On lipschitz continuity of the solution mapping to the skorokhod problem, with applications

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

“…For ε > 0 we can therefore find values t < u < 0, x < κ switch t , and y > λ s −µ s u , such that The last line follows from (9). which completes the proof of the inequality in (13).…”

“…We call a process reflected if it is the image of a process under a Skorokhod map which restricts the movements of the process to the positive orthant by shifting it at the border [9], [14]. (A definition of the onedimensional Skorokhod map is contained in Section 2.)…”

The projection dynamic and the geometry of population games

“…5.6), and later rediscovered by Dupuis and Ishii (1991) and Dupuis and Nagurney (1993). In the appendix, we briefly present a proof of Theorem 5.1 based on the Viability Theorem for differential inclusions.…”

“…Fortunately, theorems due to Henry (1973) and Aubin and Cellina (1984) (see also Dupuis and Ishii (1991) and Dupuis and Nagurney (1993)) imply that solutions to the projection dynamic exist, are unique, and are Lipschitz continuous in their initial conditions. But by virtue of its discontinuities, solutions to the projection dynamic have some properties quite different from those of standard dynamics: its solutions can merge in finite time, and can enter and exit the boundary of X repeatedly as time passes.…”

Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

“…This can be obtained in the following customary way: Set Z(t) = x(0) + It is known (see e.g. [8,13]) that we have an explicit formula for the regulator term Y in terms of Z, the so-called reflector term: for each i = 1, . .…”