211 research outputs found
Contract Adjustment under Uncertainty
Consider a contract over trade in continuous time between two players, according to which one player makes a payment to the other, in exchange for an exogenous service. At each point in time, either player may unilaterally require an adjustment of the contract payment, involving adjustment costs for both players. Playersâ payoffs from trade under the contract, as well as from trade under an adjusted contract, are exogenous and stochastic. We consider playersâ choice of whether and when to adjust the contract payment. It is argued that the optimal strategy for each player is to adjust the contract whenever the contract payment relative to the outcome of an adjustment passes a certain threshold, depending among other things of the adjustment costs. There is strategic substitutability in the choice of thresholds, so that if one player becomes more aggressive by choosing a threshold closer to unity, the other player becomes more passive. If players may invest in order to reduce the adjustment costs, there will be over-investment compared to the welfare maximizing levels.
A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions
We derive a zero-curvature formalism for a combined sine-Gordon (sG) and
modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV
hierarchy. In complete analogy to other completely integrable hierarchies of
soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV
hierarchy is recursively constructed by means of a fundamental polynomial
formalism involving a spectral parameter. We further illustrate our approach by
developing the basic algebro-geometric setting for the sGmKdV hierarchy,
including Baker-Akhiezer functions, trace formulas, Dubrovin-type equations,
and theta function representations for its algebro-geometric solutions.
Although we mainly focus on sG-type equations, our formalism also yields the
sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type
equations combined with the mKdV hierarchy.Comment: LaTeX; emphasis put on the mKdV hierarch
The Damped String Problem Revisited
We revisit the damped string equation on a compact interval with a variety of
boundary conditions and derive an infinite sequence of trace formulas
associated with it, employing methods familiar from supersymmetric quantum
mechanics. We also derive completeness and Riesz basis results (with
parentheses) for the associated root functions under less smoothness
assumptions on the coefficients than usual, using operator theoretic methods
(rather than detailed eigenvalue and root function asymptotics) only.Comment: 39 page
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