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Djehiche, B., Elie, R. & Hamadene, S. (2023). Mean-Field Reflected Backward Stochastic Differential Equations. The Annals of Applied Probability, 33(4), 2493-2518
Open this publication in new window or tab >>Mean-Field Reflected Backward Stochastic Differential Equations
2023 (English)In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 33, no 4, p. 2493-2518Article in journal (Refereed) Published
Abstract [en]

In this paper, we study a class of reflected backward stochastic differential equations (BSDEs) of mean-field type, where the mean-field interaction in terms of the distribution of the Y-component of the solution enters in both the driver and the lower obstacle. We consider in details the case where the lower obstacle is a deterministic function of (Y, E[Y]) and discuss the more general dependence on the distribution of Y. Under mild Lipschitz and in-tegrability conditions on the coefficients, we obtain the well-posedness of such a class of equations. Under further monotonicity conditions, we show convergence of the standard penalization scheme to the solution of the equa-tion, which hence satisfies a minimality property. This class of equations is motivated by applications in pricing life insurance contracts with surrender options.

Place, publisher, year, edition, pages
Institute of Mathematical Statistics, 2023
Keywords
Mean-field, backward SDEs, Snell envelope, penalization
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-333744 (URN)10.1214/20-AAP1657 (DOI)001031710500001 ()2-s2.0-85165645618 (Scopus ID)
Note

QC 20230810

Available from: 2023-08-10 Created: 2023-08-10 Last updated: 2023-08-10Bibliographically approved
Djehiche, B. & Martini, M. (2023). Time-inconsistent mean-field optimal stopping: A limit approach. Journal of Mathematical Analysis and Applications, 528(1), Article ID 127582.
Open this publication in new window or tab >>Time-inconsistent mean-field optimal stopping: A limit approach
2023 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 528, no 1, article id 127582Article in journal (Refereed) Published
Abstract [en]

We provide a characterization of an optimal stopping time for a class of finite horizon time-inconsistent optimal stopping problems (OSPs) of mean-field type, adapted to the Brownian filtration, including those related to mean-field diffusion processes and recursive utility functions. Despite the time-inconsistency of the OSP, we show that it is optimal to stop when the value-process hits the reward process for the first time, as is the case for the standard time-consistent OSP. We solve the problem by approximating the corresponding value-process with a sequence of Snell envelopes of processes, for which a sequence of optimal stopping times is constituted of the hitting times of each of the reward processes by the associated value-process. Then, under mild assumptions, we show that this sequence of hitting times converges in probability to the hitting time for the mean-field OSP and that the limit is optimal.

Place, publisher, year, edition, pages
Academic Press Inc., 2023
Keywords
Mean-field, Optimal stopping, Snell envelope, Variance
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-334345 (URN)10.1016/j.jmaa.2023.127582 (DOI)001045136900001 ()2-s2.0-85165193798 (Scopus ID)
Note

QC 20230821

Available from: 2023-08-21 Created: 2023-08-21 Last updated: 2023-09-06Bibliographically approved
Djehiche, B., Hult, H. & Nyquist, P. (2022). Importance Sampling for a Simple Markovian Intensity Model Using Subsolutions. ACM Transactions on Modeling and Computer Simulation, 32(2), 1-25, Article ID 14.
Open this publication in new window or tab >>Importance Sampling for a Simple Markovian Intensity Model Using Subsolutions
2022 (English)In: ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, E-ISSN 1558-1195, Vol. 32, no 2, p. 1-25, article id 14Article in journal (Refereed) Published
Abstract [en]

This article considers importance sampling for estimation of rare-event probabilities in a specific collection of Markovian jump processes used for, e.g., modeling of credit risk. Previous attempts at designing importance sampling algorithms have resulted in poor performance and the main contribution of the article is the design of efficient importance sampling algorithms using subsolutions. The dynamics of the jump processes cause the corresponding Hamilton-Jacobi equations to have an intricate state-dependence, which makes the design of efficient algorithms difficult. We provide theoretical results that quantify the performance of importance sampling algorithms in general and construct asymptotically optimal algorithms for some examples. The computational gain compared to standard Monte Carlo is illustrated by numerical examples.

