A stochastic calculus for continuous N-parameter strong martingales

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 ——–

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...

Continuous Martingales and Local Martingales

Throughout these notes, (Ω,F ,P ) will be a probability space and F := {Ft }t∈J a filtration indexed by J , where J is an interval, usually J = [0,∞). The filtration F is said to be complete if each Ft contains all sets of measure 0, and is right-continuous if Ft =∩s>t Fs . A standard filtration is a filtration that is both complete and right-continuous. A stochastic process {X t }t≥0 defined o...