International audienceWe introduce a generalization of the Adaptive Multilevel Splitting algorithm in the discrete time dynamic setting, namely when it is applied to sample rare events associated with paths of Markov chains. By interpreting the algorithm as a sequential sampler in path space, we are able to build an estimator of the rare event probability (and of any non-normalized quantity associated with this event) which is unbiased, whatever the choice of the importance function and the number of replicas. This has practical consequences on the use of this algorithm, which are illustrated through various numerical experiments
We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L 2 (Ω)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.
We consider a stochastic partial differential equation with two logarithmic nonlinearities, two reflections at 1 and −1, and a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of a maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche, Goudenège, and Zambotti, we obtain existence and uniqueness of a solution for initial conditions in the interval (−1, 1). Finally, we prove that the unique invariant measure is ergodic, and we give a result of exponential mixing.Introduction and main results. The Cahn-Hilliard-Cook equation is a model to describe phase separation in a binary alloy (see [6], [7], and [8]) in the presence of thermal fluctuations (see [11] and [27]). It takes the form
In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black-Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. Moreover, these two methods solve the backward dynamic programming problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite this model is only bidimensional, the whole history of the process impacts on the price, and handling all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.Pricing American options is clearly a crucial question of finance but also a challenging one since computing the optimal exercise strategy is not an evident task. This issue is even more exacting when the underling of the option is a multi-dimensional process, such as a baskets of d assets, since in this case the direct application of standard numerical schemes, such as finite difference or tree methods, is not possible because of the exponential growth of the calculation time and the required working memory.Common approaches in this field can be divided in four groups: techniques which rely on recombinant trees to discretize the underlyings (see [4], [11] and [24]), techniques which employ regression on a truncated basis of L 2 in order to compute the conditional expectations (see [28] and [32]), techniques which exploit Malliavin calculus to obtain representation formulas for the conditional expectation (see [1], [3], [9], and [27]) and techniques which make use of duality-based approaches for Bermudan option pricing (see [21], [26] and [31]). Recently, Machine Learning algorithms (Rasmussen and Williams [33]) and Deep Learning techniques (Nielsen [30]) have found great application in this sector of option pricing. Neural networks are used by Kohler et al. [25] to price American options based on several underlyings. Deep Learning techniques are nowadays widely used in solving large differential equations, which is intimately related to option pricing. In particular, Han et al. [20] introduce a Deep Learning-based approach that can handle general high-dimensional parabo...
The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile iterative method to estimate the probabilities of rare events. We prove a new central limit theorem for the associated AMS estimators introduced in [5], and which have been recently revisited in [3]-the main result there being (nonasymptotic) unbiasedness of the estimators. To prove asymptotic normality, we rely on and extend the technique presented in [3]: the (asymptotic) analysis of an integral equation. Numerical simulations illustrate the convergence and the construction of Gaussian confidence intervals.
Significance The analysis of complex systems with many degrees of freedom generally involves the definition of low-dimensional collective variables more amenable to physical understanding. Their dynamics can be modeled by generalized Langevin equations, whose coefficients have to be estimated from simulations of the initial high-dimensional system. These equations feature a memory kernel describing the mutual influence of the low-dimensional variables and their environment. We introduce and implement an approach where the generalized Langevin equation is designed to maximize the statistical likelihood of the observed data. This provides an efficient way to generate reduced models to study dynamical properties of complex processes such as chemical reactions in solution, conformational changes in biomolecules, or phase transitions in condensed matter systems.
In this paper we propose an efficient method to compute the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction technique. Specifically, the options we consider are written on a basket of assets, each of them following a Black-Scholes dynamics. In the wake of Ludkovski's approach [33], we implement here a backward dynamic programming algorithm which considers a finite number of uniformly distributed exercise dates. On these dates, the option value is computed as the maximum between the exercise value and the continuation value, which is obtained by means of Gaussian process regression technique and Monte Carlo simulations. Such a method performs well for low dimension baskets but it is not accurate for very high dimension baskets. In order to improve the dimension range, we employ the European option price as a control variate, which allows us to treat very large baskets and moreover to reduce the variance of price estimators. Numerical tests show that the proposed algorithm is fast and reliable, and it can handle also American options on very large baskets of assets, overcoming the problem of the curse of dimensionality.
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