Place, publisher, year, edition, pages
Association for Computing Machinery (ACM), 2022
Keywords
Large deviations, Monte Carlo, importance sampling, Markovian intensity models, credit risk
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-310757 (URN)10.1145/3502432 (DOI)000772649100007 ()2-s2.0-85127447384 (Scopus ID)
Note

QC 20220408

Available from: 2022-04-08 Created: 2022-04-08 Last updated: 2022-06-25Bibliographically approved
Djehiche, B., Hamadene, S., Hdhiri, I. & Zaatra, H. (2022). Infinite Horizon Stochastic Impulse Control with Delay and Random Coefficients. Mathematics of Operations Research, 47(1), 665-689
Open this publication in new window or tab >>Infinite Horizon Stochastic Impulse Control with Delay and Random Coefficients
2022 (English)In: Mathematics of Operations Research, ISSN 0364-765X, E-ISSN 1526-5471, Vol. 47, no 1, p. 665-689Article in journal (Refereed) Published
Abstract [en]

We study a class of infinite horizon impulse control problems with execution delay when the dynamics of the system is described by a general stochastic process adapted to the Brownian filtration. The problem is solved by means of probabilistic tools relying on the notion of Snell envelope and infinite horizon reflected backward stochastic differential equations. This allows us to establish the existence of an optimal strategy over all admissible strategies.

Place, publisher, year, edition, pages
Institute for Operations Research and the Management Sciences (INFORMS), 2022
Keywords
optimal impulse control, execution delay, Infinite horizon, Snell envelope, slochastic control, optimal stopping time, backward stochastic differential equations
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-310224 (URN)10.1287/moor.2021.1145 (DOI)000731950100001 ()2-s2.0-85125592706 (Scopus ID)
Note

QC 20220325

Available from: 2022-03-25 Created: 2022-03-25 Last updated: 2022-09-19Bibliographically approved
Djehiche, B., Gozzi, F., Zanco, G. & Zanella, M. (2022). Optimal portfolio choice with path dependent benchmarked labor income: A mean field model. Stochastic Processes and their Applications, 145, 48-85
Open this publication in new window or tab >>Optimal portfolio choice with path dependent benchmarked labor income: A mean field model
2022 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 145, p. 48-85Article in journal (Refereed) Published
Abstract [en]

We consider the life-cycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky assets and a riskless bond subject to a borrowing constraint. In this paper, to reflect a realistic economic setting, we propose a model where the dynamics of the labor income has two main features. First, labor income adjusts slowly to financial market shocks, a feature already considered in Biffis et al. (2015). Second, the labor income gamma i of an agent i is benchmarked against the labor incomes of a population gamma(n) := (gamma(1),gamma(2)<b>, ...,gamma(n)) of n agents with comparable tasks and/or ranks. This last feature has not been considered yet in the literature and is faced taking the limit when n -> +infinity so that the problem falls into the family of optimal control of infinite-dimensional McKean-Vlasov Dynamics, which is a completely new and challenging research field. We study the problem in a simplified case where, adding a suitable new variable, we are able to find explicitly the solution of the associated HJB equation and find the optimal feedback controls. The techniques are a careful and nontrivial extension of the ones introduced in the previous papers of Biffis et al. (2015, 0000). (C) 2021 Elsevier B.V. All rights reserved.

Place, publisher, year, edition, pages
Elsevier BV, 2022
Keywords
Life-cycle optimal portfolio with labor income following path dependent and law dependent dynamics, Dynamic programming/optimal control of SDEs in infinite dimension with Mc Kean-Vlasov dynamics and state constraints, Second order Hamilton-Jacobi-Bellman equations in infinite dimension, Verification theorems and optimal feedback controls
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-312694 (URN)10.1016/j.spa.2021.11.010 (DOI)000789706700003 ()2-s2.0-85121449800 (Scopus ID)
Note

