Papers by Benjamin Jourdain
arXiv (Cornell University), Jun 21, 2010
In this paper, we are interested in approximating the solution to scalar conservation laws using ... more In this paper, we are interested in approximating the solution to scalar conservation laws using systems of interacting stochastic particles. The scalar conservation law may involve a fractional Laplacian term of order α ∈ (0, 2]. When α ≤ 1 as well as in the absence of this term (inviscid case), its solution is characterized by entropic inequalities. The probabilistic interpretation of the scalar conservation is based on a stochastic differential equation driven by an α-stable process and involving a drift nonlinear in the sense of McKean. The particle system is constructed by discretizing this equation in time by the Euler scheme and replacing the nonlinearity by interaction. Each particle carries a signed weight depending on its initial position. At each discretization time we kill the couples of particles with opposite weights and positions closer than a threshold since the contribution of the crossings of such particles has the wrong sign in the derivation of the entropic inequalities. We prove convergence of the particle approximation to the solution of the conservation law as the number of particles tends to ∞ whereas the discretization step, the killing threshold and, in the inviscid case, the coefficient multiplying the stable increments tend to 0 in some precise asymptotics depending on whether α is larger than the critical level 1.
Bernoulli, May 1, 2022
It was shown by the authors that two one-dimensional probability measures in the convex order adm... more It was shown by the authors that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of |x -y| is smaller than twice their W1-distance (Wasserstein distance with index 1). We showed that replacing |x -y| and W1 respectively with |x -y| ρ and W ρ ρ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing W ρ ρ with the product of Wρ times the centred ρ-th moment of the second marginal to the power ρ -1. Then we study the generalisation of this new martingale Wasserstein inequality to higher dimension.
Electronic Journal of Probability, 2020
In this paper, we exhibit a new family of martingale couplings between two one-dimensional probab... more In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures µ and ν in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of µ and ν. It contains the inverse transform martingale coupling which is explicit in terms of the cumulative distribution functions of these marginal densities. The integral of |x -y| with respect to each of these couplings is smaller than twice the W1 distance between µ and ν. When the comonotonous coupling between µ and ν is given by a map T , the elements of the family minimise R |y -T (x)| M (dx, dy) among all martingale couplings between µ and ν. When µ and ν are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.
arXiv (Cornell University), Sep 15, 2017
In this paper, for µ and ν two probability measures on R d with finite moments of order ≥ 1, we d... more In this paper, for µ and ν two probability measures on R d with finite moments of order ≥ 1, we define the respective projections for the W -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 W2-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. have shown that the projections do not depend on . 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.
arXiv (Cornell University), Mar 26, 2009
We study a free energy computation procedure, introduced in , which relies on the long-time behav... more We study a free energy computation procedure, introduced in , which relies on the long-time behavior of a nonlinear stochastic differential equation. This nonlinearity comes from a conditional expectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions to this equation has been proved in , under some existence and regularity assumptions. In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear equation, and we study a particle approximation technique based on a Nadaraya-Watson estimator of the conditional expectation. The particle system converges to the solution of the nonlinear equation if the number of particles goes to infinity and then the kernel used in the Nadaraya-Watson approximation tends to a Dirac mass. We derive a rate for this convergence, and illustrate it by numerical examples on a toy model.
SIAM Journal on Numerical Analysis, Jul 13, 2022
We are interested in the Euler-Maruyama discretization of a stochastic differential equation in d... more We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension d with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order 1/2 in total variation distance. When the drift has a spatial divergence in the sense of distributions with ρ-th power integrable with respect to the Lebesgue measure in space uniformly in time for some ρ ≥ d, the order of convergence at the terminal time improves to 1 up to some logarithmic factor. In dimension d = 1, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.
arXiv (Cornell University), Jan 7, 2021
Our main result is to establish stability of martingale couplings: suppose that π is a martingale... more Our main result is to establish stability of martingale couplings: suppose that π is a martingale coupling with marginals µ, ν. Then, given approximating marginal measures μ ≈ µ, ν ≈ ν in convex order, we show that there exists an approximating martingale coupling π ≈ π with marginals μ, ν. In mathematical finance, prices of European call / put option yield information on the marginal measures of the arbitrage free pricing measures. The above result asserts that small variations of call / put prices lead only to small variations on the level of arbitrage free pricing measures. While these facts have been anticipated for some time, the actual proof requires somewhat intricate stability results for the adapted Wasserstein distance. Notably the result has consequences for several related problems. Specifically, it is relevant for numerical approximations, it leads to a new proof of the monotonicity principle of martingale optimal transport and it implies stability of weak martingale optimal transport as well as optimal Skorokhod embedding. On the mathematical finance side this yields continuity of the robust pricing problem for exotic options and VIX options with respect to market data. These applications will be detailed in two companion papers.
