Spitzer's Identity Research Articles

Convex minorants and the fluctuation theory of Lévy processes

We establish a novel characterisation of the law of the convex minorant of any L\'evy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of L\'evy processes. Our main result provides a new simple and self-contained approach to the fluctuation theory of L\'evy processes, circumventing local time and excursion theory. Easy corollaries include classical theorems, such as Rogozin's regularity criterion, Spitzer's identities and the Wiener-Hopf factorisation, as well as a novel factorisation identity.

Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise boundary for early-exercise options. Our approach is based on the Spitzer identities for general Lévy processes and on the Wiener–Hopf method. Our direct calculation of the price of α-quantile options combines for the first time the Dassios–Port–Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Lévy processes.

Pricing Methods for Alpha-Quantile and Perpetual Early Exercise Options Based on Spitzer Identities

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise barrier for early-exercise options. Our approach is based on the Spitzer identities for general Levy processes and on the Wiener-Hopf method. Our direct calculation of the price of alpha-quantile options combines for the first time the Dassios-Port-Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Levy processes.

Fluctuation identities with continuous monitoring and their application to the pricing of barrier options

We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential Lévy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener–Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-z domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.

Fluctuation Identities with Continuous Monitoring and Their Application to Price Barrier Options

We present a numerical scheme to calculate fluctuation identities for exponential L'evy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential L'evy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener-Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-$z$ domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.

Series expansions for the all-time maximum of α-stable random walks

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Levy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Levy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.

Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.

Pseudoprocesses on a circle and related Poisson kernels

Pseudoprocesses, constructed by means of the solutions of higher-order heat-type equations, have been developed by several authors and many related functionals have been analysed by applying the Feynman–Kac functional or by means of the Spitzer identity. We here examine pseudoprocesses wrapped up on circles and derive their explicit signed density measures. By composing the circular pseudoprocesses with positively skewed stable processes, we arrive at genuine circular processes whose distribution is obtained in the form of Poisson kernels. The distribution of circular even-order pseudoprocesses is similar to the Von Mises (or Fisher) circular normal law and to the wrapped up law of Brownian motion. Time-fractional and space-fractional equations related to processes and pseudoprocesses on the unit radius circumference are introduced and analysed.

On One Identity for Distribution of Sums of Independent Random Variables

In a 1957 paper F. Spitzer published, among other things, one estimate of the distribution function of sums of independent symmetric random variables with a common absolutely continuous distribution. This estimation was constructed with the help of one identity for the distribution of a sum of the above-mentioned random variables, the proof of which was not given. In this paper we prove the Spitzer identity for any independent random variables, and by using it we construct an estimate of the distribution of a sum of independent identically distributed random variables. The Spitzer estimate can be derived as a particular case of the proposed estimation.

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota–Baxter algebras which is shown to encode, among others, this process in the context of Connes–Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier–Rota theory of classical Spitzer-type identities for commutative Rota–Baxter algebras. In the classical, commutative, case these identities can be understood as deriving from the theory of symmetric functions. Here we show that an analogous property holds for noncommutative Rota–Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent, play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

A one-parameter family of dendriform identities

We prove a q-identity in the dendriform algebra of colored free quasi-symmetric functions. For q = 1 , we recover identities due to Ebrahimi-Fard, Manchon, and Patras, in particular the noncommutative Bohnenblust–Spitzer identity.

New identities in dendriform algebras

Dendriform structures arise naturally in algebraic combinatorics (where they allow, for example, the splitting of the shuffle product into two pieces) and through Rota–Baxter algebra structures (the latter appear, among others, in differential systems and in the renormalization process of pQFT). We prove new combinatorial identities in dendriform algebras that appear to be strongly related to classical phenomena, such as the combinatorics of Lyndon words, rewriting rules in Lie algebras, or the fine structure of the Malvenuto–Reutenauer algebra. One of these identities is an abstract noncommutative, dendriform, generalization of the Bohnenblust–Spitzer identity and of an identity involving iterated Chen integrals due to C.S. Lam.

