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Aurélien Alfonsi

Publications and source records attributed to Aurélien Alfonsi.

At least 37 records · Page 2Linked to original sources
arXiv2021

Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach)

We are concerned with a mixture of Boltzmann and McKean-Vlasov type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself,and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (1978) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of endomorphisms on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the "flow solution" is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.

math.PR↗
arXiv2021

Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests

This paper studies the multilevel Monte-Carlo estimator for the expectation of a maximum of conditional expectations. This problem arises naturally when considering many stress tests and appears in the calculation of the interest rate module of the standard formula for the SCR. We obtain theoretical convergence results that complements the recent work of Giles and Goda and gives some additional tractability through a parameter that somehow describes regularity properties around the maximum. We then apply the MLMC estimator to the calculation of the SCR at future dates with the standard formula for an ALM savings business on life insurance. We compare it with estimators obtained with Least Square Monte-Carlo or Neural Networks. We find that the MLMC estimator is computationally more efficient and has the main advantage to avoid regression issues, which is particularly significant in the context of projection of a balance sheet by an insurer due to the path dependency. Last, we discuss the potentiality of this numerical method and analyze in particular the effect of the portfolio allocation on the SCR at future~dates.

q-fin.CP↗
arXiv2021

Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation

The Strictly Correlated Electrons (SCE) limit of the Levy-Lieb functional in Density Functional Theory (DFT) gives rise to a symmetric multi-marginal optimal transport problem with Coulomb cost, where the number of marginal laws is equal to the number of electrons in the system, which can be very large in relevant applications. In this work, we design a numerical method, built upon constrained overdamped Langevin processes to solve Moment Constrained Optimal Transport (MCOT) relaxations (introduced in A. Alfonsi, R. Coyaud, V. Ehrlacher and D. Lombardi, Math. Comp. 90, 2021, 689--737) of symmetric multi-marginal optimal transport problems with Coulomb cost. Some minimizers of such relaxations can be written as discrete measures charging a low number of points belonging to a space whose dimension, in the symmetrical case, scales linearly with the number of marginal laws. We leverage the sparsity of those minimizers in the design of the numerical method and prove that any local minimizer to the resulting problem is actually a \emph{global} one. We illustrate the performance of the proposed method by numerical examples which solves MCOT relaxations of 3D systems with up to 100 electrons.

math.NA↗
arXiv2019

Squared quadratic Wasserstein distance : optimal couplings and Lions differentiability

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance $W^2_2(μ,ν)$ between two probability measures $μ$ and $ν$ with finite second order moments on $\mathbb{R}^d$ is the composition of a martingale coupling with an optimal transport map ${\mathcal T}$. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between $μ$ and ${\mathcal T}\#μ$. Next, we give a direct proof that $σ\mapsto W_2^2(σ,ν)$ is differentiable at $μ$ in the Lions sense iff there is a unique optimal coupling between $μ$ and $ν$ and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré and Ambrosio and Gangbo that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu. Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant function $F$ on $L^2(Ω,\mathbb{P};\mathbb{R}^d)$ is enough for the Fréchet differential at $X$ to be a measurable function of $X$.

math.PR↗
arXiv2019

A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula

The aim of this paper is to introduce a synthetic ALM model that catches the main specificity of life insurance contracts. First, it keeps track of both market and book values to apply the regulatory profit sharing rule. Second, it introduces a determination of the crediting rate to policyholders that is close to the practice and is a trade-off between the regulatory rate, a competitor rate and the available profits. Third, it considers an investment in bonds that enables to match a part of the cash outflow due to surrenders, while avoiding to store the trading history. We use this model to evaluate the Solvency Capital Requirement (SCR) with the standard formula, and show that the choice of the interest rate model is important to get a meaningful model after the regulatory shocks on the interest rate. We discuss the different values of the SCR modules first in a framework with moderate interest rates using the shocks of the present legislation, and then we consider a low interest framework with the latest recommandation of the EIOPA on the shocks. In both cases, we illustrate the importance of matching cash-flows and its impact on the SCR.

q-fin.RM↗
arXiv2019

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 $P_{h_{l}}=Q_{l}+O(h_{l}^{1+α})$ for some $α>0$. Then, we consider random time grids $Π(ω)=\{t_0(ω)=0<t_{1}(ω)<...<t_{m}(ω)=T\}$ such that for all $1\le k\le m$, $t_{k}(ω)-t_{k-1}(ω)=h_{l_{k}}$ for some $l_{k}\in \mathbb{N}$, and we associate the approximation discrete semigroup $P_{T}^{Π(ω)}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the following: for any approximation order $ν$, we can construct random grids $Π_{i}(ω)$ and coefficients $c_{i}$, with $i=1,...,r$ such that \[ P_{t}f=\sum_{i=1}^{r}c_{i}\mathbb{E}(P_{t}^{Π_{i}(ω)}f(x))+O(n^{-ν}) \]% with the expectation concerning the random grids $Π_{i}(ω).$ Besides, $\text{Card}(Π_{i}(ω))=O(n)$ and the complexity of the algorithm is of order $n$, for any order of approximation $ν$. 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 $α$ lead to the same random grids $Π_{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions.

