1
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Cui F, Zhao W. Well-posedness of mean reflected BSDEs with non-Lipschitz coefficients. Stat Probab Lett 2023. [DOI: 10.1016/j.spl.2022.109718] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
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2
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Saouli MA. Fractional backward SDEs with locally monotone coefficient and application to PDEs. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2023. [DOI: 10.1515/rose-2022-2095] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023]
Abstract
Abstract
In this work, we will try to weaken the hypothesis imposed by Hu and Peng. We will be concerned with finding the solution of locally monotone BSDEs associated to fBm. As an auxiliary
step, we study the existence and uniqueness of a solution to the monotone
backward SDEs associated to fBm. Then we connect these two kinds of
fractional backward SDEs with the corresponding semilinear partial differential equations (PDEs for short).
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Affiliation(s)
- Mostapha Abdelouahab Saouli
- Laboratory of Applied Mathematics , Department of Mathematics , University of Kasdi Merbah Ouargla , Ouargla , Algeria
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3
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Existence and Uniqueness of Solutions for Multi-dimensional Reflected Backward Stochastic Differential Equations with Diagonally Quadratic Generators. J THEOR PROBAB 2022. [DOI: 10.1007/s10959-022-01224-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
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4
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Reflected Backward Stochastic Differential Equations Associated to Jump Markov Processes and Application to Partial Differential Equations. J THEOR PROBAB 2022. [DOI: 10.1007/s10959-022-01212-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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5
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Optimal stopping under g-expectation with -integrable reward process. J Appl Probab 2022. [DOI: 10.1017/jpr.2022.35] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
Abstract
In this paper we study a class of optimal stopping problems under g-expectation, that is, the cost function is described by the solution of backward stochastic differential equations (BSDEs). Primarily, we assume that the reward process is
$L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$
-integrable with
$\mu>\mu_0$
for some critical value
$\mu_0$
. This integrability is weaker than
$L^p$
-integrability for any
$p>1$
, so it covers a comparatively wide class of optimal stopping problems. To reach our goal, we introduce a class of reflected backward stochastic differential equations (RBSDEs) with
$L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$
-integrable parameters. We prove the existence, uniqueness, and comparison theorem for these RBSDEs under Lipschitz-type assumptions on the coefficients. This allows us to characterize the value function of our optimal stopping problem as the unique solution of such RBSDEs.
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6
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Reflected BSDEs in non-convex domains. Probab Theory Relat Fields 2022. [DOI: 10.1007/s00440-022-01125-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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7
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A Multi-Step Algorithm for BSDEs Based On a Predictor-Corrector Scheme and Least-Squares Monte Carlo. Methodol Comput Appl Probab 2022. [DOI: 10.1007/s11009-022-09943-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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8
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Cohen SN, Treetanthiploet T. Gittins’ theorem under uncertainty. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp742] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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9
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Hu Y, Moreau R, Wang F. Quadratic mean-field reflected BSDEs. PROBABILITY, UNCERTAINTY AND QUANTITATIVE RISK 2022. [DOI: 10.3934/puqr.2022012] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
<p style='text-indent:20px;'>In this paper, we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown <inline-formula><tex-math id="M1">\begin{document}$ z $\end{document}</tex-math></inline-formula>. Using a linearization technique and the BMO martingale theory, we first apply a fixed-point argument to establish the uniqueness and existence result for the case with bounded terminal condition and obstacle. Then, with the help of the <inline-formula><tex-math id="M2">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> -method, we develop a successive approximation procedure to remove the boundedness condition on the terminal condition and obstacle when the generator is concave (or convex) with respect to the second unknown <inline-formula><tex-math id="M3">\begin{document}$ z $\end{document}</tex-math></inline-formula>.</p>
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Affiliation(s)
- Ying Hu
- CNRS, IRMAR-UMR 6625, Univ. Rennes, F-35000, Rennes, France
- School of Mathematical Sciences, Fudan University, Shanghai 200433, China
| | - Remi Moreau
- CNRS, IRMAR-UMR 6625, Univ. Rennes, F-35000, Rennes, France
| | - Falei Wang
- Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China
- School of Mathematics, Shandong University, Jinan 250100, China
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10
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Abstract
We prove some new results on reflected BSDEs and doubly reflected BSDEs driven by a multi-dimensional RCLL martingale. The goal is to develop a general multi-asset framework encompassing a wide spectrum of nonlinear financial models, including as particular cases the setups studied by Peng and Xu [BSDEs with random default time and their applications to default risk, working paper, preprint (2009), arXiv:0910.2091] and Dumitrescu et al. [BSDEs with default jump, in Computation and Combinatorics in Dynamics, Stochastics and Control, Abel Symposia, Vol. 13, eds. E. Celledoni, G. Di Nunno, K. Ebrahimi-Fard and H. Munthe-Kaas (Springer, Cham, 2018), pp. 233–263] who examined BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. Our results are not covered by existing literature on reflected and doubly reflected BSDEs driven by a Brownian motion and a Poisson random measure.
