arXiv subjects
Publications and source records attributed to Benjamin Jourdain.
We prove existence and uniqueness for some nonlinear stochastic differential equation used in molecular dynamics, whose nonlinearity comes from a conditional expectation term. We also introduce an interacting particle system in order to approximate this conditional expectation, providing a discretization scheme for this equation.
In this paper, we are interested in continuous time models in which the index level induces some feedback on the dynamics of its composing stocks. More precisely, we propose a model in which the log-returns of each stock may be decomposed into a systemic part proportional to the log-returns of the index plus an idiosyncratic part. We show that, when the number of stocks in the index is large, this model may be approximated by a local volatility model for the index and a stochastic volatility model for each stock with volatility driven by the index. This result is useful in a calibration perspective : it suggests that one should first calibrate the local volatility of the index and then calibrate the dynamics of each stock. We explain how to do so in the limiting simplified model and in the original model.
Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly dependent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator: in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor between 3 and 15.
This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (i.e. when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators.
The Metropolis Hastings algorithm and its multi-proposal extensions are aimed at the computation of the expectation $<π,f>$ of a function $f$ under a probability measure $π$ difficult to simulate. They consist in constructing by an appropriate acceptation/rejection procedure a Markov chain $(X_k,k\geq 0)$ with transition matrix $P$ such that $π$ is reversible with respect to $P$ and in estimating $<π,f>$ by the empirical mean $I_n(f)=\inv{n}\sum_{k=1}^n f(X_k)$. The waste-recycling Monte Carlo (WR) algorithm introduced by physicists is a modification of the Metropolis-Hastings algorithm, which makes use of all the proposals in the empirical mean, whereas the standard Metropolis-Hastings algorithm only uses the accepted proposals. In this paper, we extend the WR algorithm into a general control variate technique and exhibit the optimal choice of the control variate in terms of asymptotic variance. We also give an example which shows that in contradiction to the intuition of physicists, the WR algorithm can have an asymptotic variance larger than the one of the Metropolis-Hastings algorithm. However, in the particular case of the Metropolis-Hastings algorithm called Boltzmann algorithm, we prove that the WR algorithm is asymptotically better than the Metropolis-Hastings algorithm.
In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincaré inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.
In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction. And our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm.
In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a Lévy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily 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.
The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a {\bf fixed} number of random walkers evolving according to a Stochastic Differential Equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove the convergence of the method for a fixed number of reconfigurations when the number of walkers tends to $+\infty$ while the timestep tends to 0. We confirm our theoretical rates of convergence by numerical experiments. Various resampling algorithms are investigated, both theoretically and numerically
It is well known that in models with time-homogeneous local volatility functions and constant interest and dividend rates, the European Put prices are transformed into European Call prices by the simultaneous exchanges of the interest and dividend rates and of the strike and spot price of the underlying. This paper investigates such a Call Put duality for perpetual American options. It turns out that the perpetual American Put price is equal to the perpetual American Call price in a model where, in addition to the previous exchanges between the spot price and the strike and between the interest and dividend rates, the local volatility function is modified. We prove that equality of the dual volatility functions only holds in the standard Black-Scholes model with constant volatility. Thanks to these duality results, we design a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices for a fixed spot price $x_0$. The knowledge of the Put (resp. Call) prices for all strikes enables to recover the local volatility function on the interval $(0,x_0)$ (resp. $(x_0,+\infty)$).
In this paper, we investigate the generalization of the Call-Put duality equality obtained in [1] for perpetual American options when the Call-Put payoff $(y-x)^+$ is replaced by $ϕ(x,y)$. It turns out that the duality still holds under monotonicity and concavity assumptions on $ϕ$. The specific analytical form of the Call-Put payoff only makes calculations easier but is not crucial unlike in the derivation of the Call-Put duality equality for European options. Last, we give some examples for which the optimal strategy is known explicitly.
The probabilistic equivalent formulation of Dupire's PDE is the Put-Call duality equality. In local volatility models including exponential Lévy jumps, we give a direct probabilistic proof for this result based on stochastic flows arguments. This approach also enables us to check the probabilistic equivalent formulation of various generalizations of Dupire's PDE recently obtained by Pironneau by the adjoint equation technique in the case of complex options.