FRIKHA Noufel

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In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.

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In this article, we are interested in the strong well-posedness together with the numerical approximation of some one-dimensional stochastic differential equations with a non-linear drift, in the sense of McKean-Vlasov, driven by a spectrally-positive Lévy process and a Brownian motion. We provide criteria for the existence of strong solutions under non-Lipschitz conditions of Yamada-Watanabe type without non-degeneracy assumption. The strong convergence rate of the propagation of chaos for the associated particle system and of the corresponding Euler-Maruyama scheme are also investigated. In particular, the strong convergence rate of the Euler-Maruyama scheme exhibits an interplay between the regularity of the coefficients and the order of singularity of the Lévy measure around zero.

This thesis is devoted to the theoretical and numerical study of the weak error of time and particle discretization of nonlinear Stochastic Differential Equations in the McKean sense. In the first part, we analyze the weak convergence speed of the time discretization of standard SDEs. More specifically, we study the convergence in total variation of the Euler-Maruyama scheme applied to d-dimensional DEs with a measurable drift coefficient and additive noise. We obtain, assuming that the drift coefficient is bounded, a weak order of convergence 1/2. By adding more regularity on the drift, namely a spatial divergence in the sense of L[rho]-space-uniform distributions in time for some [rho] greater than or equal to d, we reach a convergence order equal to 1 (within a logarithmic factor) at terminal time. In dimension 1, this result is preserved when the spatial derivative of the drift is a measure in space with a total mass bounded uniformly in time. In the second part of the thesis, we analyze the weak discretization error in both time and particles of two classes of nonlinear DHSs in the McKean sense. The first class consists of multi-dimensional SDEs with regular drift and diffusion coefficients in which the law dependence occurs through moments. The second class consists of one-dimensional SDEs with a constant diffusion coefficient and a singular drift coefficient where the law dependence occurs through the distribution function. We approximate the EDS by the Euler-Maruyama schemes of the associated particle systems and we obtain for both classes a weak order of convergence equal to 1 in time and in particles. In the second class, we also prove a result of chaos propagation of optimal order 1/2 in particles and a strong order of convergence equal to 1 in time and 1/2 in particles. All our theoretical results are illustrated by numerical simulations.

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $α$-stable Lévy processes with values in $R^d$ under some mild Hölder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $\alpha \in (0, 1)$. New strong well-posedness results are also derived from the previous analysis.

In this paper, we investigate the well-posedness of the martingale problem associated to non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov under mild assumptions on the coefficients as well as classical solutions for a class of associated linear partial differential equations (PDEs) defined on $[0,T] \times \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d)$, for any $T>0$, $\mathcal{P}_2(\mathbb{R}^d)$ being the Wasserstein space (\emph{i.e.} the space of probability measures on $\mathbb{R}^d$ with a finite second-order moment). In this case, the derivative of a map along a probability measure is understood in the Lions' sense. The martingale problem is addressed by a fixed point argument on a suitable complete metric space, under some mild regularity assumptions on the coefficients that covers a large class of interaction. Also, new well-posedness results in the strong sense are obtained from the previous analysis. Under additional assumptions, we then prove the existence of the associated density and investigate its smoothness property. In particular, we establish some Gaussian type bounds for its derivatives. We eventually address the existence and uniqueness for the related linear Cauchy problem with irregular terminal condition and source term.

In this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li's type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.

In this article, we provide some new quantitative estimates for propagation of chaos of non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov. We obtain explicit error estimates, at the level of the trajectories, at the level of the semi-group and at the level of the densities, for the mean-field approximation by systems of interacting particles under mild regularity assumptions on the coefficients. A first order expansion for the difference between the densities of one particle and its mean-field limit is also established. Our analysis relies on the well-posedness of classical solutions to the backward Kolmogorov partial differential equations (PDEs) defined on the strip [0, T ] × R d × P 2 (R d), P 2 (R d) being the Wasserstein space, that is, the space of probability measures on R d with a finite second-order moment and also on the existence and uniqueness of a fundamental solution for the related parabolic linear operator here stated on [0, T ] × P 2 (R d).

