scholarly journals On the Robustness of the Snell Envelope

2011 ◽  
Vol 2 (1) ◽  
pp. 587-626 ◽  
Author(s):  
Pierre Del Moral ◽  
Peng Hu ◽  
Nadia Oudjane ◽  
Bruno Rémillard
Keyword(s):  
Snell Envelope

Approximations of the optimal stopping problem in partial observation

1986 ◽  
Vol 23 (2) ◽  
pp. 341-354 ◽  
Author(s):  
G. Mazziotto
Keyword(s):  
Optimal Stopping ◽  
Measure Space ◽  
Stopping Time ◽  
Optimal Stopping Problem ◽  
Snell Envelope ◽  
Infinite Dimensional ◽  
Markov State ◽  
State Process ◽  
Partially Observed ◽  
True State

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.

scholarly journals An iterative method for multiple stopping: convergence and stability

2006 ◽  
Vol 38 (3) ◽  
pp. 729-749 ◽  
Author(s):  
Christian Bender ◽  
John Schoenmakers
Keyword(s):  
Monte Carlo ◽  
Dynamic Programming ◽  
Iterative Method ◽  
Discrete Time ◽  
Iterative Procedure ◽  
Snell Envelope ◽  
Conditional Expectations ◽  
Multiple Stopping ◽  
The Stability ◽  
Monte Carlo Implementation

We present a new iterative procedure for solving the multiple stopping problem in discrete time and discuss the stability of the algorithm. The algorithm produces monotonically increasing approximations of the Snell envelope which coincide with the Snell envelope after finitely many steps. Unlike backward dynamic programming, the algorithm allows us to calculate approximative solutions with only a few nestings of conditional expectations and is, therefore, tailor-made for a plain Monte Carlo implementation.

Optimal Stopping Theory and American Options

2019 ◽  
pp. 376-396
Author(s):  
Tomas Björk
Keyword(s):  
Dynamic Programming ◽  
Optimal Stopping ◽  
Continuous Time ◽  
American Options ◽  
Programming Approach ◽  
Free Boundary Value Problem ◽  
Snell Envelope ◽  
Dynamic Programming Approach ◽  
Optimal Stopping Theory ◽  
Time Theory

In this chapter we present the dynamic programming approach to optimal stopping problems. We start by presenting the discrete time theory, deriving the relevant Bellman equation. We present the Snell envelope and prove the Snell Envelope Theorem. For Markovian models we explore the connection to alpha-excessive functions. The continuous time theory is presented by deriving the free boundary value problem connected to the stopping problem, and we also derive the associated system of variational inequalities. American options are discussed in some detail.

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 Lp-SOLUTIONS FOR REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 12 (02) ◽  
pp. 1150016 ◽  
Author(s):  
SAÏD HAMADÈNE ◽  
ALEXANDRE POPIER
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Snell Envelope ◽  
Equation Problem ◽  
Class Of Functions ◽  
Uniqueness Of A Solution ◽  
Lp Solutions

This paper deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation (BSDE for short) with one reflecting barrier in the case when the terminal value, the generator and the obstacle process are Lp-integrable with p ∈ ]1, 2[. To construct the solution we use two methods: penalization and Snell envelope. As an application we broaden the class of functions for which the related obstacle partial differential equation problem has a unique viscosity solution.

scholarly journals Generalized Snell envelope as a minimal solution of BSDE with lower barriers

2013 ◽  
Vol 137 (4) ◽  
pp. 498-508 ◽  
Author(s):  
E.H. Essaky ◽  
M. Hassani ◽  
Y. Ouknine
Keyword(s):  
Minimal Solution ◽  
Snell Envelope

Mean-field optimal multi-modes switching problem: A balance sheet

2019 ◽  
Vol 19 (04) ◽  
pp. 1950026 ◽  
Author(s):  
M’hamed Eddahbi ◽  
Imade Fakhouri ◽  
Youssef Ouknine
Keyword(s):  
Balance Sheet ◽  
Mean Field ◽  
Field Approximation ◽  
Expected Profit ◽  
Picard Iteration ◽  
Mean Field Approximation ◽  
Coupling Term ◽  
Snell Envelope ◽  
The Mean ◽  
Backward Sdes

We study a finite horizon optimal multi-modes switching problem with many nodes. The switching is based on the optimal expected profit and cost yields, moreover both sides of the balance sheet are considered. The profit and cost yields per unit time are respectively assumed to be coupled through a coupling term which is the average of profit and cost yields. The corresponding system of Snell envelopes is highly complex, so we consider the aggregated yields where a mean-field approximation is used for the coupling term. First, the problem is formulated by the mean of the Snell envelope of processes. Then, in terms of backward SDEs, the problem is equivalent to a system of mean-field reflected backward SDEs with interconnected and nonlinear obstacles. More precisely, the driver function depends also on the mean of the unknown process (expected profit or cost yields) which makes the mean-field interaction in the driver nonlinear. The first main result of this paper, is to show the existence of a continuous minimal solution of the system of mean-field reflected backward SDEs, which is done by using the Picard iteration method. The second main result concerns the optimality of the switching strategies which we fully characterize.

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