Emmanuel Gobet

École Polytechnique

Tackling climate change is one of the biggest challenges of today. Limiting cli- mate change translates to drastically cutting carbon emissions to net zero as soon as possible. More and more commitments have been made by various authorities and companies to mitigate their GHG emissions accordingly, notably the Paris Agreement in 2015 that sets the...

We examine climate-related exposure within a large credit portfolio, addressing transition and physical risks. We design a modeling methodology that begins with the Shared Socioeconomic Pathways (SSP) scenarios and ends with describing the losses of a portfolio of obligors. The SSP scenarios impact the physical risk of each obligor via a DICE-inspi...

We propose new parametrizations for neural networks in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between...

We provide explicit approximation formulas for VIX futures and options in forward variance models, with particular emphasis on the family of so-called Bergomi models: the one-factor Bergomi model, the rough Bergomi model, and an enhanced version of the rough model that can generate realistic positive skew for VIX smiles–introduced simultaneously by...

Accounting for climate transition risks is one of the most important challenges in the transition to a low-carbon economy. Banks are encouraged to align their investment portfolios to CO2 trajectories fixed by international agreements, showing the necessity of a quantitative methodology to implement it. We propose a mathematical formulation for thi...

In financial risk management, modelling dependency within a random vector X is crucial, a standard approach is the use of a copula model. Say the copula model can be sampled through realizations of Y having copula function C: had the marginals of Y been known, sampling X(i), the i-th component of X, would directly follow by composing Y(i) with its...

This paper investigates the impact of transition risk on a firm’s low-carbon production. As the world is facing global climate change, the Intergovernmental Panel on Climate Change (IPCC) has set the idealized carbon-neutral scenario around 2050. In the meantime, many carbon reduction scenarios, known as Shared Socioeconomic Pathways (SSPs) have be...

We develop a new iterative method based on the Pontryagin principle to solve stochastic control problems. This method is nothing other than the Newton method extended to the framework of stochastic optimal control, where the state dynamics are given by an ODEs with stochastic coefficients and the cost is random. Each iteration of the method is made...

We study the mathematical modeling of the energy management system of a smart grid, related to a aggregated consumer equipped with renewable energy production (PV panels e.g.), storage facilities (batteries), and connected to the electrical public grid. He controls the use of the storage facilities in order to diminish the random fluctuations of hi...

We provide a large probability bound on the uniform approximation of fractional Brownian motion with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a N-dimensional Gaussian vector, with bounds on the network design (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error...

We provide explicit approximation formulas for VIX futures and options in stochastic forward variance models, with particular emphasis on the family of so-called Bergomi models: the one-factor Bergomi model [Bergomi, Smile dynamics II, Risk, 2005], the rough Bergomi model [Bayer, Friz, and Gatheral, Pricing under rough volatility, Quantitative Fina...

This paper investigates the impact of transition risk on a firm’s low-carbon production. As the world is facing global climate changes, the Intergovernmental Panel on Climate Change (IPCC) has set the idealized carbon-neutral scenario around 2050. In the meantime, many carbon reduction scenarios, known as Shared Socioeconomic Pathways (SSPs) have b...

In this paper we present a series of results that permit to extend in a direct manner uniform deviation inequalities of the empirical process from the independent to the dependent case characterizing the additional error in terms of $\beta-$mixing coefficients associated to the training sample. We then apply these results to some previously obtaine...

We propose a stochastic control problem to control cooperatively Thermostatically Controlled Loads (TCLs) to promote power balance in electricity networks. We develop a method to solve this stochastic control problem with a decentralized architecture, in order to respect privacy of individual users and to reduce both the telecommunications and the...

We develop a new iterative method based on Pontryagin principle to solve stochastic control problems. This method is nothing else than the Newton method extended to the framework of stochastic controls, where the state dynamics is given by an ODE with stochastic coefficients. Each iteration of the method is made of two ingredients: computing the Ne...

Discrete-time hedging produces a residual P&L, namely the tracking error. The major problem is to get valuation/hedging policies minimising this error. We evaluate the risk between trading dates through a function penalising profits and losses asymmetrically. After deriving the asymptotics from a discrete-time risk measurement for a large number of...

Kernel techniques are among the most widely-applied and influential tools in machine learning with applications at virtually all areas of the field. To combine this expressive power with computational efficiency numerous randomized schemes have been proposed in the literature, among which probably random Fourier features (RFF) are the simplest and...

This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatilit...

In this work we design a novel and efficient quasi-regression Monte Carlo algorithm in order to approximate the solution of discrete time backward stochastic differential equations (BSDEs), and we analyze the convergence of the proposed method. With the challenge of tackling problems in high dimensions we propose suitable projections of the solutio...

We study the mathematical modeling of the energy management system of a smart grid, related to a aggregated consumer equipped with renewable energy production (PV panels e.g.), storage facilities (batteries), and connected to the electrical public grid. He controls the use of the storage facilities in order to diminish the random fluctuations of hi...

In this article we design a novel quasi-regression Monte Carlo algorithm in order to approximate the solution of discrete time backward stochastic diferential equations, and we analyze the convergence of the proposed method. The algorithm also approximates the solution to the related semi-linear parabolic partial diferential equation obtained throu...

Power producers are interested in valuing their power plant production. By trading into forward contracts, we propose to reduce the contingency of the associated income considering the fixed costs and using an asymmetric risk criterion. In an asymptotic framework, we provide an optimal hedging strategy through a solution of a nonlinear partial diff...

