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

Stochastic differential equations and diffusion processes

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models.

This textbook forms the basis of a graduate course on the theory and applications of Levy processes, from the perspective of their path fluctuations. Central to the presentation are decompositions of the paths of Levy processes in terms of their local maxima and an understanding of their short- and long-term behaviour.

The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Levy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical transparency and explicitness. Each chapter has a comprehensive set of exercises with complete solutions.

Introductory Lectures on Fluctuations of Lévy Processes with Applications

Applied Probability and Queues.

INTRODUCTION DISCRETE-TIME MODELS Discrete-time formalism Martingales and arbitrage opportunities Complete markets and option pricing Problem: Cox, Ross and Rubinstein model OPTIMAL STOPPING PROBLEM AND AMERICAN OPTIONS Stopping time The Snell envelope Decomposition of supermartingales Snell envelope and Markov chains Application to American options BROWNIAN MOTION AND STOCHASTIC DIFFERENTIAL EQUATIONS General comments on continuous-time processes Brownian motion Continuous-time martingales Stochastic integral and Ito calculus Stochastic differential equations THE BLACK-SCHOLES MODEL Description of the model Change of probability: Representation of martingales Pricing and hedging options in the Black-Scholes model American options Implied volatility and local volatility models The Black-Scholes model with dividends and call/put symmetry Problems OPTION PRICING AND PARTIAL DIFFERENTIAL EQUATIONS European option pricing and diffusions Solving parabolic equations numerically American options INTEREST RATE MODELS Modeling principles Some classical models ASSET MODELS WITH JUMPS Poisson process Dynamics of the risky asset Martingales in a jump-diffusion model Pricing options in a jump-diffusion model CREDIT RISK MODELS Structural models Intensity-based models Copulas SIMULATION AND ALGORITHMS FOR FINANCIAL MODELS Simulation and financial models Introduction to variance reduction methods Computer experiments APPENDIX Normal random variables Conditional expectation Separation of convex sets BIBLIOGRAPHY INDEX Exercises appear at the end of each chapter.

Introduction to stochastic calculus applied to finance

This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called Brennan-Schwartz algorithm for the valuation of American put options.

Variational inequalities and the pricing of American options

Introduction au calcul stochastique appliqué à la finance

https://hal.archives-ouvertes.fr/hal-00017929/document

Residual risks and hedging strategies in Markovian markets

We propose a recursive stochastic algorithm with decreasing step to compute the invariant distribution $v$ of a Brownian diffusion process, in which we approximate $ν(f)$ for a wide class of possibly unbounded continuous functions $f$. We consider a somewhat general setting which includes cases where the diffusion may have several invariant distributions. Our main convergence result contains as a corollary the almost sure central limit theorem. Further, we investigate the weak rate of convergence of the algorithm. We show, in the class of polynomial steps$\gamma_n=n^{-\alpha}$, that it can be at most $n^{1/3}$ when the white noise has third moment zero and $n^{1/4}$ otherwise, where $n$ denotes the number of iterations of the algorithm.

Damien Lamberton

Papers

Introduction to stochastic calculus applied to finance

Variational inequalities and the pricing of American options

Introduction au calcul stochastique appliqué à la finance

Residual risks and hedging strategies in Markovian markets

Recursive computation of the invariant distribution of a diffusion