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In this survey, we show that various stochastic optimization problems arising in
option theory, in dynamical allocation problems, and in the microeconomic theory
of intertemporal consumption choice can all be reduced to the same problem of
representing a given stochastic process in terms of running maxima of another
process. We describe recent results of Bank and El Karoui (2002) on the general
stochastic representation problem, derive results in closed form for Lévy processes
and diffusions, present an algorithm for explicit computations, and discuss some
applications.
This paper is concerned with the effficient implementation of transparent boundary conditions
(TBCs) for wide angle parabolic equations (WAPEs) assuming cylindrical symmetry.
In [1] a discrete TBC of convolution type was derived from the fully discretized whole?space
problem that is reflection?free and yields an unconditionally stable scheme. Since the discrete
TBC includes a convolution with respect to range with a weakly decaying kernel, its
numerical evaluation becomes very costly for long-range simulations.
As a remedy we construct new approximative transparent boundary conditions involving
exponential sums as an approximation to the convolution kernel. This special approximation
enables us to use a fast evaluation of the convolution type boundary condition.
This new approach was outlined in detail in [2] for the standard "parabolic" equation.
Differential algebraic equations with properly stated leading term are equations of the form A(x(t),t)(d(x(t),t))'+b(x(t),t)=0 with in some sense well-matched coefficients. Systems resulting from the modified nodal analysis (MNA) in circuit simulation promptly fit into this form. Recent results concerning solvability and numerical treatment of those equations are discussed. An index notion that works via linearization is given. This allows for index criteria just in terms of the coefficients A,d,b and their first partial derivatives, no further derivative arrays are used.
We present a way to efficiently treat the well-known transparent boundary
conditions for the Schrödinger equation. Our approach is based on two ideas:
firstly, to derive a discrete transparent boundary condition (DTBC) based on the
Crank-Nicolson finite difference scheme for the governing equation. And, secondly,
to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for
a rapid recursive calculation of the convolution. We illustrate the efficiency of the
proposed method on several examples.
The index of DAE systems arising from linear quadratic optimal control problems is considered. Necessary and sufficient conditions ensuring regularity with tractability index one are proved. Then, it is shown that if the control problem DAE is regular with index one and if the leading term of the DAE to be controlled is given by one full-column-rank and one full-row-rank matrix, then it has a Hamiltonian inherent explicit ODE. For problems with regular index zero or index one DAEs to be controlled, the DAE of the control problem is shown to be regular with tractability index one or three, depending on whether the control coefficient R is singular.
By the use of the corresponding shift matrix, the paper gives a criterion for the unique solvability of linear boundary value problems posed for linear differential algebraic equations up to index 2 with well-matched leading coefficients. The solution is constructed by a proper Green function. Another characterization of the solutions is based upon the description of arbitrary affine linear subspaces of solutions to linear differential algebraic equations in terms of solutions to the adjoint equation. When applied to boundary value problems, the result provides a constructive criterion for unique solvability and allows reducing the problem to initial value problems and linear algebraic equations.
We consider a particle constrained to a submanifold ? of the
configuration space Rm. Using that the notion of holonomic constraints coincides
with integrability of the corresponding vector field, we show how this property
naturally determines local coordinates on ?. We give a rigorous justification for
the calculation of the mean force along a constrained coordinate, and we provide
a concise geometrical interpretation of the different contributions to the mean
force in terms of the unconstrained vector field and extrinsic curvature properties
of ? in Rm. Our approach gives rise to a Hybrid Monte-Carlo based algorithm
that can be used to compute the mean force acting on selected coordinates in the
context of thermodynamic free energy statistics.
We analyze an interactive model of credit ratings where external shocks, initially
affecting only a small number of firms, spread by a contagious chain reaction to the
entire economy. Counterparty relationships along with discrete adjustments of credit
ratings generate a transition mechanism that allows the financial distress of one firm
to spill over to its business partners. Such a contagious infectious of financial distress
constitutes a source of intrinsic risk for large portfolios of credit sensitive securities that
cannot be “diversified away.” We provide a complete characterization of the fluctuations
of credit ratings in large economies when adjustments follow a threshold rule. We also
analyze the effects of downgrading cascades on aggregate losses of credit portfolios. We
show that the loss distribution has a power-law tail if the interaction between different
companies is strong enough.
Stability of Linear Stochastic Difference Equations in Strategically Controlled Random Environments
(2004)
We consider the stochastic sequence fYtgt2N defined recursively by the linear relation
Yt+1 = AtYt+Bt in a random environment. The environment is described by the stochastic
process f(At;Bt)gt2N and is under the simultaneous control of several agents playing a
discounted stochastic game. We formulate sufficient conditions on the game which ensure
the existence of Nash equilibrium in Markov strategies which has the additional property
that, in equilibrium, the process fYtgt2N converges in distribution to a stationary regime.