9 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
European Options in a Nonlinear Incomplete Market Model with Default
We study the superhedging prices and the associated superhedging strategies for European options in a nonlinear incomplete market model with default. The underlying market model consists of one risk-free asset and one risky asset, whose price may admit a jump at the default time. The portfolio processes follow nonlinear dynamics with a nonlinear driver . By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum, over a suitable set of equivalent probability measures , of the -evaluation/expectation under of the payoff. We also establish a characterization of the seller's (superhedging) price as the initial value of the minimal supersolution of a constrained backward stochastic differential equation with default. Moreover, we provide some properties of the terminal profit made by the seller, and some results related to replication and no-arbitrage issues. Our results rely on first establishing a nonlinear optional and a nonlinear predictable decomposition for processes which are -strong supermartingales under for all
BSDEs with Default Jump
We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λâ=â(λt). The driver of the BSDEs can be of a generalized form involving a singular optional finite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale; for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singular optional process. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default
American options in a non-linear incomplete market model with default
We study the superhedging prices and the associated superhedging strategies for American options in a non-linear incomplete market model with default. The points of view of the seller and of the buyer are presented. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow non-linear dynamics with a non-linear driver f. We give a dual representation of the seller's (superhedging) price for the American option associated with a completely irregular payoff (not necessarily cĂ dlĂ g) in terms of the value of a non-linear mixed control/stopping problem. The dual representation involves a suitable set of equivalent probability measures, which we call f-martingale probability measures. We also provide two infinitesimal characterizations of the seller's price process: in terms of the minimal supersolution of a constrained reflected BSDE and in terms of the minimal supersolution of an optional reflected BSDE. Under some regularity assumptions on , we also show a duality result for the buyer's price in terms of the value of a non-linear control/stopping game problem