scholarly journals Numerical approximations of McKean anticipative backward stochastic differential equations arising in initial margin requirements

2019 ◽  
Vol 65 ◽  
pp. 1-26 ◽  
Author(s):  
A. Agarwal ◽  
S. De Marco ◽  
E. Gobet ◽  
J. G. López-Salas ◽  
F. Noubiagain ◽  
...  
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Value At Risk ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Equations ◽  
Conditional Value At Risk ◽  
General Regularity ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results

We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relatively general regularity assumptions on the coefficients. We show how such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) requires the members to deposit variation and initial margins to cover their exposure. In the case when the initial margin is proportional to the Conditional Value-at-Risk (CVaR) of the contract price, we apply our general result to define the price as a solution of a MKABSDE. We provide several linear and non-linear simpler approximations, which we solve using different numerical (deterministic and Monte-Carlo) methods.

scholarly journals A Note on Fractional Differential Equations with Fractional Separated Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Separated Boundary Conditions ◽  
Fractional Differential

We consider a new class of boundary value problems of nonlinear fractional differential equations with fractional separated boundary conditions. A connection between classical separated and fractional separated boundary conditions is developed. Some new existence and uniqueness results are obtained for this class of problems by using standard fixed point theorems. Some illustrative examples are also discussed.

Comparison Theorem for any Solutions of Backward Stochastic Differential Equations and its Application

2012 ◽  
Vol 524-527 ◽  
pp. 3801-3804
Author(s):  
Shi Yu Li ◽  
Wu Jun Gao ◽  
Jin Hui Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Equations ◽  
Local Martingale ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
The One ◽  
Uniformly Lipschitz

ƒIn this paper, we study the one-dimensional backward stochastic equations driven by continuous local martingale. We establish a generalized the comparison theorem for any solutions where the coefficient is uniformly Lipschitz continuous in z and is equi-continuous in y.

scholarly journals Reflected Backward Stochastic Differential Equations Driven by Countable Brownian Motions

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Pengju Duan ◽  
Min Ren ◽  
Shilong Fei
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Brownian Motions ◽  
Snell Envelope ◽  
New Class

This paper deals with a new class of reflected backward stochastic differential equations driven by countable Brownian motions. The existence and uniqueness of the RBSDEs are obtained via Snell envelope and fixed point theorem.

scholarly journals SPDIEs and BSDEs Driven by Lévy Processes and Countable Brownian Motions

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Pengju Duan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Cauchy Sequence ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Brownian Motions ◽  
New Class ◽  
Probabilistic Solution

The paper is devoted to solving a new class of backward stochastic differential equations driven by Lévy process and countable Brownian motions. We prove the existence and uniqueness of the solutions to the backward stochastic differential equations by constructing Cauchy sequence and fixed point theorem. Moreover, we give a probabilistic solution of stochastic partial differential integral equations by means of the solution of backward stochastic differential equations. Finally, we give an example to illustrate.

scholarly journals Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Mean Field ◽  
Partial Differential ◽  
Existence And Uniqueness Results ◽  
Doubly Stochastic ◽  
Uniqueness Results

Mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) are studied, which extend many important equations well studied before. Under some suitable monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of a method of continuation. Furthermore, the probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations (SPDEs) combined with algebra equations is given.

scholarly journals New class of boundary value problem for nonlinear fractional differential equations involving Erdélyi-Kober derivative

2019 ◽  
Vol 27 (2) ◽  
pp. 113-141
Author(s):  
Yacine Arioua ◽  
Maria Titraoui
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Nonlinear Fractional Differential Equations ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Nonlinear Alternative ◽  
Fractional Differential

AbstractIn this paper, we introduce a new class of boundary value problem for nonlinear fractional differential equations involving the Erdélyi-Kober differential operator on an infinite interval. Existence and uniqueness results for a positive solution of the given problem are obtained by using the Banach contraction principle, the Leray-Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in a special Banach space. To that end, some examples are presented to illustrate the usefulness of our main results.

scholarly journals On stochastic differential equations driven by the renormalized square of the Gaussian white noise

2015 ◽  
Vol 18 (04) ◽  
pp. 1550025
Author(s):  
Bilel Kacem Ben Ammou ◽  
Alberto Lanconelli
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Gaussian White Noise ◽  
Wick Product ◽  
Nonlinear Functions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Definition Of ◽  
White Noises

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Itô-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

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