math.PR↗
arXiv2019

Approximation of Optimal Transport problems with marginal moments constraints

Optimal Transport (OT) problems arise in a wide range of applications, from physics to economics. Getting numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff's theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. This approximation method is also relevant for Martingale OT problems. We show the convergence of the MCOT problem toward the corresponding OT problem. In some fundamental cases, we obtain rates of convergence in $O(1/n)$ or $O(1/n^2)$ where $n$ is the number of moments, which illustrates the role of the moment functions. Last, we present algorithms exploiting the fact that the MCOT is reached by a finite discrete measure and provide numerical examples of approximations.

math.PR↗
arXiv2019

Sampling of probability measures in the convex order by Wasserstein projection

In this paper, for $μ$ and $ν$ two probability measures on $\mathbb{R}^d$ with finite moments of order $ρ\ge 1$, we define the respective projections for the $W_ρ$-Wasserstein distance of $μ$ and $ν$ on the sets of probability measures dominated by $ν$ and of probability measures larger than $μ$ in the convex order. The $W_2$-projection of $μ$ can be easily computed when $μ$ and $ν$ have finite support by solving a quadratic optimization problem with linear constraints. In dimension $d=1$, Gozlan et al.~(2018) have shown that the projections do not depend on $ρ$. We explicit their quantile functions in terms of those of $μ$ and $ν$. The motivation is the design of sampling techniques preserving the convex order in order to approximate Martingale Optimal Transport problems by using linear programming solvers. We prove convergence of the Wasserstein projection based sampling methods as the sample sizes tend to infinity and illustrate them by numerical experiments.

math.PR↗
arXiv2018

Long-time large deviations for the multi-asset Wishart stochastic volatility model and option pricing

We prove a large deviations principle for the class of multidimensional affine stochastic volatility models considered in (Gourieroux, C. and Sufana, R., J. Bus. Econ. Stat., 28(3), 2010), where the volatility matrix is modelled by a Wishart process. This class extends the very popular Heston model to the multivariate setting, thus allowing to model the joint behaviour of a basket of stocks or several interest rates. We then use the large deviation principle to obtain an asymptotic approximation for the implied volatility of basket options and to develop an asymptotically optimal importance sampling algorithm, to reduce the number of simulations when using Monte-Carlo methods to price derivatives.

q-fin.PR↗
arXiv2016

Evolution of the Wasserstein distance between the marginals of two Markov processes

In this paper, we are interested in the time derivative of the Wasserstein distance between the marginals of two Markov processes. As recalled in the introduction, the Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, we show that this remains true for Piecewise Deterministic Markov Processes. We apply the formula to estimate the exponential decrease rate of the Wasserstein distance between the marginals of two birth and death processes with the same generator in terms of the Wasserstein curvature.

math.PR↗
arXiv2016

Maximum Likelihood Estimation for Wishart processes

In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli and are of independent interest.

math.ST↗
arXiv2015

Smile with the Gaussian term structure model

We propose an affine extension of the Linear Gaussian term structure Model (LGM) such that the instantaneous covariation of the factors is given by an affine process on semidefinite positive matrices. First, we set up the model and present some important properties concerning the Laplace transform of the factors and the ergodicity of the model. Then, we present two main numerical tools to implement the model in practice. First, we obtain an expansion of caplets and swaptions prices around the LGM. Such a fast and accurate approximation is useful for assessing the model behavior on the implied volatility smile. Second, we provide a second order scheme for the weak error, which enables to calculate exotic options by a Monte-Carlo algorithm.

q-fin.MF↗
arXiv2015

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.

q-fin.TR↗
arXiv2015

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.

q-fin.TR↗
arXiv2015

Dynamic optimal execution in a mixed-market-impact Hawkes price model

We study a linear price impact model including other liquidity takers, whose flow of orders either follows a Poisson or a Hawkes process. The optimal execution problem is solved explicitly in this context, and the closed-formula optimal strategy describes in particular how one should react to the orders of other traders. This result enables us to discuss the viability of the market. It is shown that Poissonian arrivals of orders lead to quite robust Price Manipulation Strategies in the sense of Huberman and Stanzl. Instead, a particular set of conditions on the Hawkes model balances the self-excitation of the order flow with the resilience of the price, excludes Price Manipulation Strategies and gives some market stability.

q-fin.TR↗
arXiv2015

Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and H{ö}lder continuous with exponent $γ$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C \left(1+\mathbf{1}\_{γ=1} \sqrt{\ln(N)}\right)N^{-γ}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio, Gigli and Savar{é} to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.

math.PR↗
arXiv2013

A remark on the optimal transport between two probability measures sharing the same copula

We are interested in the Wasserstein distance between two probability measures on $\R^n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension $n=1$. It turns out that for cost functions $c(x,y)$ equal to the $p$-th power of the $L^q$ norm of $x-y$ in $\R^n$, this coupling is optimal only when $p=q$ i.e. when $c(x,y)$ may be decomposed as the sum of coordinate-wise costs.

math.PR↗