24 research outputs found
Maximum of the resolvent over matrices with given spectrum
In numerical analysis it is often necessary to estimate the condition number
and the norm of the resolvent
of a given matrix . We derive new
spectral estimates for these quantities and compute explicit matrices that
achieve our bounds. We recover the well-known fact that the supremum of
over all matrices with and minimal absolute eigenvalue
is the Kronecker bound .
This result is subsequently generalized by computing the corresponding supremum
of for any . We find that the supremum
is attained by a triangular Toeplitz matrix. This provides a simple class of
structured matrices on which condition numbers and resolvent norm bounds can be
studied numerically. The occuring Toeplitz matrices are so-called model
matrices, i.e. matrix representations of the compressed backward shift operator
on the Hardy space to a finite-dimensional invariant subspace
Hedging of Financial Derivative Contracts via Monte Carlo Tree Search
The construction of approximate replication strategies for derivative
contracts in incomplete markets is a key problem of financial engineering.
Recently Reinforcement Learning algorithms for pricing and hedging under
realistic market conditions have attracted significant interest. While
financial research mostly focused on variations of -learning, in Artificial
Intelligence Monte Carlo Tree Search is the recognized state-of-the-art method
for various planning problems, such as the games of Hex, Chess, Go,... This
article introduces Monte Carlo Tree Search as a method to solve the stochastic
optimal control problem underlying the pricing and hedging of financial
derivatives. As compared to -learning it combines reinforcement learning
with tree search techniques. As a consequence Monte Carlo Tree Search has
higher sample efficiency, is less prone to over-fitting to specific market
models and generally learns stronger policies faster. In our experiments we
find that Monte Carlo Tree Search, being the world-champion in games like Chess
and Go, is easily capable of directly maximizing the utility of investor's
terminal wealth without an intermediate mathematical theory.Comment: Added figures. Added references. Corrected typos. 15 pages, 5 figure
A decoupling approach to classical data transmission over quantum channels
Most coding theorems in quantum Shannon theory can be proven using the
decoupling technique: to send data through a channel, one guarantees that the
environment gets no information about it; Uhlmann's theorem then ensures that
the receiver must be able to decode. While a wide range of problems can be
solved this way, one of the most basic coding problems remains impervious to a
direct application of this method: sending classical information through a
quantum channel. We will show that this problem can, in fact, be solved using
decoupling ideas, specifically by proving a "dequantizing" theorem, which
ensures that the environment is only classically correlated with the sent data.
Our techniques naturally yield a generalization of the
Holevo-Schumacher-Westmoreland Theorem to the one-shot scenario, where a
quantum channel can be applied only once