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Mathematics Colloquium


Date:  Tuesday, November 18, 2008

Title:  Ziwet Lectures: Some Stochastic Control Problems in Mathematical Finance

Abstract:  We formulate and review a class of stochastic control problems, collectively known under the rubric of “portfolio optimization”, that arise in the mathematics of finance. Ideas from convex duality play a prominent role in the resolution of these problems; so does the theory of parabolic partial differential equations, under certain strong conditions on the financial market structure. Under less stringent conditions, stochastic analogues of the classical Hamilton-Jacobi-Bellman equation emerge as particularly relevant in this context, in connection with ideas and results from ‘backwards’ stochastic equations and the Ito-Wentzell formula for random fields. Using such tools, feedback formulae become available for the investor’s optimal strategies, based on his current level of wealth. Recent progress on these issues will be surveyed, and some open questions will be mentioned.

Speaker:  Ioannis Karatzas
Institution:  University of Columbia


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