STAZHYNSKI Uladzislau

Stochastic Approximation is an iterative procedure for computing a zero θ of a function that is not explicit but defined as an expectation. It is, for example, a numerical tool for computing maximum likelihood in "regular" latent data models. If the definition of the statistical model is tainted with an uncertainty τ , of which only an a priori dπ(τ ) is known, then the zeros depend on τ and the natural question is to explore their distribution when τ ∼ dπ. In this paper, we propose an iterative algorithm based on a Stochastic Approximation scheme that,in the limit, computes θ (τ) for any τ and produces a characterization of its distribution. and weenounce sufficient conditions for the convergence of this algorithm.

We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs.

We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs.

This thesis consists of two parts which study two separate subjects. Chapters 1-4 are devoted to the problem of processes discretization at stopping times. In Chapter 1 we study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a path wise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales and we prove that the asymptotic lower bound is attainable. In Chapter 2 we study the model-adaptive optimal discretization error of stochastic integrals. In Chapter 1 the construction of the optimal strategy involved the knowledge about the diffusion coefficient of the semimartingale under study. In this work we provide a model-adaptive asymptotically optimal discretization strategy that does not require any prior knowledge about the model. In Chapter 3 we study the convergence in distribution of renormalized discretization errors of Ito processes for a concrete general class of random discretization grids given by stopping times. Previous works on the subject only treat the case of dimension 1. Moreover they either focus on particular cases of grids, or provide results under quite abstract assumptions with implicitly specified limit distribution. At the contrast we provide explicitly the limit distribution in a tractable form in terms of the underlying model. The results hold both for multidimensional processes and general multidimensional error terms. In Chapter 4 we study the problem of parametric inference for diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids. Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp. In Chapters 5-6 we study the problem of uncertainty quantification for stochastic approximation limits. In Chapter 5 we analyze the uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm. In our setup, this limit is defined as the zero of a function given by an expectation. The expectation is taken w.r.t. a random variable for which the model is assumed to depend on an uncertain parameter. We consider the SA limit as a function of this parameter. We introduce the so-called Uncertainty for SA (USA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of this function on an orthogonal basis of a suitable Hilbert space. The almost-sure and Lp convergences of USA, in the Hilbert space, are established under mild, tractable conditions. In Chapter 6 we analyse the L2-convergence rate of the USA algorithm designed in Chapter 5.The analysis is non-trivial due to infinite dimensionality of the procedure. Moreover, our setting is not covered by the previous works on infinite dimensional SA. The obtained rate depends non-trivially on the model and the design parameters of the algorithm. Its knowledge enables optimization of the dimension growth speed in the USA algorithm, which is the key factor of its efficient performance.

This thesis contains two parts that study two different topics. Chapters 1-4 are devoted to problems of discretization of processes with stopping times. In Chapter 1 we study the optimal discretization error for stochastic integrals with respect to a continuous multidimensional Brownian semimartingale. In this framework we establish a trajectory lower bound for the renormalized quadratic variation of the error. We provide a sequence of stopping times that gives an asymptotically optimal discretization. This sequence is defined as the output time of random ellipsoids by the semimartingale. Compared to the previous results we allow a rather large class of semimartingales. We prove that the lower bound is exact. In Chapter 2 we study the adaptive version of the model of the optimal discretization of stochastic integrals. In Chapter 1 the construction of the optimal strategy uses the knowledge of the diffusion coefficient of the considered semimartingale. In this work we establish an asymptotically optimal discretization strategy that is adaptive to the model and does not use any information about the model. We prove the optimality for a rather general class of discretization grids based on kernel techniques for adaptive estimation. In Chapter 3 we study the convergence of renormalized discretization error laws of Itô processes for a concrete and rather general class of discretization grids given by stopping times. Previous works on the subject consider only the case of dimension 1. Moreover they concentrate on particular cases of grids, or prove results under abstract assumptions. In our work the boundary distribution is given explicitly in a clear and simple form, the results are shown in the multidimensional case for the process and for the discretization error. In Chapter 4 we study the parametric estimation problem for diffusion processes based on time-lapse observations. Previous works on the subject consider deterministic, strongly predictable or random observation times independent of the process. Under weak assumptions, we construct a suite of consistent estimators for a large class of observation grids given by stopping times. An asymptotic analysis of the estimation error is performed. Furthermore, for the parameter of dimension 1, for any sequence of estimators that verifies an unbiased LCT, we prove a uniform lower bound for the asymptotic variance. We show that this bound is exact. Chapters 5-6 are devoted to the uncertainty quantification problem for stochastic approximation bounds. In Chapter 5 we analyze the uncertainty quantification for stochastic approximation limits (SA). In our framework the limit is defined as a zero of a function given by an expectation. This expectation is taken with respect to a random variable for which the model is supposed to depend on an uncertain parameter. We consider the limit of SA as a function of this parameter. We introduce an algorithm called USA (Uncertainty for SA). It is a procedure in increasing dimension to compute the basic chaos expansion coefficients of this function in a basis of a well chosen Hilbert space. The convergence of USA in this Hilbert space is proved. In Chapter 6 we analyze the convergence rate in L2 of the USA algorithm developed in Chapter 5. The analysis is non-trivial because of the infinite dimension of the procedure. The rate obtained depends on the model and the parameters used in the USA algorithm. Its knowledge allows to optimize the rate of growth of the dimension in USA.

In this paper we study the problem of parametric inference for multidimensional diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject (such as [Doh87, GJ93, Gob01, AM04] among others) consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids (studied in [GL14, GS18]). Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp.

No summary available.

We study the convergence in distribution of the renormalized error arising from the discretization of a Brownian semimartingale sampled at stopping times. Our mild assumptions on the form of stopping times allow the time grid to be a combination of hitting times of stochastic domains and of Poisson-like random times. Remarkably, a Functional Central Limit Theorem holds under great generality on the semimartingale and on the form of stopping times. Furthermore, the asymptotic characteristics are quite explicit. Along the derivation of such results, we also establish some key estimates related to approximations and sensitivities of hitting time/position with respect to model and domain perturbations.

The uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm is analyzed. In our setup, this limit $\targetfn$ is defined as a zero of an intractable function and is modeled as uncertain through a parameter $\param$. We aim at deriving the function $\targetfn$, as well as the probabilistic distribution of $\targetfn(\param)$ given a probability distribution $\pi$ for $\param$. We introduce the so-called Uncertainty Quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of $\param \mapsto \targetfn(\param)$ on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations $K$, returns a finite set of coefficients, providing an approximation $\widehat{\targetfn_K}(\cdot)$ of $\targetfn(\cdot)$. We establish the almost-sure and $L^p$-convergences in the Hilbert space of the sequence of functions $\widehat{\targetfn_K}(\cdot)$ when the number of iterations $K$ tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix and of higher order moments of the quantity $\widehat{\targetfn_K}(\param)$ when $\param$ is random with distribution $\pi$. UQSA is illustrated and the role of its design parameters is discussed numerically.

We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretiza- tion stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.