575 research outputs found
Doubly Reflected BSDEs and -Dynkin games: beyond the right-continuous case
We formulate a notion of doubly reflected BSDE in the case where the barriers
and do not satisfy any regularity assumption and with a general
filtration. Under a technical assumption (a Mokobodzki-type condition), we show
existence and uniqueness of the solution. In the case where is right
upper-semicontinuous and is right lower-semicontinuous, the solution is
characterized in terms of the value of a corresponding -Dynkin
game, i.e. a game problem over stopping times with (non-linear)
-expectation, where is the driver of the doubly reflected BSDE. In the
general case where the barriers do not satisfy any regularity assumptions, the
solution of the doubly reflected BSDE is related to the value of ''an
extension'' of the previous non-linear game problem over a larger set of
''stopping strategies'' than the set of stopping times. This characterization
is then used to establish a comparison result and \textit{a priori} estimates
with universal constants
On the optimal dividend problem for a spectrally negative L\'{e}vy process
In this paper we consider the optimal dividend problem for an insurance
company whose risk process evolves as a spectrally negative L\'{e}vy process in
the absence of dividend payments. The classical dividend problem for an
insurance company consists in finding a dividend payment policy that maximizes
the total expected discounted dividends. Related is the problem where we impose
the restriction that ruin be prevented: the beneficiaries of the dividends must
then keep the insurance company solvent by bail-out loans. Drawing on the
fluctuation theory of spectrally negative L\'{e}vy processes we give an
explicit analytical description of the optimal strategy in the set of barrier
strategies and the corresponding value function, for either of the problems.
Subsequently we investigate when the dividend policy that is optimal among all
admissible ones takes the form of a barrier strategy.Comment: Published at http://dx.doi.org/10.1214/105051606000000709 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Optimal probabilities and controls for reflecting diffusion processes
A solution to the optimal problem for determining vector fields which
maximize (resp. minimize) the transition probabilities from one location to
another for a class of reflecting diffusion processes is obtained in the
present paper. The approach is based on a representation for the transition
probability density functions. The optimal transition probabilities under the
constraint that the drift vector field is bounded by a constant are studied in
terms of the HJB equation. In dimension one, the optimal reflecting diffusion
processes and the bang-bang diffusion processes are considered. We demonstrate
by simulations that, even in this special case, the optimal diffusion processes
exhibit an interesting feature of phase transitions. We also solve an optimal
stochastic control problem for a class of stochastic control problems involving
diffusion processes with reflection.Comment: 20 Pages, 2 figure
The slow regime of randomly biased walks on trees
We are interested in the randomly biased random walk on the supercritical
Galton--Watson tree. Our attention is focused on a slow regime when the biased
random walk is null recurrent, making a maximal displacement of order
of magnitude in the first steps. We study the localization
problem of and prove that the quenched law of can be approximated
by a certain invariant probability depending on and the random environment.
As a consequence, we establish that upon the survival of the system,
converges in law to some non-degenerate limit on
whose law is explicitly computed.Comment: 43 pages. We added a recent work by Jim Pitman ([38]) for the
limiting la
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