Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

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Lévy Processes and Stochastic Calculus

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and a Euler discretisation, the computational cost to achieve an accuracy of O(e) is reduced from O(e-3) to O(e-2 (log e)2). The analysis is supported by numerical results showing significant computational savings.

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Multilevel Monte Carlo Path Simulation

Stochastic differential equations and diffusion processes

Given the solution (Xt ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(Xt ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (Xt ) by simulating a simple t,rajectory. For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E. Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy

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Expansion of the global error for numerical schemes solving stochastic differential equations

We study the approximation problem ofE
f(X

 T
 ) byE
f(X

 ), where (X

 t
 ) is the solution of a stochastic differential equation, (X

 ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE
f(X

 T
 ) −f(X

 ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hormander type for the infinitesimal generator of (X

 t
 ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX

 and compare it to the density of the law ofX

 T
 .

/pdf/the-law-of-the-euler-scheme-for-stochastic-differential-5fudrb7c38.pdf

The law of the euler scheme for stochastic differential equations: i. convergence rate of the distribution function

In relation with Monte Carlo methods to solve some integro-differential equations, we study the approximation problem of $\mathbb{E}g(X_T)$ by $\mathbb{E}g(\overline{X}_T^n)$, where $(X_t, 0 \leq t \leq T)$ is the solution of a stochastic differential equation governed by a Levy process $(Z_t), (\overline{X}_t^n)$ is defined by the Euler discretization scheme with step $T/n$. With appropriate assumptions on $g(\cdot)$, we show that the error of $\mathbb{E}g(X_T) - \mathbb{E}g(\overline{X}_T^n)$ can be expanded in powers of $1/n$ if the Levy measure of $Z$ has finite moments of order high enough. Otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the Levy measure.

/pdf/the-euler-scheme-for-levy-driven-stochastic-differential-2y0ua1do14.pdf

The Euler scheme for Lévy driven stochastic differential equations

In the first part of this work~\cite{Bally-Talay-94-1} we have studied the approximation problem of $\ee f(X_T)$ by $\ee f(X_T^n)$, where $(X_t)$ is the solution of a stochastic differential equation, $(X^n_t)$ is defined by the Euler discretization scheme with step $\fracTn$, and $f(\cdot)$ is a given function, only supposed measurable and bounded. We have proven that the discretization error can be expanded in terms of powers of $\frac1n$ under a nondegeneracy condition of Hormander type for the infinitesimal generator of $(X_t)$. In this second part, we consider the density of the law of a small perturbation of $X_T^n$ and we compare it to the density of the law of $X_T$: we prove that the difference between the densities can also be expanded in terms of $\frac1n$. \noindent{\bf AMS(MOS) classification}: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.

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The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density

In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is three-fold. First, we consider a McKean-Vlasov equation in [0,T] x R with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure μ t , the solution to the McKean-Vlasov equation. The simulation of the stochastic system with N particles provides a discrete measure which approximates μkΔt for each time kΔt (where Δt is a discretization step of the time interval [0, T]). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of μkΔt. We show that the convergence rate is O (1/√N+ for the approximation in L i (Ω x R) of the cumulative distribution function at time T, and of order O (e 2 +1/e) (1/√N + √Δt for the approximation in L l (Ω x R) of the density at time T (Ω is the underlying probability space, e is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rate O (1/√N+√Δt). This part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.

https://www-sop.inria.fr/members/Denis.Talay/fichiers-pdf/bossy-talay-mathcomp.pdf

Denis Talay

Papers

Expansion of the global error for numerical schemes solving stochastic differential equations

The law of the euler scheme for stochastic differential equations: i. convergence rate of the distribution function

The Euler scheme for Lévy driven stochastic differential equations

The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density

A stochastic particle method for the McKean-Vlasov and the Burgers equation