Rota–Baxter Algebras and New Combinatorial Identities

The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are indicated.

A note on Spitzer identity for random walk

Let ( S n ) n ⩾ 0 be a random walk evolving on the real line and introduce the first hitting time of the half-line ( a , + ∞ ) for any real a: τ a = min { n ⩾ 1 : S n > a } . The classical Spitzer identity (1960) supplies an expression for the generating function of the couple ( τ 0 , S τ 0 ) . In 1998, Nakajima [Joint distribution of the first hitting time and first hitting place for a random walk. Kodai Math. J. 21 (1998) 192–200.] derived a relationship between the generating functions of the random couples ( τ 0 , S τ 0 ) and ( τ a , S τ a ) for any positive number a. In this note, we propose a new and shorter proof for this relationship and complement this analysis by considering the case of an increasing random walk. We especially investigate the Erlangian case and provide an explicit expression for the joint distribution of ( τ a , S τ a ) in this situation.

Discrete extrema of Brownian motion and pricing of exotic options

We provide a closed-form expression for the distribution of the extrema of a Brownian motion observed at discrete times. We reduce the evaluation problem to a Wiener–Hopf integral equation that we solve analytically. Then, we apply the result to price in closed-form discrete monitored exotic options (lookback, quantile and barrier) in the Black–Scholes setting. Numerical results from our formulae are then compared with those from other numerical methods available in the literature. Finally, we discuss the relationship of our result with the well-known Spitzer identity.

Exact results for the roughness of a finite size random walk

We consider the role of finite size effects on the value of the effective Hurst exponent H. This problem is motivated by the properties of the high-frequency daily stock-prices. For a finite size random walk we derive some exact results based on Spitzer's identity. The conclusion is that finite size effects strongly enhance the value of H and the convergency to the asymptotic value ( H = 1 2 ) is rather slow. This result has a series of conceptual and practical implication which we discuss.

A DISCRETE QUEUE, FOURIER SAMPLING ON SZEGÖ CURVES AND SPITZER FORMULAS

We consider a discrete-time multi-server queue for which the moments of the stationary queue length can be expressed in terms of series over the zeros in the closed unit disk of a queue-specific characteristic function. In many important cases these zeros can be considered to be located on a queue-specific curve, called generalized Szegö curve. By adopting a special parametrization of these Szegö curves, the relevant zeros occur as equidistant samples of a 2π-periodic function whose Fourier coefficients can be determined analytically. Thus the series occurring in the expressions for the moments can be written as Fourier aliasing series with terms given in analytic form. This gives rise to formulas for e.g. the mean and variance of the queue length that are reminiscent of Spitzer's identity for the moment generating function of the steady-state waiting time for a G/G/1 queue. Indeed, by considering the queue under investigation as a G/G/1 queue, the new formulas for the mean and variance also follow from Spitzer's identity. The approach in this paper can also be used to compute the probability distribution function of the queue length in analytic form.

Relaxation Time for the Discrete D/G/1 Queue

When queueing models are used for performance analysis of some stochastic system, it is usually assumed that the system is in steady-state. Whether or not this is a realistic assumption depends on the speed at which the system tends to its steady-state. A characterization of this speed is known in the queueing literature as relaxation time. The discrete D/G/1 queue has a wide range of applications. We derive relaxation time asymptotics for the discrete D/G/1 queue in a purely analytical way, mostly relying on the saddle point method. We present a simple and useful approximate upper bound which is sharp in case the load on the system is not very high. A sharpening of this upper bound, which involves the complementary error function, is then developed and this covers both the cases of low and high loads. For the discrete D/G/1 queue, the stationary waiting time distribution can be expressed in terms of infinite series that follow from Spitzer's identity. These series involve convolutions of the probability distribution of a discrete random variable, which makes them suitable for computation. For practical purposes, though, the infinite series should be truncated. The relaxation time asymptotics can be applied to determine an appropriate truncation level based on a sharp estimate of the error caused by truncating.

Spitzer's identity and the algebraic Birkhoff decomposition in pQFT

In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analyzed in terms of complete filtered Rota-Baxter algebras.