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

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

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 $P_{t},t\geq 0$. In order to do it, we fix a time horizon $T$ and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in \mathbb{N}$ and we suppose that we have at hand some short time approximation operators $Q_{l}$ such that $P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })$ for some $\alpha >0$. Then, we consider random time grids $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<...<t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in \mathbb{N}$, and we associate the approximation discrete semigroup $P_{T}^{\Pi (\omega )}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the following: for any approximation order $\nu$, we can construct random grids $\Pi_{i}(\omega )$ and coefficients $c_{i}$, with $i=1,...,r$ such that $$P_{t}f=\sum_{i=1}^{r}c_{i}\mathbb{E}(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu})$$% with the expectation concerning the random grids $\Pi _{i}(\omega ).$ Besides, $\text{Card}(\Pi _{i}(\omega ))=O(n)$ and the complexity of the algorithm is of order $n$, for any order of approximation $\nu$. The standard example concerns diffusion processes, using the Euler approximation for~$Q_l$. In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of $P_tf$ with finite variance. However, an important feature of our approach is its universality in the sense that it works for every general semigroup $P_{t}$ and approximations. Besides, approximation schemes sharing the same $\alpha$ lead to the same random grids $\Pi_{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions

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 $K\times K$ 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