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Affiliation(s)
- Tianyang Nie
- School of Mathematics, Shandong University, Jinan, Shandong 250100, P. R. China
| | - Marek Rutkowski
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
- Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warszawa, Poland
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11
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Dai Y, Li R. Transportation cost inequality for backward stochastic differential equations with mean reflection. Stat Probab Lett 2021. [DOI: 10.1016/j.spl.2021.109167] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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12
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Beck C, Hutzenthaler M, Jentzen A. On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. STOCH DYNAM 2021. [DOI: 10.1142/s0219493721500489] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The classical Feynman–Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this paper, we extend the classical Feynman–Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic fixed point equations (SFPEs), which arise when the classical Feynman–Kac identity is formally applied to semilinear Kolmorogov PDEs, as viscosity solutions of the corresponding PDEs. This justifies, in particular, employing full-history recursive multilevel Picard (MLP) approximation algorithms, which have recently been shown to overcome the curse of dimensionality in the numerical approximation of solutions of SFPEs, in the numerical approximation of semilinear Kolmogorov PDEs.
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Affiliation(s)
- Christian Beck
- Applied Mathematics Münster: Institute for Analysis and Numerics, University of Münster, Einsteinstraße 62, 48149 Münster, Germany
- Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zürich, Switzerland
| | - Martin Hutzenthaler
- Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
| | - Arnulf Jentzen
- Applied Mathematics Münster: Institute for Analysis and Numerics, University of Münster, Einsteinstraße 62, 48149 Münster, Germany
- Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zürich, Switzerland
- School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, 2001 Longxiang Road, 518172 Shenzhen, China
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13
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Bouhadou S, Ouknine Y. Reflected BSDEs when the obstacle is predictable and nonlinear optimal stopping problem. STOCH DYNAM 2021. [DOI: 10.1142/s0219493721500490] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In the first part of this paper, we study RBSDEs in the case where the filtration is non-quasi-left-continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part, we introduce an optimal stopping problem indexed by predictable stopping times with the nonlinear predictable [Formula: see text] expectation induced by an appropriate backward stochastic differential equation (BSDE). We establish some useful properties of [Formula: see text]-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.
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Affiliation(s)
- Siham Bouhadou
- LIBMA Laboratory, Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Morocco
| | - Youssef Ouknine
- LIBMA Laboratory, Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Morocco
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14
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Matoussi A, Possamaï D, Zhou C. Corrigendum for “Second-order reflected backward stochastic differential equations” and “Second-order BSDEs with general reflection and game options under uncertainty”. ANN APPL PROBAB 2021. [DOI: 10.1214/20-aap1622] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | | | - Chao Zhou
- Department of Mathematics, National University of Singapore
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15
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El Otmani M. Reflected generalized BSDEs with discontinuous barriers driven by a Lévy process. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2021. [DOI: 10.1515/rose-2021-2060] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
Abstract
This article deals with the reflected and doubly reflected generalized backward stochastic differential equations when the noise is given
by Brownian motion and Teugels martingales associated with an independent pure jump Lévy process. We prove the
existence and the uniqueness of the solution for these equations with monotone generators and right continuous left limited obstacles.
