stochastic calculus and applications in finance

STOCHASTIC CALCULUS AND

APPLICATIONS IN FINANCEUniversity of Tasmania Staff Seminars

SOME LITERATURE

� Mikosch,T., Elementary Stochastic Calculus with Finance in View, World Scientific 1998. Based on his notes from Stcohastic Calculus course he was teaching at Victoria University in Wellington.

� Fries, C.P., Mathematical Finance: Theory, Modeling and Implementation, 2006?.

� van Handel, R., Stochastic Calculus, Filtering, and Stochastic Control, Lecture notes, 2007.

� Shreve, S., Stochastic Calculus and Finance, Lecture notes, 1997.

Steele, J.M., , Springer 2000.� Steele, J.M., Stochastic Calculus and Financial Applications, Springer 2000.

� Kuo, H.-H., Introduction to Stochastic Integration, Springer 2006.

� Quastel,J., Notes for Stochastic calculus for Mathematical Finance, University of Toronto

� Kuo, H.-H., Introduction to Stochastic Integration, Springer 2006. (Thank you Jet)

� The Mathematics of Finance, Lecture notes. (Thank you Shane)

� Varadhan,S.R.S., Stochastic Processes, Lecture notes, AMS 2007. (Thank you Mardi)

� Bass,R., Lecture notes for Stochastic calculus, with applications to finance.

PRELIMINARIES

ILLUSTRATION OF MEASURABILITY

MARTINGALES

LEBESGUE INTEGRAL (VS. RIEMANN)

NON-INTEGRABLE UNDER RIEMANN

(DIRICHLET FN)

BROWNIAN MOTION

BROWNIAN MOTION

� Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors

GEOMETRIC BROWNIAN MOTION (GBM)

ITO LEMMA

ITO LEMMA (GENERAL)

GIRSANOV THEOREM

� Visualisation of the Girsanov theorem — The left side shows a Wiener process with negative drift under a canonical measure P; on the right side each path of the process is colored according to its likelihood under the martingale measure Q. The density transformation from P to Q is given by the Girsanov theorem.