Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

We define three types of non causal stochastic integrals: forward, backward and symmetric. Our approach consists in approximating the integrator. Two optics are considered: the first one is based on traditional usual stochastic calculus and the second one on Wiener distributions.

Forward, backward and symmetric stochastic integration

The quadratic variation of a continuous process (when it exists) is defined through a regularization procedure. A large class of finite quadratic variation processes is provided, with a particular emphasis on Gaussian processes. For such processes a calculus is developed with application to the study of some stochastic differential equations.

Stochastic calculus with respect to continuous finite quadratic variation processes

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Ito formula for "tame'' functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the nonanticipating nature of the SDDE, the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel.

https://opensiuc.lib.siu.edu/cgi/viewcontent.cgi?article=1054&context=math_articles

Discrete-time approximations of stochastic delay equations: The Milstein scheme

In a unified model-free framework that includes long-expiry, short-expiry, extreme-strike, and jointly-varying strike-expiry regimes, we find asymptotic implied volatility and implied variance formulas in terms of L, with rigorous error estimates of order 1/L to any given power, where L denotes the absolute log of an option price that approaches zero. Our results therefore sharpen, to arbitrarily-high order of accuracy, the model-free asymptotics of implied volatility in extreme regimes. We then apply these general formulas to particular examples: Levy and Heston.

Asymptotics of Implied Volatility to Arbitrary Order

This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012). The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes. Rama Cont's notes provide an introduction to the Functional Ito Calculus, a non-anticipative functional calculus that extends the classical Ito calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations. This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.

Stochastic Integration by Parts and Functional Itô Calculus

A new expression for the characteristic function of log-spot in Heston model is presented This expression more clearly exhibits its properties as an analytic characteristic function and allows us to compute the exact domain of the moment generating function This result is then applied to the volatility smile at extreme strikes and to the control of the moments of spot We also give a factorization of the moment generating function as product of Bessel type factors, and an approximating sequence to the law of log-spot is deduced

A new look at the Heston characteristic function

In this work, we give a closed form and a recurrence relation for a family of time-space harmonic poly nomials relative to a Levy process. We also state the relationship with the Kailath-Segall (orthogonal) polynomials associated to the process.

/pdf/time-space-harmonic-polynomials-relative-to-a-levy-process-2h6d0u0mkv.pdf

Time-space harmonic polynomials relative to a Levy process

Let X = {X t , t ≥ 0} be a cadlag Levy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with X. On one hand, the Kailath-Segall formula gives the relationship between the iterated integrals and the variations of order n of X, and defines a family of polynomials P 1 (x 1 ), P 2 (x 1 ,x 2 ),... that are orthogonal with respect to the joint law of the variations of X. On the other hand, we can construct a sequence of orthogonal polynomials p σ n (x) with respect to the measure σ 2 δ 0 (dx) + x 2 ν(dx), where σ 2 is the variance of the Gaussian part of X and v its Levy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the Levy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the Levy processes such that the associated polynomials P n (x 1 ,..., x n ) depend on a fixed number of variables are characterized. Also, we give a sequence of Levy processes that converge in the Skorohod topology to X, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of X.

/pdf/on-the-orthogonal-polynomials-associated-with-a-levy-process-429hid542k.pdf

On the orthogonal polynomials associated with a Lévy process

In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257–1283], we present an Ito multiple integral and a Stratonovich multiple integral with respect to a Levy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the Ito multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu–Meyer formula that gives the relationship between both integrals is proved. As particular cases, the classical Hu–Meyer formulas for the Brownian motion and for the Poisson process are deduced. Furthermore, a pathwise interpretation for the multiple integrals with respect to a subordinator is given.

/pdf/multiple-stratonovich-integral-and-hu-meyer-formula-for-levy-hwojjc72cb.pdf

Multiple stratonovich integral and Hu-Meyer formula for Lévy processes

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