225 research outputs found
A New Result for Second Order BSDEs with Quadratic Growth and its Applications
In this paper, we study a class of second order backward stochastic
differential equations (2BSDEs) with quadratic growth in coefficients. We first
establish solvability for such 2BSDEs and then give their applications to
robust utility maximization problems
Generalized Wasserstein distance and weak convergence of sublinear expectations
In this paper, we define the generalized Wasserstein distance for sets of
Borel probability measures and demonstrate that the weak convergence of
sublinear expectations can be characterized by means of this distance
Reflected stochastic differential equations driven by -Brownian motion in non-convex domains
In this paper, we first review the penalization method for solving
deterministic Skorokhod problems in non-convex domains and establish estimates
for problems with -H\"older continuous functions. With the help of
these results obtained previously for deterministic problems, we pathwisely
define the reflected -Brownian motion and prove its existence and uniqueness
in a Banach space. Finally, multi-dimensional reflected stochastic differential
equations driven by -Brownian motion are investigated via a fixed-point
argument
On the dual problem of utility maximization in incomplete markets
In this paper, we study the dual problem of the expected utility maximization
in incomplete markets with bounded random endowment. We start with the problem
formulated in the paper of Cvitani\'{c}-Schachermayer-Wang (2001) and prove the
following statement: in the Brownian framework, the countably additive part
of the dual optimizer obtained in that paper can be
represented by the terminal value of a supermartingale deflator defined in
the paper of Kramkov-Schachermayer (1999), which is a local martingale
Utility maximization problem under transaction costs: optimal dual processes and stability
This paper discusses the num\'eraire-based utility maximization problem in
markets with proportional transaction costs. In particular, the investor is
required to liquidate all her position in stock at the terminal time. We first
observe the stability of the primal and dual value functions as well as the
convergence of the primal and dual optimizers when perturbations occur on the
utility function and on the physical probability. We then study the properties
of the optimal dual process (ODP), that is, a process from the dual domain that
induces the optimality of the dual problem. When the market is driven by a
continuous process, we construct the ODP for the problem in the limiting market
by a sequence of ODPs corresponding to the problems with small misspecificated
parameters. Moreover, we prove that this limiting ODP defines a shadow price
A note on utility maximization with transaction costs and random endoment: num\'eraire-based model and convex duality
In this note, we study the utility maximization problem on the terminal
wealth under proportional transaction costs and bounded random endowment. In
particular, we restrict ourselves to the num\'eraire-based model and work with
utility functions only supporting R+. Under the assumption of existence of
consistent price systems and natural regularity conditions, standard convex
duality results are established. Precisely, we first enlarge the dual domain
from the collection of martingale densities associated with consistent price
systems to a set of finitely additive measures; then the dual formulation of
the utility maximization problem can be regarded as an extension of the paper
of Cvitani\'c-Schachermayer-Wang (2001) to the context under proportional
transaction costs
On the existence of shadow prices for optimal investment with random endowment
In this paper, we consider a num\'eraire-based utility maximization problem
under constant proportional transaction costs and random endowment. Assuming
that the agent cannot short sell assets and is endowed with a strictly positive
contingent claim, a primal optimizer of this utility maximization problem
exists. Moreover, we observe that the original market with transaction costs
can be replaced by a frictionless shadow market that yields the same
optimality. On the other hand, we present an example to show that in some case
when these constraints are relaxed, the existence of shadow prices is still
warranted
Quadratic backward stochastic differential equations driven by -Brownian motion: discrete solutions and approximation
In this paper, we consider backward stochastic differential equations driven
by -Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We
prove the existence and uniqueness of solution for such equations. On the one
hand, a priori estimates are obtained by applying the Girsanov type theorem in
the -framework, from which we deduce the uniqueness. On the other hand, to
prove the existence of solutions, we first construct solutions for discrete
GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate
solutions for general quadratic GBSDEs in Banach spaces
Causal transport in discrete time and applications
Loosely speaking, causal transport plans are a relaxation of adapted
processes in the same sense as Kantorovich transport plans extend Monge-type
transport maps. The corresponding causal version of the transport problem has
recently been introduced by Lassalle. Working in a discrete time setup, we
establish a dynamic programming principle that links the causal transport
problem to the transport problem for general costs recently considered by
Gozlan et al. Based on this recursive principle, we give conditions under which
the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal
analogue to the Brenier's map. Moreover, these considerations provide
transport-information inequalities for the nested distance between stochastic
processes pioneered by Pflug and Pichler, and so serve to gauge the discrepancy
between stochastic programs driven by different noise distributions.Comment: We added a characterization of the Knothe-Rosenblatt rearrangement in
terms of increasing triangular transformations, 25 page
Lyapunov-type conditions and stochastic differential equations driven by -Brownian motion
This paper studies the solvability and the stability of stochastic
differential equations driven by G-Brownian motion (GSDEs). In particular, the
existence and uniqueness of the solution for locally Lipschitz GSDEs is
obtained by localization methods, also the stability of such GSDEs are
discussed with Lyapunov-type conditions
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