In this thesis, we study several financial mathematics problems related to the valuation of derivatives. Through different asymptotic approaches, we develop methods to compute accurate approximations of the price of certain types of options in cases where no explicit formula exists.In the first chapter, we focus on the valuation of options whose payoff depends on the trajectory of the underlying by Monte Carlo methods, when the underlying is modeled by an affine process with stochastic volatility. We prove a principle of large trajectory deviations in long time, which we use to compute, using Varadhan's lemma, an asymptotically optimal change of measure, allowing to significantly reduce the variance of the Monte-Carlo estimator of option prices.The second chapter considers the valuation by Monte-Carlo methods of options depending on multiple underlyings, such as basket options, in Wishart's stochastic volatility model, which generalizes the Heston model. Following the same approach as in the previous chapter, we prove that the process vérifie a principle of large deviations in long time, which we use to significantly reduce the variance of the Monte Carlo estimator of option prices, through an asymptotically optimal change of measure. In parallel, we use the principle of large deviations to characterize the long-time behavior of the Black-Scholes implied volatility of basket options.In the third chapter, we study the valuation of realized variance options, when the spot volatility is modeled by a constant volatility diffusion process. We use recent asymptotic results on the densities of hypo-elliptic diffusions to compute an expansion of the realized variance density, which we integrate to obtain the expansion of the option price, and then their Black-Scholes implied volatility.The final chapter is devoted to the valuation of interest rate derivatives in the Lévy model of the Libor market, which generalizes the classical Libor market model (log-normal) by adding jumps. By writing the former as a perturbation of the latter and using the Feynman-Kac representation, we explicitly compute the asymptotic expansion of the price of interest rate derivatives, in particular, caplets and swaptions.

In this paper, we develop a general methodology to prove weak uniqueness for stochastic differential equations with coefficients depending on some path-functionals of the process. As an extension of the technique developed by Bass & Perkins [BP09] in the standard diffusion case, the proposed methodology allows one to deal with processes whose probability laws are singular with respect to the Lebesgue measure. To illustrate our methodology, we prove weak existence and uniqueness in two examples : a diffusion process with coefficients depending on its running symmetric local time and a diffusion process with coefficients depending on its running maximum. In each example, we also prove the existence of the associated transition density and establish some Gaussian upper-estimates.

This paper studies multi-level stochastic approximation algorithms. Our aim is to extend the scope of the multilevel Monte Carlo method recently introduced by Giles (Giles 2008) to the framework of stochastic optimization by means of stochastic approximation algorithm. We first introduce and study a two-level method, also referred as statistical Romberg stochastic approximation algorithm. Then, its extension to multi-level is proposed. We prove a central limit theorem for both methods and describe the possible optimal choices of step size sequence. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.

In this article, we obtain properties of the law associated to the first hitting time of a threshold by a one-dimensional uniformly elliptic diffusion process and to the associated process stopped at the threshold. Our methodology relies on the parametrix method that we apply to the associated Markov semigroup. It allows to obtain explicit expressions for the corresponding transition densities and to study its regularity properties up to the boundary under mild assumptions on the coefficients. As a by product, we also provide Gaussian upper estimates for these laws and derive a probabilistic representation that may be useful for the construction of an unbiased Monte Carlo path simulation method, among other applications.

We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Frikha2013]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte Carlo linear estimator (introduced in [Talay & Tubaro 1990] and deeply studied in [Pagès 2007]) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.

We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Frikha2013]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte Carlo linear estimator (introduced in [Talay & Tubaro 1990] and deeply studied in [Pagès 2007]) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.

