13 research outputs found
Stochastic dominance with respect to a capacity and risk measures
Pursuing our previous work in which the classical notion of increasing convex stochastic dominance relation with respect to a probability has been extended to the case of a normalised monotone (but not necessarily additive) set function also called a capacity, the present paper gives a generalization to the case of a capacity of the classical notion of increasing stochastic dominance relation. This relation is characterized by using the notions of distribution function and quantile function with respect to the given capacity. Characterizations, involving Choquet integrals with respect to a distorted capacity, are established for the classes of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to a given generalized stochastic dominance relation. Moreover, under suitable assumptions, a "Kusuoka-type" characterization is proved for the class of monetary risk measures having the properties of comonotonic additivity and consistency with respect to the generalized increasing convex stochastic dominance relation. Generalizations to the case of a capacity of some well-known risk measures (such as the Value at Risk or the Tail Value at Risk) are provided as examples. It is also established that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.Choquet integral ; stochastic orderings with respect to a capacity ; distortion risk measure ; quantile function with respect to a capacity ; distorted capacity ; Choquet expected utility ; ambiguity ; non-additive probability ; Value at Risk ; Rank-dependent expected utility ; behavioural finance ; maximal correlation risk measure ; quantile-based risk measure ; Kusuoka's characterization theorem
Stochastic dominance with respect to a capacity and risk measures
Pursuing our previous work in which the classical notion of increasing convex stochastic dominance relation with respect to a probability has been extended to the case of a normalised monotone (but not necessarily additive) set function also called a capacity, the present paper gives a generalization to the case of a capacity of the classical notion of increasing stochastic dominance relation. This relation is characterized by using the notions of distribution function and quantile function with respect to the given capacity. Characterizations, involving Choquet integrals with respect to a distorted capacity, are established for the classes of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to a given generalized stochastic dominance relation. Moreover, under suitable assumptions, a "Kusuoka-type" characterization is proved for the class of monetary risk measures having the properties of comonotonic additivity and consistency with respect to the generalized increasing convex stochastic dominance relation. Generalizations to the case of a capacity of some well-known risk measures (such as the Value at Risk or the Tail Value at Risk) are provided as examples. It is also established that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity
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
Non-linear non-zero-sum Dynkin games with Bermudan strategies
In this paper, we study a non-zero-sum game with two players, where each of
the players plays what we call Bermudan strategies and optimizes a general
non-linear assessment functional of the pay-off. By using a recursive
construction, we show that the game has a Nash equilibrium point
Reflected BSDEs when the obstacle is not right-continuous and optimal stopping
In the first part of the paper, we study reflected backward stochastic
differential equations (RBSDEs) with lower obstacle which is assumed to be
right upper-semicontinuous but not necessarily right-continuous. We prove
existence and uniqueness of the solutions to such RBSDEs in appropriate Banach
spaces. The result is established by using some tools from the general theory
of processes such as Mertens decomposition of optional strong (but not
necessarily right-continuous) supermartingales, some tools from optimal
stopping theory, as well as an appropriate generalization of It{\^o}'s formula
due to Gal'chouk and Lenglart. In the second part of the paper, we provide some
links between the RBSDE studied in the first part and an optimal stopping
problem in which the risk of a financial position is assessed by an
-conditional expectation (where is a Lipschitz
driver). We characterize the "value function" of the problem in terms of the
solution to our RBSDE. Under an additional assumption of left
upper-semicontinuity on , we show the existence of an optimal stopping
time. We also provide a generalization of Mertens decomposition to the case of
strong -supermartingales
Superhedging prices of European and American options in a non-linear incomplete market with default
Grigorova M, Quenez M-C, Sulem A. Superhedging prices of European and American options in a non-linear incomplete market with default. Center for Mathematical Economics Working Papers. Vol 607. Bielefeld: Center for Mathematical Economics; 2018.This paper studies the superhedging prices and the associated superhedging strategies
for European and American options in a non-linear incomplete market with default.
We present the seller's and the buyer's point of view. 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 process follows non-linear dynamics
with a non-linear driver ƒ. 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 *Q* ∈ of
the ƒ-evaluation/expectation under *Q* of the payoff. We also provide an infinitesimal
characterization of this price as the minimal supersolution of a constrained BSDE with
default. By a form of symmetry, we derive corresponding results for the buyer. We also
give a dual representation of the seller's (superhedging) price for the American option
associated with an irregular payoff (ξ*t*) (not necessarily càdlàg) in terms of the value of
a non-linear mixed control/stopping problem. We also provide an infinitesimal characterization
of this price in terms of a constrained reflected BSDE. When ξ is càdlàg,
we show a duality result for the buyer's price. These results rely on first establishing a
non-linear optional decomposition for processes which are ƒ -strong supermartingales
under *Q*, for all *Q* ∈
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 càdlàg) 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
Optimal Stopping With ƒ-Expectations: the irregular case
Grigorova M, Imkeller P, Ouknine Y, Quenez M-C. Optimal Stopping With ƒ-Expectations: the irregular case. Center for Mathematical Economics Working Papers. Vol 587. Bielefeld: Center for Mathematical Economics; 2017.We consider the optimal stopping problem with non-linear ƒ-expectation (induced by a BSDE) without making any regularity assumptions on the reward process ξ. We show that the value family can be aggregated by an optional process *Y* . We characterize the process *Y* as the ƒ-Snell envelope of ξ. We also establish an infinitesimal characterization of the value process *Y* in terms of a Reflected BSDE with ξ as the obstacle.
To do this, we first establish a comparison theorem for irregular RBS
DEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model
Doubly Reflected BSDEs and -Dynkin games: beyond the right-continuous case
Grigorova M, Imkeller P, Quenez M-C, Ouknine Y. Doubly Reflected BSDEs and -Dynkin games: beyond the right-continuous case. Center for Mathematical Economics Working Papers. Vol 598. Bielefeld: Center for Mathematical Economics; 2018.We formulate a notion of doubly reflected BSDE in the case where the barriers ξ
and ζ do not satisfy any regularity assumption. Under a technical assumption (a
Mokobodzki-type condition), we show existence and uniqueness of the solution. In the
case where ξ is right upper-semicontinuous and ζ is right lower-semicontinuous, the
solution is characterized in terms of the value of a corresponding ƒ -Dynkin game,
i.e. a game problem over stopping times with (non-linear) ƒ-expectation, where ƒ 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 a priori estimates with universal
constants