scholarly journals On Variant Reflected Backward SDEs, with Applications

2009 ◽  
Vol 2009 ◽  
pp. 1-26
Author(s):  
Jin Ma ◽  
Yusun Wang
Keyword(s):  
Small Time ◽  
Well Posedness ◽  
Time Duration ◽  
Boundary Process ◽  
New Type ◽  
Optimal Stopping Problems ◽  
Reflected Bsdes ◽  
Stability Results ◽  
Backward Sdes ◽  
Skorohod Problem

We study a new type of reflected backward stochastic differential equations (RBSDEs), where the reflecting process enters the drift in a nonlinear manner. This type of the reflected BSDEs is based on a variance of the Skorohod problem studied recently by Bank and El Karoui (2004), and is hence named the “Variant Reflected BSDEs” (VRBSDE) in this paper. The special nature of the Variant Skorohod problem leads to a hidden forward-backward feature of the BSDE, and as a consequence this type of BSDE cannot be treated in a usual way. We shall prove that in a small-time duration most of the well-posedness, comparison, and stability results are still valid, although some extra conditions on the boundary process are needed. We will also provide some possible applications where the VRBSDE can be potentially useful. These applications show that the VRBSDE could become a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable.

Solvability of coupled FBSDEs with diagonally quadratic generators

2017 ◽  
Vol 17 (06) ◽  
pp. 1750043 ◽  
Author(s):  
Peng Luo ◽  
Ludovic Tangpi
Keyword(s):  
Existence And Uniqueness ◽  
Small Time ◽  
Coupled Systems ◽  
Time Interval ◽  
Well Posedness ◽  
Initial Value ◽  
Small Time Interval ◽  
Uniqueness Of The Solution ◽  
Backward Sdes

We study the well-posedness for multi-dimensional and coupled systems of forward–backward SDEs when the generator can be separated into a quadratic and a subquadratic part. We obtain the existence and uniqueness of the solution on a small time interval. Moreover, the continuity and differentiability with respect to the initial value are presented.

scholarly journals Evolution of populations with strategy-dependent time delays

2019 ◽  
Author(s):  
Jacek Miȩkisz ◽  
Marek Bodnar
Keyword(s):  
Stationary State ◽  
Time Delays ◽  
Replicator Dynamics ◽  
Evolutionarily Stable Strategy ◽  
Evolutionary Game ◽  
Evolutionary Games ◽  
Small Time ◽  
Stochastic Perturbations ◽  
New Type ◽  
Novel Behavior

AbstractWe address the issue of stability of coexistence of two strategies with respect to time delays in evolving populations. It is well known that time delays may cause oscillations. Here we report a novel behavior. We show that a microscopic model of evolutionary games with a unique mixed evolutionarily stable strategy (a globally asymptotically stable interior stationary state in the standard replicator dynamics) and with strategy-dependent time delays leads to a new type of replicator dynamics. It describes the time evolution of fractions of the population playing given strategies and the size of the population. Unlike in all previous models, an interior stationary state of such dynamics depends continuously on time delays and at some point it might disappear, no cycles are present. In particular, this means that an arbitrarily small time delay changes an interior stationary state. Moreover, at certain time delays, there may appear another interior stationary state.Author summarySocial and biological processes are usually described by ordinary or partial differential equations, or by Markov processes if we take into account stochastic perturbations. However, interactions between individuals, players or molecules, naturally take time. Results of biological interactions between individuals may appear in the future, and in social models, individuals or players may act, that is choose appropriate strategies, on the basis of the information concerning events in the past. It is natural therefore to introduce time delays into evolutionary game models. It was usually observed, and expected, that small time delays do not change the behavior of the system and large time delays may cause oscillations. Here we report a novel behavior. We show that microscopic models of evolutionary games with strategy-dependent time delays, in which payoffs appear some time after interactions of individuals, lead to a new type of replicator dynamics. Unlike in all previous models, interior stationary states of such dynamics depend continuously on time delays. This shows that effects of time delays are much more complex than it was previously thought.

scholarly journals A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data

2019 ◽  
Vol 62 (3) ◽  
pp. 417-444
Author(s):  
A. Chambolle ◽  
M. Holler ◽  
T. Pock
Keyword(s):  
Image Reconstruction ◽  
Inverse Problems ◽  
Incomplete Data ◽  
Variational Model ◽  
Well Posedness ◽  
Optimal Solutions ◽  
Analytical Properties ◽  
Numerical Performance ◽  
Stability Results ◽  
Continuous Setting

AbstractA variational model for learning convolutional image atoms from corrupted and/or incomplete data is introduced and analyzed both in function space and numerically. Building on lifting and relaxation strategies, the proposed approach is convex and allows for simultaneous image reconstruction and atom learning in a general, inverse problems context. Further, motivated by an improved numerical performance, also a semi-convex variant is included in the analysis and the experiments of the paper. For both settings, fundamental analytical properties allowing in particular to ensure well-posedness and stability results for inverse problems are proven in a continuous setting. Exploiting convexity, globally optimal solutions are further computed numerically for applications with incomplete, noisy and blurry data and numerical results are shown.

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