dynamic program and such that optimizers can be represented in terms
of the solution of the dynamic program. Using an approximate solution to
the dynamic program, which can be precomputed by any algorithm, then
leads to `close-to-optimal' controls for these optimization problems and
to `tight' lower and upper bounds for the time-discretized BSDE,
provided that the algorithm for constructing the approximate solution
was `successful'.
The talk is based on joint work with N. Schweizer, J. Zhuo, and C.
Gärtner.
| Uhrzeit: |
14:15 |
| Ort: |
N16 |
| Gruppe: |
OS Numerik |
| Einladender: |
Lubich, Prohl |
Donnerstag, 22.10.2015: A method to approximate Lyapunov exponents of switching systems
Nicola Guglielmi (Università degli Studi dell'Aquila)
We discuss a new approach for constructing polytope Lyapunov
functions for continuous-time linear switching systems. The
method we propose allows to decide the uniform stability of
a switching system and to compute the Lyapunov exponent with
an arbitrary precision.
The method relies on the discretization of the system and
provides - for any given discretization stepsize - a lower
and an upper bound for the Lyapunov exponent.
The efficiency of the new method is illustrated by numerical
examples.
| Uhrzeit: |
15:15 |
| Ort: |
N16 |
| Gruppe: |
Oberseminar Numerik |
| Einladender: |
Lubich, Prohl |