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Affiliation(s)
- Mohamed El Otmani
- Laboratory of Analysis and Applied Mathematics (LAMA) , Faculty of Sciences Agadir , Ibn Zohr University , Agadir , Morocco
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16
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Quasilinear Stochastic PDEs with two obstacles: Probabilistic approach. Stoch Process Their Appl 2021. [DOI: 10.1016/j.spa.2020.11.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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17
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Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting. J THEOR PROBAB 2021. [DOI: 10.1007/s10959-020-01070-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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18
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Klimsiak T, Rzymowski M. Reflected BSDEs with two optional barriers and monotone coefficient on general filtered space. ELECTRON J PROBAB 2021. [DOI: 10.1214/21-ejp655] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Tomasz Klimsiak
- Institute of Mathematics, Polish Academy of Sciences, Poland
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19
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Affiliation(s)
- Edward Kim
- School of Mathematics and Statistics, University of Sydney, Australia
| | - Tianyang Nie
- Corresponding author. School of Mathematics, Shandong University, Jinan, China
| | - Marek Rutkowski
- School of Mathematics and Statistics, University of Sydney, Australia and Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
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20
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BSDEs with jumps and two completely separated
irregular barriers in a general filtration. LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS 2021. [DOI: 10.30757/alea.v18-28] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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21
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Reflected BSDEs with jumps in time-dependent convex càdlàg domains. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2020.06.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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22
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Reflected backward stochastic differential equation driven by G-Brownian motion with an upper obstacle. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2020.06.002] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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23
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Wu Z, Xu Z. A kind of stochastic recursive Zero-Sum differential game problem with double obstacles constraint. COMMUN STAT-THEOR M 2020. [DOI: 10.1080/03610926.2019.1618477] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Zhen Wu
- School of Mathematics, Shandong University, Jinan, P. R. China
| | - Zhenda Xu
- Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, P. R. China
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24
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Richard A, Tanré E, Torres S. Penalisation techniques for one-dimensional reflected rough differential equations. BERNOULLI 2020. [DOI: 10.3150/20-bej1212] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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25
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26
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Luo P. Reflected BSDEs with time-delayed generators and nonlinear resistance. Stat Probab Lett 2020. [DOI: 10.1016/j.spl.2020.108765] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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27
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Falkowski A, Słomiński L. Backward stochastic differential equations with two barriers and generalized reflection. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2020.01.015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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28
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Chen ZQ, Feng X. Reflected Backward Stochastic Differential Equation with Rank-Based Data. J THEOR PROBAB 2020. [DOI: 10.1007/s10959-020-01026-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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29
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A new numerical method for 1-D backward stochastic differential equations without using conditional expectations. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2020. [DOI: 10.1515/rose-2020-2030] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
AbstractIn this paper, we propose a new numerical method for 1-D
backward stochastic differential equations (BSDEs for
short) without using
conditional expectations. The approximations of the solutions
are obtained as solutions of a backward linear
system generated by the terminal conditions. Our idea is inspired
from the extended Kalman filter to non-linear system models by using
a linear approximation around deterministic nominal reference
trajectories.
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30
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El Jamali M, El Otmani M. BSDE with rcll reflecting barrier driven by a Lévy process. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2020. [DOI: 10.1515/rose-2020-2029] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
AbstractIn this paper, we study the solution of a backward stochastic differential equation driven by a Lévy process with one rcll reflecting barrier.
We show the existence and uniqueness of a solution by means of the penalization method when the coefficient is stochastic Lipschitz. As an application, we give a fair price of an American option.
