Stochastic calculus and martingales on trees

Stochastic Calculus of Variations for Martingales

The stochastic calculus of variations for the Wiener process, initiated in Malliavin , aims to obtain conditions for the regularity of the density of Wiener functionals given by the values of diffusion processes. It also developed as an extension to anticipating processes of the Itô calculus, by means of the Skorohod integral, cf. Nualart-Pardoux , Üstünel . In the case of point processes we ca...

Functional Ito calculus and stochastic integral representation of martingales

We develop a non-anticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by B Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative ad...

Stochastic Calculus ——– an introduction to option pricing with martingales ——–

Malliavin calculus and asymptotic expansion for martingales

We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the Crame r condition did in the case of independent observations. Applications to stati...

Ieor E4707: Financial Engineering: Continuous-time Models Introduction to Stochastic Calculus 1 Martingales and Brownian Motion Introduction to Stochastic Calculus 2 Quadratic Variation Introduction to Stochastic Calculus 3 Stochastic Integrals Introduction to Stochastic Calculus

These notes provide an introduction to stochastic calculus, the branch of mathematics that is most identified with financial engineering and mathematical finance. We will ignore most of the “technical” details and take an “engineering” approach to the subject. We will cover more material than is strictly necessary for this course. Any material that is not required, however, should be of value f...