[grads] [Sem-coll] AM Sem/Coll April 23 [Taskar], April 24 [Moffit], April 25 [Bielecki]

George Skontos skougeo at iit.edu
Thu Apr 19 17:22:27 CDT 2007

Please join us for this round of seminars/colloquia. As always
refreshments will be served 10-15 min prior to each talk.

AM colloqium: Monday, April 23, 4:40pm E1 106
Speaker: Michael Taksar (University of Missouri-Columbia)
Title: Singular stochastic control and related pdes with gradient
constraints in portfolio optimization models in mathematical finance

 In modern mathematical finance stock prices are modeled by stochastic
differential equations whose solutions produce logarithmic Brownian
motions. This is the backbone of what has become the classical
Black-Scholes option pricing theory and Merton's investment/consumption
theory. We consider a dynamical portfolio optimization model in the
spirit of the latter. The portfolio consists of several risky assets
(stocks) and one risk-free asset (bond). The rate of return on the Bond
is constant while the rates of return of stocks are governed by SDEs of
the logarithmic Brownian motion type. Funds can be transferred from one
asset to another. However such transactions involve penalties (brokerage
fees) proportional to the size of the transaction. The objective is to
find the policy which maximizes the expected rate of growth of funds.

The main mathematical tool in the solution of this problem is singular
stochastic control theory. In this theory the control functionals are
represented by processes of bounded variation, and the optimal control
consists of functionals which reflect the process from an a priori
unknown boundary. They are continuous but singular (not absolutely
continuous) with respect to time. The analytical part of the solution to
the singular control is related to a free boundary problem for an
elliptic PDE with gradient constraints, similar to those encountered in
elastic-plastic torsion problems. The existence of the classical C^2
solution cannot be proved in general but one can show existence of a
viscosity solution to this equation.

The optimal policy is to keep the vector of fractions of funds invested
in different assets in an optimal (a priori unknown) boundary. This
gives practical guidance on when and how to rebalance a growth oriented
portfolio. We show how to find these boundaries explicitly in the case
of one risky and one risk-free asset when the problem becomes one
dimensional. In this case the free boundary problem can be reduced to a
Stephan problem for an ODE.

Open problems of both theoretical and practical importance will be

AM seminar Tuesday, April 24 3:30pm E1 242
Speaker: Steven Moffit (W.H. Trading, LLC)
Title: Tsallis Statistics - A New Paradigm In Statistical Mechanics With
Applications in Finance

Abstract to follow soon...

AM colloquium Wednesday, April 25 4:40pm E1 242
Speaker: Tomasz R. Bielecki (IIT Applied Mathematics)
Title: Valuation and Hedging of Credit Index Derivatives

Abstract to follow soon...
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