Social Science Research Network, 2017
By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market pr... more By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in , provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to [12], we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called regime switching local volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. Finally, we deduce existence of the calibrated model by extending the results in .
arXiv (Cornell University), Jun 9, 2016
In this paper, we are interested in the time derivative of the Wasserstein distance between the m... more 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.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 5, 2020
In this work, a generalised version of the central limit theorem is proposed for nonlinear functi... more In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivative. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. We use this result to deal with the contribution of the initialisation in the convergence of the fluctuations between the empirical measure of interacting diffusion and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. A complementary contribution related to the time evolution is treated using the master equation, a parabolic PDE involving L-derivatives with respect to the measure component, which is a stronger notion of derivative that is nonetheless related to the linear functional derivative.
Electronic Journal of Probability, 2019
In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte... more In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.
Mathematics and Financial Economics, Mar 19, 2016
This paper deals with the problem of outsourcing the debt for a big investment, according two sit... more This paper deals with the problem of outsourcing the debt for a big investment, according two situations: either the firm outsources both the investment (and the associated debt) and the exploitation to another firm (for example a private consortium), or the firm supports the debt and the investment but outsources the exploitation. We prove the existence of Stackelberg and Nash equilibria between the firms, in both situations. We compare the benefits of these contracts, theorically and numerically. We conclude with a study of what happens in case of incomplete information, in the sense that the risk aversion coefficient of each partner may be unknown by the other partner.
Stochastic Processes and their Applications, Sep 1, 2012
We analyze the regularity of the value function and of the optimal exercise boundary of the Ameri... more We analyze the regularity of the value function and of the optimal exercise boundary of the American Put option when the underlying asset pays a discrete dividend at known times during the lifetime of the option. The ex-dividend asset price process is assumed to follow the Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. This function is assumed to be non-negative, non-decreasing and with growth rate not greater than 1. We prove that the exercise boundary is continuous and that the smooth contact property holds for the value function at any time but the dividend dates. We thus extend and generalize the results obtained in [JV11] when the dividend function is also positive and concave. Lastly, we give conditions on the dividend function ensuring that the exercise boundary is locally monotonic in a neighborhood of the corresponding dividend date.
Proceedings of the Edinburgh Mathematical Society, Oct 1, 2004
We are interested in proving the convergence of Monte Carlo approximations for vortex equations i... more We are interested in proving the convergence of Monte Carlo approximations for vortex equations in bounded domains of R 2 with Neumann's condition on the boundary. This work is the first step towards justifying theoretically some numerical algorithms for Navier-Stokes equations in bounded domains with no-slip conditions. We prove that the vortex equation has a unique solution in an appropriate energy space and can be interpreted from a probabilistic point of view through a nonlinear reflected process with space-time random births on the boundary of the domain. Next, we approximate the solution w of this vortex equation by the weighted empirical measure of interacting diffusive particles with normal reflecting boundary conditions and space-time random births on the boundary. The weights are related to the initial data and to the Neumann condition. We prove a trajectorial propagation-of-chaos result for these systems of interacting particles. We can deduce a simple stochastic particle algorithm to simulate w.
Social Science Research Network, 2017
Strassen's theorem : (1965) Assume Multi-marginal case : payoff c(S T 1 , . . . , S Tn ) with c :... more Strassen's theorem : (1965) Assume Multi-marginal case : payoff c(S T 1 , . . . , S Tn ) with c : (R d ) n → R. Beiglböck, Henry-Labordère, Penkner (2013) : Duality and connection with super/subhedging strategies. Many theoretical contributions since. Lower bound (on 100 indep runs of ((μ 100 ) 2 P(ν 100 ) , ν100 ) : mean 0.2293, 95% confidence interval half-width 0.017, Upper bound : mean 0.4111, 95% CI half-width 0.0284
Electronic Journal of Probability, 2015
In this paper, we prove that the time supremum of the Wasserstein distance between the time-margi... more 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 Hölder continuous with exponent γ with respect to the time variable and its Euler scheme with N uniform time-steps is smaller than C(1 + 1γ=1 ln(N ))N -γ . To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio et al. 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.
Stochastics And Partial Differential Equations: Analysis And Computations, Jun 22, 2013
We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the re... more We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition. We call 'probabilistic solution' a weak solution which remains a cumulative distribution function at all times. We prove the uniqueness of such a solution and we deduce the existence from a propagation of chaos result on a system of scalar diffusion processes, the interactions of which only depend on their ranking. We then investigate the long time behaviour of the solution. Using a probabilistic argument and under weak assumptions, we show that the flow of the Wasserstein distance between two solutions is contractive. Under more stringent conditions ensuring the regularity of the probabilistic solutions, we finally derive an explicit formula for the time derivative of the flow and we deduce the convergence of solutions to equilibrium.
CiteSeer X (The Pennsylvania State University), Jul 18, 2007
In this paper we study general nonlinear stochastic differential equations, where the usual Brown... more In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a Lévy process. Moreover, we do not suppose that the coefficient multiplying the increments of this process is linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Lévy process with a smooth but unbounded Lévy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are absolutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian. This paper studies the following nonlinear stochastic differential equation: ∀s ∈ [0, T ], P s denotes the probability distribution of X s . (0.1)
arXiv (Cornell University), Jun 30, 2016
By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market pr... more By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in , provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to [12], we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called regime switching local volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. Finally, we deduce existence of the calibrated model by extending the results in .
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Papers by Benjamin Jourdain