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Affiliation(s)
- Mohamed El Jamali
- Laboratory of Analysis and Applied Mathematics (LAMA), Faculty of Sciences Agadir, Ibn Zohr University, Agadir, Morocco
| | - Mohamed El Otmani
- Laboratory of Analysis and Applied Mathematics (LAMA), Faculty of Sciences Agadir, Ibn Zohr University, Agadir, Morocco
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31
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32
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A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2019.05.013] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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33
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Liu Y, Yang A, Lin J, Yao J. A new method of valuing American options based on Brownian models. COMMUN STAT-THEOR M 2020. [DOI: 10.1080/03610926.2020.1725053] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Affiliation(s)
- Yue Liu
- Faculty of Finance Management and Accounting, School of Finance and Economics, Jiangsu University, Zhenjiang, China
| | - Aijun Yang
- Department of Applied Economics, College of Economics and Management, Nanjing Forestry University, Nanjing, China
| | - Jinguan Lin
- Department of Statistics, School of Statistics and Mathematics, Nanjing Audit University, Nanjing, China
| | - Jingjing Yao
- Faculty of Finance Management and Accounting, School of Finance and Economics, Jiangsu University, Zhenjiang, China
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34
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35
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Xiao L, Fan S, Tian D. A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems. ESAIM-PROBAB STAT 2020. [DOI: 10.1051/ps/2019023] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.
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36
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Krzhizhanovskaya VV, Závodszky G, Lees MH, Dongarra JJ, Sloot PMA, Brissos S, Teixeira J. Multidimensional BSDEs with Mixed Reflections and Balance Sheet Optimal Switching Problem. LECTURE NOTES IN COMPUTER SCIENCE 2020. [PMCID: PMC7304763 DOI: 10.1007/978-3-030-50436-6_43] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
In this paper, we study a system of multidimensional coupled reflected backward stochastic differential equations (RBSDEs) with interconnected generators and barriers and mixed reflections, i.e. oblique and normal reflections. This system of equations is arising in the context of optimal switching problem when both sides of the balance sheet are considered. This problem incorporates both the action of switching between investment modes and the action of abandoning the investment project before its maturity once it becomes unprofitable. Pricing such real options (switch option and abandon option) is equivalent to solve the system of coupled RBSDEs considered in the paper, for which we show the existence of a continuous adapted minimal solution via a Picard iteration method.
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37
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Yao S. 𝕃 p solutions of reflected backward stochastic differential equations with jumps. ESAIM-PROBAB STAT 2020. [DOI: 10.1051/ps/2020026] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
Given p ∈ (1, 2), we study 𝕃p-solutions of a reflected backward stochastic differential equation with jumps (RBSDEJ) whose generator g is Lipschitz continuous in (y, z, u). Based on a general comparison theorem as well as the optimal stopping theory for uniformly integrable processes under jump filtration, we show that such a RBSDEJ with p-integrable parameters admits a unique 𝕃p solution via a fixed-point argument. The Y -component of the unique 𝕃p solution can be viewed as the Snell envelope of the reflecting obstacle 𝔏 under g-evaluations, and the first time Y meets 𝔏 is an optimal stopping time for maximizing the g-evaluation of reward 𝔏.
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38
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Bartl D, Tangpi L. Functional inequalities for forward and backward diffusions. ELECTRON J PROBAB 2020. [DOI: 10.1214/20-ejp495] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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39
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Lin Y, Ren Z, Touzi N, Yang J. Second order backward SDE with random terminal time. ELECTRON J PROBAB 2020. [DOI: 10.1214/20-ejp498] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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40
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Yang X, Zhang J. The obstacle problem for quasilinear stochastic PDEs with degenerate operator. Stoch Process Their Appl 2019. [DOI: 10.1016/j.spa.2018.08.009] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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41
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Dong Y, Yang X, Zhang J. The obstacle problem for quasilinear stochastic PDEs with Neumann boundary condition. STOCH DYNAM 2019. [DOI: 10.1142/s0219493719500394] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We prove the existence and uniqueness of solution to obstacle problem for quasilinear stochastic partial differential equations with Neumann boundary condition. Our method is based on the analytical techniques coming from parabolic potential theory. The solution is expressed as a pair [Formula: see text] where [Formula: see text] is a predictable continuous process which takes values in a proper Sobolev space and [Formula: see text] is a random regular measure satisfying minimal Skohorod condition.
