# Stochastic Calculus

Imagine the real-line movement of a Brownian particle (stock price) started at 0 during the time interval [0,1]. Let S_1 denote the maximal positive height that the particle ever reaches during the time interval. As S_1 is a random quantity depending on the entire Brownian path over [0,1], its ultimate value at any given time in [0,1) is unknown. Following the Brownian particle from the initial time 0 onward, the question arises naturally as to determine the time when the movement should be terminated so that the position of the particle is as "close" as possible to the ultimate maximum S_1.

Stochastic calculus makes it possible to solve problems of these kind. If "closeness" is measured by a mean-square distance, it can be proved that the optimal stopping time is the first time t at which S_t-B_t hits c √(1-t), where B_t denotes the position of the particle at time t, the symbol S_t denotes the maximal positive height of the particle up to time t, and c is a computable constant. This is achieved by representing S_1 as a stochastic integral, changing the speed of time and the shape of space, and formulating a free-boundary problem. The latter involves a differential equation and boundary conditions at a point which is unknown a priori and needs to be determined ("free boundary"). Solving the free-boundary problem and making use of stochastic calculus it is possible to complete the proof.

The methodology described above extends to other important problems in applied sciences. Most notably, in Financial Mathematics every American option (in a complete market) is equivalent to a free-boundary problem and the technique illustrated above can be used to compute the arbitrage-free price and find an optimal exercise time for the option. It is remarkable that these problems are equivalent to melting ice problems as well as obstacle problems in mathematical physics (the latter describing the shape of a membrane).

Professor Peskir is interested in this circle of facts and ideas. Applications in financial mathematics include "Option Pricing Theory" (optimal stopping) and "Portfolio Selection Theory" (optimal stochastic control). "Stochastic Calculus" provides basic tools needed for these problems to be successfully tackled.

Research contacts and collaborations exist worldwide.

### Academic contact

Professor Goran Peskir, Tel: +44 (0)161 306 3215, E-mail: Goran.Peskir (@manchester.ac.uk)