541 research outputs found
Optimal Multi-Modes Switching Problem in Infinite Horizon
This paper studies the problem of the deterministic version of the
Verification Theorem for the optimal m-states switching in infinite horizon
under Markovian framework with arbitrary switching cost functions. The problem
is formulated as an extended impulse control problem and solved by means of
probabilistic tools such as the Snell envelop of processes and reflected
backward stochastic differential equations. A viscosity solutions approach is
employed to carry out a finne analysis on the associated system of m
variational inequalities with inter-connected obstacles. We show that the
vector of value functions of the optimal problem is the unique viscosity
solution to the system. This problem is in relation with the valuation of firms
in a financial market
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Quadratic BSDEs driven by a continuous martingale and application to utility maximization problem
In this paper, we study a class of quadratic Backward Stochastic Differential
Equations (BSDEs) which arises naturally when studying the problem of utility
maximization with portfolio constraints. We first establish existence and
uniqueness results for such BSDEs and then, we give an application to the
utility maximization problem. Three cases of utility functions will be
discussed: the exponential, power and logarithmic ones
A general comparison theorem for 1-dimensional anticipated BSDEs
Anticipated backward stochastic differential equation (ABSDE) studied the
first time in 2007 is a new type of stochastic differential equations. In this
paper, we establish a general comparison theorem for 1-dimensional ABSDEs with
the generators depending on the anticipated term of .Comment: 8 page
Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems
This paper deals with existence and uniqueness, in viscosity sense, of a
solution for a system of m variational partial differential inequalities with
inter-connected obstacles. A particular case of this system is the
deterministic version of the Verification Theorem of the Markovian optimal
m-states switching problem. The switching cost functions are arbitrary. This
problem is connected with the valuation of a power plant in the energy market.
The main tool is the notion of systems of reflected BSDEs with oblique
reflection.Comment: 36 page
Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators
The aim of the present paper is to study the regularity properties of the
solution of a backward stochastic differential equation with a monotone
generator in infinite dimension. We show some applications to the nonlinear
Kolmogorov equation and to stochastic optimal control
An overview of Viscosity Solutions of Path-Dependent PDEs
This paper provides an overview of the recently developed notion of viscosity
solutions of path-dependent partial di erential equations. We start by a quick
review of the Crandall- Ishii notion of viscosity solutions, so as to motivate
the relevance of our de nition in the path-dependent case. We focus on the
wellposedness theory of such equations. In partic- ular, we provide a simple
presentation of the current existence and uniqueness arguments in the
semilinear case. We also review the stability property of this notion of
solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme
approximation method. Our results rely crucially on the theory of optimal
stopping under nonlinear expectation. In the dominated case, we provide a
self-contained presentation of all required results. The fully nonlinear case
is more involved and is addressed in [12]
On Markovian solutions to Markov Chain BSDEs
We study (backward) stochastic differential equations with noise coming from
a finite state Markov chain. We show that, for the solutions of these equations
to be `Markovian', in the sense that they are deterministic functions of the
state of the underlying chain, the integrand must be of a specific form. This
allows us to connect these equations to coupled systems of ODEs, and hence to
give fast numerical methods for the evaluation of Markov-Chain BSDEs
Inf-convolution of G-expectations
In this paper we will discuss the optimal risk transfer problems when risk
measures are generated by G-expectations, and we present the relationship
between inf-convolution of G-expectations and the inf-convolution of drivers G.Comment: 23 page
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