Santa Clara University

Mathematics and Computer Science department

SCU Mathematics/CS Colloquium Series

Winter 2010

All talks will be at 3:50 PM.

There will be refreshments before each talk in O'Connor 31 at 3:40 PM.

Tuesday, January 19th


Speaker: Debra Lewis, UC- Santa Cruz, Mathematics Dept.


Title: Moderation Incentives in Optimal Control


Abstract: Control theory is a branch of dynamical systems theory in
which the evolution of the control variables is explicitly specified,
while the state variables obey a control-dependent system of
differential equations. Solution of a control problem involves the
identification of curves in the set of admissible controls determining
the desired behavior of the state variables. In optimal control, the
solution minimizing a given cost function is sought; this additional
requirement brings variational methods and Hamiltonian mechanics into
play. Apparently minor changes in the cost function can significantly
influence the optimal solutions. The cost function for a generalized
time minimization problem, with a purely state-dependent cost
function, can be modified by the addition of a control-dependent term
rewarding submaximal control utilization. Interpretation of this term
as an incentive that drops to zero on the boundary of the admissible
control region, rather than a penalty (e.g. an infinite potential wall
on the boundary), suggests some novel cost functions.



Tuesday, January 26th


Speaker: Aaron Melman, Santa Clara University, Applied Mathematics


Title: Variations, Generalizations, and Applications of Gershgorin


Abstract: Gershgorin disks are a remarkably simple way to obtain
regions in the complex plane that are guaranteed to contain all the
eigenvalues of a matrix. They are not the only such regions and we
begin with a brief review and history of related results. We then
prove Gershgorin's theorem in a standard and elementary way and show
how it can be modified to obtain smaller eigenvalue inclusion
regions. Although more complicated than the Gershgorin disks, these
can be useful when applied to matrices with special structure, such as
sparse matrices or matrices with particular symmetries. We present
some applications to companion matrices, and to centrohermitian,
persymmetric, and perhermitian matrices.





Thursday, February 4th


Speaker: Sanjiv Das, Santa Clara University, Finance Dept.


Title: The Principal Principle: Optimal Modificationof Distressed Home
Loans (Why Lenders should Forgive, not Forsake, Mortgages)

Abstract: Lenders will often restructure a loan rather than foreclose
on a property because it is less value-destroying. A loan modification
primarily entails a change in the loan rate, principal balance and/or
remaining time to maturity; other loan features may be modified too. We
analyze optimal loan modification schemes in a stochastic home price
environment. Lenders maximize their loan values by minimizing the
value of the borrower's option to default on the loan. Depending on
the level of interest rates and home price volatility, different
prescriptions apply. We argue that, controlling for the borrower's
ability to pay, loan modifications via rate reductions and maturity
extensions are suboptimal, leading to dissipation in loan value to the
lender, and resulting in a high probability of re-default by
homeowners even after modification of their loans. In contrast, loan
write-downs (the Principal Principle), not a favored recipe, and
sometimes prohibited by covenants, are mostly optimal. A recent
innovation, the shared appreciation mortgage, enhances the ability to
pay, mitigates adverse selection against lenders, and reduces the
present value of expected deadweight foreclosure costs.


If you have a disability and require a reasonable accommodation,
please call/email Dan Ostrov 408-554-4551/dostrov at scu dot edu (or
use 1-800-735-2929 TTY—California Relay).

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