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Affiliation(s)
- Yuchao Dong
- School of Mathematical Sciences, Fudan University, 220 Handan Rd., Yangpu District, Shanghai 200433, P. R. China
| | - Xue Yang
- School of Mathematics, Tianjin University, 135 Yaguan Road, Haihe Education Park, Tianjin 300350, P. R. China
| | - Jing Zhang
- School of Mathematical Sciences, Fudan University, 220 Handan Rd., Yangpu District, Shanghai 200433, P. R. China
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42
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Eddahbi M, Fakhouri I, Ouknine Y. Mean-field optimal multi-modes switching problem: A balance sheet. STOCH DYNAM 2019. [DOI: 10.1142/s0219493719500266] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
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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Affiliation(s)
- M’hamed Eddahbi
- Mathematics Department, College of Sciences, King Saud University, P. O. Box 2455, Riyadh, Z.C. 11451, Riyadh, Saudi Arabia
| | - Imade Fakhouri
- Complex Systems Engineering and Human Systems, Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid, Ben Guerir 43150, Morocco
| | - Youssef Ouknine
- Complex Systems Engineering and Human Systems, Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid, Ben Guerir 43150, Morocco
- Mathematics Department, Faculty of Sciences Semalalia, Cadi Ayyad University, Boulevard Prince Moulay Abdellah, P. O. Box 2390, Marrakesh 40000, Morocco
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Marzougue M, El Otmani M. BSDEs with right upper-semicontinuous reflecting obstacle and stochastic Lipschitz coefficient. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2019. [DOI: 10.1515/rose-2019-2005] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
Abstract
This paper proves the existence and uniqueness of a solution to reflected backward stochastic differential equations with a lower obstacle, which is assumed to be right upper-semicontinuous. The result is established where the coefficient is stochastic Lipschitz by using some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales and other tools from optimal stopping theory.
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Fan S. $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions. ELECTRON J PROBAB 2019. [DOI: 10.1214/19-ejp345] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Huang Z, Wang H, Wu Z, Yu Z. Quadratic reflected BSDEs and related obstacle problems for PDEs. COMMUN STAT-THEOR M 2018. [DOI: 10.1080/03610926.2018.1543778] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Zongyuan Huang
- School of Mathematics, Shandong University, Jinan, China
| | - Haiyang Wang
- School of Mathematics and Statistics, Shandong Normal University, Jinan, China
| | - Zhen Wu
- School of Mathematics, Shandong University, Jinan, China
| | - Zhiyong Yu
- School of Mathematics, Shandong University, Jinan, China
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Lv S, Wu Z. Infinite horizon reflected backward stochastic differential equations with Markov chains. COMMUN STAT-THEOR M 2018. [DOI: 10.1080/03610926.2017.1353629] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Siyu Lv
- School of Mathematics, Southeast University, Nanjing, P. R. China
| | - Zhen Wu
- School of Mathematics, Shandong University, Jinan, P. R. China
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Aazizi S, El Mellali T, Fakhouri I, Ouknine Y. Optimal switching problem and related system of BSDEs with left-Lipschitz coefficients and mixed reflections. Stat Probab Lett 2018. [DOI: 10.1016/j.spl.2018.01.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Esmaeeli N, Imkeller P. American options with asymmetric information and reflected BSDE. BERNOULLI 2018. [DOI: 10.3150/16-bej902] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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