The recent liberalization of the electricity and gas markets has resulted in the growth of energy exchanges and modelling problems. In this paper, we modelize jointly gas and electricity spot prices using a mean-reverting model which fits the correlations structures for the two commodities. The dynamics are based on Ornstein processes with parameterized diffusion coefficients. Moreover, using the empirical distributions of the spot prices, we derive a class of such parameterized diffusions which captures the most salient statistical properties: stationarity, spikes and heavy-tailed distributions. The associated calibration procedure is based on standard and efficient statistical tools. We calibrate the model on French market for electricity and on UK market for gas, and then simulate some trajectories which reproduce well the observed prices behavior. Finally, we illustrate the importance of the correlation structure and of the presence of spikes by measuring the risk on a power plant portfolio.

We obtain new transport-entropy inequalities and, as a by-product, new deviation estimates for the laws of two kinds of discrete stochastic approximation schemes. The first one refers to the law of an Euler like discretization scheme of a diffusion process at a fixed deterministic date and the second one concerns the law of a stochastic approximation algorithm at a given time-step. Our results notably improve and complete those obtained in [Frikha, Menozzi,2012]. The key point is to properly quantify the contribution of the diffusion term to the concentration regime. We also derive a general non-asymptotic deviation bound for the difference between a function of the trajectory of a continuous Euler scheme associated to a diffusion process and its mean. Finally, we obtain non-asymptotic bound for stochastic approximation with averaging of trajectories, in particular we prove that averaging a stochastic approximation algorithm with a slow decreasing step sequence gives rise to optimal concentration rate.

We obtain new transport-entropy inequalities and, as a by-product, new deviation estimates for the laws of two kinds of discrete stochastic approximation schemes. The first one refers to the law of an Euler like discretization scheme of a diffusion process at a fixed deterministic date and the second one concerns the law of a stochastic approximation algorithm at a given time-step. Our results notably improve and complete those obtained in [Frikha, Menozzi,2012]. The key point is to properly quantify the contribution of the diffusion term to the concentration regime. We also derive a general non-asymptotic deviation bound for the difference between a function of the trajectory of a continuous Euler scheme associated to a diffusion process and its mean. Finally, we obtain non-asymptotic bound for stochastic approximation with averaging of trajectories, in particular we prove that averaging a stochastic approximation algorithm with a slow decreasing step sequence gives rise to optimal concentration rate.

This thesis is devoted to probabilistic numerical problems related to modeling, control and risk management and motivated by applications in energy markets. The main tool used is the theory of stochastic algorithms and simulation methods. This thesis consists of three parts. The first part is devoted to the estimation of two risk measures of the L-distribution of losses in a portfolio: the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR). This estimation is performed using a stochastic algorithm combined with an adaptive variance reduction method. The first part of this chapter deals with the case of finite dimension, the second extends the first to the case of a function of the trajectory of a process and the last one deals with the case of sequences with low discrepancy. The second chapter is dedicated to methods for hedging risk in CVaR in an incomplete market operating in discrete time using stochastic algorithms and optimal vector quantization. Theoretical results on CVaR hedging are presented and then numerical aspects are discussed in a Markovian framework. The last part is devoted to the joint modeling of spot gas and electricity prices. The multi-factor model presented is based on stationary Ornstein processes with a parametric diffusion coefficient.

This thesis is devoted to probabilistic numerical problems related to modeling, control and risk management and motivated by applications in energy markets. The main tool used is the theory of stochastic algorithms and simulation methods. This thesis consists of three parts. The first part is devoted to the estimation of two risk measures of the L-distribution of losses in a portfolio: the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR). This estimation is performed using a stochastic algorithm combined with an adaptive variance reduction method. The first part of this chapter deals with the case of finite dimension, the second extends the first to the case of a function of the trajectory of a process and the last one deals with the case of sequences with low discrepancy. The second chapter is dedicated to methods for hedging risk in CVaR in an incomplete market operating in discrete time using stochastic algorithms and optimal vector quantization. Theoretical results on CVaR hedging are presented and then numerical aspects are discussed in a Markovian framework. The last part is devoted to the joint modeling of spot gas and electricity prices. The multi-factor model presented is based on stationary Ornstein processes with a parametric diffusion coefficient.