5,332 research outputs found
Strong convergence of some drift implicit Euler scheme. Application to the CIR process
We study the convergence of a drift implicit scheme for one-dimensional SDEs
that was considered by Alfonsi for the Cox-Ingersoll-Ross (CIR) process. Under
general conditions, we obtain a strong convergence of order 1. In the CIR case,
Dereich, Neuenkirch and Szpruch have shown recently a strong convergence of
order 1/2 for this scheme. Here, we obtain a strong convergence of order 1
under more restrictive assumptions on the CIR parameters
A Mean-Reverting SDE on Correlation matrices
We introduce a mean-reverting SDE whose solution is naturally defined on the
space of correlation matrices. This SDE can be seen as an extension of the
well-known Wright-Fisher diffusion. We provide conditions that ensure weak and
strong uniqueness of the SDE, and describe its ergodic limit. We also shed
light on a useful connection with Wishart processes that makes understand how
we get the full SDE. Then, we focus on the simulation of this diffusion and
present discretization schemes that achieve a second-order weak convergence.
Last, we explain how these correlation processes could be used to model the
dependence between financial assets
A generic construction for high order approximation schemes of semigroups using random grids
Our aim is to construct high order approximation schemes for general
semigroups of linear operators . In order to do it, we fix a
time horizon and the discretization steps and we suppose that we have at hand some short time approximation
operators such that for some
. Then, we consider random time grids such that for all ,
for some , and
we associate the approximation discrete semigroup Our main result is the following: for any
approximation order , we can construct random grids
and coefficients , with such that % with the expectation concerning the random grids
Besides, and the complexity of the
algorithm is of order , for any order of approximation . The standard
example concerns diffusion processes, using the Euler approximation for~.
In this particular case and under suitable conditions, we are able to gather
the terms in order to produce an estimator of with finite variance.
However, an important feature of our approach is its universality in the sense
that it works for every general semigroup and approximations. Besides,
approximation schemes sharing the same lead to the same random grids
and coefficients . Numerical illustrations are given for
ordinary differential equations, piecewise deterministic Markov processes and
diffusions
Extension and calibration of a Hawkes-based optimal execution model
We provide some theoretical extensions and a calibration protocol for our
former dynamic optimal execution model. The Hawkes parameters and the
propagator are estimated independently on financial data from stocks of the
CAC40. Interestingly, the propagator exhibits a smoothly decaying form with one
or two dominant time scales, but only so after a few seconds that the market
needs to adjust after a large trade. Motivated by our estimation results, we
derive the optimal execution strategy for a multi-exponential Hawkes kernel and
backtest it on the data for round trips. We find that the strategy is
profitable on average when trading at the midprice, which is in accordance with
violated martingale conditions. However, in most cases, these profits vanish
when we take bid-ask costs into account
Multivariate transient price impact and matrix-valued positive definite functions
We consider a model for linear transient price impact for multiple assets
that takes cross-asset impact into account. Our main goal is to single out
properties that need to be imposed on the decay kernel so that the model admits
well-behaved optimal trade execution strategies. We first show that the
existence of such strategies is guaranteed by assuming that the decay kernel
corresponds to a matrix-valued positive definite function. An example
illustrates, however, that positive definiteness alone does not guarantee that
optimal strategies are well-behaved. Building on previous results from the
one-dimensional case, we investigate a class of nonincreasing, nonnegative and
convex decay kernels with values in the symmetric matrices. We show
that these decay kernels are always positive definite and characterize when
they are even strictly positive definite, a result that may be of independent
interest. Optimal strategies for kernels from this class are well-behaved when
one requires that the decay kernel is also commuting. We show how such decay
kernels can be constructed by means of matrix functions and provide a number of
examples. In particular we completely solve the case of matrix exponential
decay
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