This paper is devoted to the study of the deviation of the (random) average L ² −error associated to the least-squares regressor over a family of functions F n (with controlled complexity) obtained from n independent, but not necessarily identically distributed, samples of explanatory and response variables, from the minimal (deterministic) average...

We study the optimal discretization error of stochastic integrals driven by a multidimensional continuous Brownian semimartingale. In the previous works a pathwise lower bound for the renormalized quadratic variation of the error was provided together with an asymptotically optimal discretization strategy, i.e. for which the lower bound is attained...

This paper is devoted to the study of the deviation of the (random) average L 2 −error associated to the least-squares regressor over a family of functions F_n (with controlled complexity) obtained from n independent, but not necessarily identically distributed, samples of explanatory and response variables, from the minimal (deterministic) average...

Discrete-time hedging produces a residual risk, i.e., the tracking error. The major problem is to get valuation/hedging policies minimizing this error. We evaluate the risk between trading dates through a function penalizing profits and losses asymmetrically. After deriving the asymptotics within a discrete-time risk measurement for a large number...

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 ex- pected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the �rm, which we account for by nested Monte Carlo �a la Gordy and Junej...

In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a probabilistic forecast: the input parameters of the SDE model are the AROME numerical weather predictions co...

We consider a random map x → F (ω, x) and a random variable Θ(ω), and we denote by F N (ω, x) and Θ N (ω) their approximations: We establish a strong convergence result, in Lp-norms, of the compound approximation F N (ω, Θ N (ω)) to the compound variable F (ω, Θ(ω)), in terms of the approximations of F and Θ. Two applications of this result are the...

We derive expansion results in order to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approximation is based on the use of proxys with normal distribution or log-normal distribution, so that the expansion terms are explici...

We establish general moment estimates for the discrete and continuous exit times of a general Itô process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove th...

Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(...

We relate the LpLp-variation, 2≤p<∞2≤p<∞, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness takes into account the quantitative propagation of singularities in time.

This paper presents new approximation formulae of European options in a local volatility model with stochastic interest rates. This is a companion paper to our work on perturbation methods for local volatility models for the case of stochastic interest rates. The originality of this approach is to model the local volatility of the discounted spot a...

This overview article concerns the notion of fractional smoothness of random variables of the form g(X T ), where X=(X t )t∈[0,T] is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete-ti...

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of dis...

We present and analyze an algorithm to solve numerically BSDEs based on Picard's iterations and on a sequential control variate technique. Its convergence is geometric. Moreover, the solution provided by our algorithm is regular both w.r.t. time and space.

For a stopped diffusion process in a multidimensional time-dependent domain D, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size Δ and stopping it at discrete times in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward no...

This note is a complement of the paper "Solving BSDE with adaptive control variate". It deals with the convergence of the approximating operator P, based on a non parametric regression technique called local averaging. Although the computations are quite standard (see Hardle '92, Gyorfi etal. '02), the specificities of the paper are the following:...

The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [Hes93] or piecewise constant [MN03]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Hesto...

Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greek...

This article deals with the numerical resolution of backward stochastic differential equations. Firstly, we consider a rather general case where the filtration is generated by a Brownian motion and a Poisson random measure. We provide a simulation algorithm based on iterative regressions on function bases, which coefficients are evaluated using Mon...

Since the beginning of the 1990s, mathematics, and more particularly the theory of probability, have taken an increasing role in the banking and insurance industries. This motivated the authors to present here some interactions between Mathematics and Finance and their consequences at the level of research and training in France in these domains.

In this work, we present advanced Monte Carlo techniques applied to the pricing of barrier options and other related exotic contracts. It covers in particular the Brownian bridge approaches, the barrier shifting techniques (BAST) and their extensions as well. We leverage the link between discrete and continuous monitoring to design efficient scheme...

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump...

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump...

We present and analyze a numerical algorithm for solving BSDEs based on Picard iterations and on a sequential control variate method. Its convergence is geometric. Moreover, our algorithm provides a regular solution w.r.t. time and space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

For a multidimensional Itô process (Xt)t ≥ 0 driven by a Brownian motion, we are interested in approximating the law of ω ((Xs)s ∈ [0, T]), T > 0 deterministic, for a given functional φ using a discrete sample of the process X. For various functionals (related to the maximum, to the integral of the process, or to the killed/stopped path) we extend...

For a stopped diffusion process in a multidimensional time-dependent domain $\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $\Delta$ and stopping it at discrete times $(i\Delta)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally...

This study focuses on the numerical resolution of backward stochastic differential equations with data dependent on a jump-diffusion process. We propose and analyse a numerical scheme based on iterative regression functions which are approximated by projections on vector spaces of functions, with coefficients evaluated using Monte Carlo simulations...

We study the error induced by the time discretization of a decoupled forward-backward stochastic differential equations $(X,Y,Z)$. The forward component $X$ is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme $X^N$ with $N$ time steps. The backward component is approximated by a backward scheme. Firs...

This paper is concerned with numerical approximations for stochastic partial dif- ferential Zakai equation of nonlinear filtering problem. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp...

We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, con...

This paper is concerned with numerical approximations for a class of nonlinear stochastic partial differential equations: Zakai equation of nonlinear filtering problem and McKean-Vlasov type equations. The approximation scheme is based on the re-presentation of the solutions as weighted conditional distributions. We first accurately analyse the err...

We are interested in approximating a multidimensional hypoelliptic diffusion process (Xt)t[greater-or-equal, slanted]0 killed when it leaves a smooth domain D. When a discrete Euler scheme with time step h is used, we prove under a noncharacteristic boundary condition that the weak error is upper bounded by , generalizing the result obtained by Gob...