The need to simulate stochastic processes numerically arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness. This paper addresses the question of how to use quasi-Monte Carlo methods to improve this simulation. Special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps is required. Two such techniques, the generalized Brownian bridge and particle reordering, are described here. These methods are applied to a problem from finance, the valuation of a 30-year bond with monthly coupon payments assuming a mean reverting stochastic interest rate. When expressed as an integral, this problem is nominally 360 dimensional. The analysis of the integrand presented here explains the effectiveness of the quasi-random sequences on this high-dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi-random sequences.

Generating Quasi-Random Paths for Stochastic Processes

In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncertainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.

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Statistical analysis of differential equations: introducing probability measures on numerical solutions

https://hal.inria.fr/inria-00000118/document

A central limit theorem and improved error bounds for a hybrid-Monte Carlo sequence with applications in computational finance

This paper is concerned with the problem of ruin in the classical compound binomial and compound Poisson risk models. Our primary purpose is to extend to those models an exact formula derived by Picard and Lefevre (24) for the probability of (non-)ruin within finite time. First, a standard method based on the ballot theorem and an argument of Seal-type provides an initial (known) formula for that probability. Then, a concept of pseudo-distributions for the cumulated claim amounts, combined with some simple implications of the ballot theorem, leads to the desired formula. Two expressions for the (non-)ruin probability over an infinite horizon are also deduced as corollaries. Finally, an illustration within the framework of Solvency II is briefly presented.

/pdf/on-finite-time-ruin-probabilities-for-classical-risk-models-46w7e1ohs7.pdf

On finite-time ruin probabilities for classical risk models

A random Euler scheme for Carathéodory differential equations

A multirisks model is constructed that describes the evolution in discrete-time of an insurance portfolio covering several interdependent risks. The main problem under study is the determination of the probabilities of ruin over a finite horizon, for one or more risks. An underlying polynomial structure in the expression of these probabilities is exhibited. This result is then used to provide a simple recursive method for their numerical evaluation. Furthermore, it is shown qualitatively that a stronger positive-type dependence between the risks increases the non-ruin probabilities. Some illustrations enhance the efficiency, in time and precision, of the developed algorithm.

Multirisks Model and Finite-Time Ruin Probabilities

A quasi-Monte Carlo particle simulation for solving a linear Boltzmann equation is constructed and a convergence proof is given. The analysis is restricted to the three-dimensional equation in the space homogeneous case and the velocity domain is normalized to be I 3 = [0,1)3. A particle simulation is described. It combines a Euler scheme in time with quasi-Monte Carlo integration in velocity space. The quadratures use (0,m,s)-nets, which are sets with a very regular distribution behavior. The particles are reordered according to the components of their velocity at each time step. A deterministic error bound is obtained. Finally, numerical experiments are presented for some test problems where an exact solution is known. The accuracy of the quasi-Monte Carlo scheme is compared with a standard Monte Carlo algorithm. The results show an improvement in both magnitude of error and convergence rate of quasi-Monte Carlo over Monte Carlo approach.

A Quasi-Monte Carlo Scheme Using Nets for a Linear Boltzmann Equation

We consider a discrete-time risk model which describes the evolution of the reserves of an insurance company at periodic dates fixed in advance. The amount of loss per unit of time corresponds to independent and identically distributed random variables with arithmetic distribution, and the process of the receipt of premiums is assumed to be deterministic, nonnegative but not uniform (instead of being constant and equal to 1 as in the standard, compound binomial model). For this model, we determine the probability of ruin (or of non-ruin), as well as the distribution of the severity of the eventual ruin, with some finite horizon. A compact and efficient exact expression is found by bringing up-to-date a generalised family of Appell polynomials. The method used is illustrated with some numerical examples.

https://www.jstor.org/stable/info/3215934

Problèmes de ruine en théorie du risque à temps discret avec horizon fini

We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations y'(t) = f(t,y(t)). The function f is smooth in y and we suppose that f and D 1 y f are of bounded variation in t and that D 2 y f is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte Carlo estimate of integrals. The error bound involves the square of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. Numerical experiments show that the quasi-randomized method outperforms a recently proposed randomized numerical method.

/pdf/a-quasi-randomized-runge-kutta-method-1bhk700rec.pdf

A quasi-randomized Runge-Kutta method

This note discusses a simple quasi-Monte Carlo method to evaluate numerically the ultimate ruin probability in the classical compound Poisson risk model. The key point is the Pollaczek–Khintchine representation of the non-ruin probability as a series of convolutions. Our suggestion is to truncate the series at some appropriate level and to evaluate the remaining convolution integrals by quasi-Monte Carlo techniques. For illustration, this approximation procedure is applied when claim sizes have an exponential or generalized Pareto distribution.

Ibrahim Coulibaly

Papers

Multirisks Model and Finite-Time Ruin Probabilities

A Quasi-Monte Carlo Scheme Using Nets for a Linear Boltzmann Equation

Problèmes de ruine en théorie du risque à temps discret avec horizon fini

A quasi-randomized Runge-Kutta method

On a simple quasi-Monte Carlo approach for classical ultimate ruin probabilities