QC 20220524

Available from: 2022-05-24 Created: 2022-05-24 Last updated: 2022-06-25Bibliographically approved
Barreiro-Gomez, J., Choutri, S. E. & Djehiche, B. (2022). Stability Via Adversarial Training of Neural Network Stochastic Control of Mean-Field Type. In: 2022 IEEE 61ST CONFERENCE ON DECISION AND CONTROL (CDC): . Paper presented at IEEE 61st Conference on Decision and Control (CDC), DEC 06-09, 2022, Cancun, MEXICO (pp. 7547-7552). Institute of Electrical and Electronics Engineers (IEEE)
Open this publication in new window or tab >>Stability Via Adversarial Training of Neural Network Stochastic Control of Mean-Field Type
2022 (English)In: 2022 IEEE 61ST CONFERENCE ON DECISION AND CONTROL (CDC), Institute of Electrical and Electronics Engineers (IEEE) , 2022, p. 7547-7552Conference paper, Published paper (Refereed)
Abstract [en]

In this paper, we present an approach to neural network mean-field-type control and its stochastic stability analysis by means of adversarial inputs (aka adversarial attacks). This is a class of data-driven mean-field-type control where the distribution of the variables such as the system states and control inputs are incorporated into the problem. Besides, we present a methodology to validate the feasibility of the approximations of the solutions via neural networks and evaluate their stability. Moreover, we enhance the stability by enlarging the training set with adversarial inputs to obtain a more robust neural network. Finally, a worked-out example based on the linear-quadratic mean-field type control problem (LQ-MTC) is presented to illustrate our methodology.

Place, publisher, year, edition, pages
Institute of Electrical and Electronics Engineers (IEEE), 2022
Series
IEEE Conference on Decision and Control, ISSN 0743-1546
Keywords
Neural networks, data-driven control, stability, robustness, supervised machine learning, adversarial training
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-326492 (URN)10.1109/CDC51059.2022.9993216 (DOI)000948128106047 ()2-s2.0-85146987197 (Scopus ID)
Conference
IEEE 61st Conference on Decision and Control (CDC), DEC 06-09, 2022, Cancun, MEXICO
Note

QC 20230503

Available from: 2023-05-03 Created: 2023-05-03 Last updated: 2023-05-03Bibliographically approved
Djehiche, B., Mazhar, O. & Rojas, C. R. (2021). Finite impulse response models: A non-asymptotic analysis of the least squares estimator. Bernoulli, 27(2), 976-1000
Open this publication in new window or tab >>Finite impulse response models: A non-asymptotic analysis of the least squares estimator
2021 (English)In: Bernoulli, ISSN 1350-7265, E-ISSN 1573-9759, Vol. 27, no 2, p. 976-1000Article in journal (Refereed) Published
Abstract [en]

We consider a finite impulse response system with centered independent sub-Gaussian design covariates and noise components that are not necessarily identically distributed. We derive non-asymptotic near-optimal estimation and prediction bounds for the least squares estimator of the parameters. Our results are based on two concentration inequalities on the norm of sums of dependent covariate vectors and on the singular values of their covariance operator that are of independent value on their own and where the dependence arises from the time shift structure of the time series. These results generalize the known bounds for the independent case.

Place, publisher, year, edition, pages
Bernoulli Society for Mathematical Statistics and Probability, 2021
Keywords
Finite impulse response, least squares, non-asymptotic estimation, shifted random vector, random covariance Toeplitz matrix, concentration inequality
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-293397 (URN)10.3150/20-BEJ1262 (DOI)000634567600011 ()2-s2.0-85104247506 (Scopus ID)
Note

QC 20210423

Available from: 2021-04-23 Created: 2021-04-23 Last updated: 2022-06-25Bibliographically approved
Chen, Y., Djehiche, B. & Hamadène, S. (2021). Mean-field backward-forward stochastic differential equations and nonzero sum stochastic differential games. Stochastics and Dynamics, 21(06), Article ID 2150036.
Open this publication in new window or tab >>Mean-field backward-forward stochastic differential equations and nonzero sum stochastic differential games
2021 (English)In: Stochastics and Dynamics, ISSN 0219-4937, E-ISSN 1793-6799, Vol. 21, no 06, article id 2150036Article in journal (Refereed) Published
Abstract [en]

We study a general class of fully coupled backward-forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system of MF-BFSDE. We apply these results to derive an explicit form of open-loop Nash equilibrium strategies for nonzero sum mean-field linear-quadratic stochastic differential games with random coefficients. These strategies are valid for any time horizon of the game.

Place, publisher, year, edition, pages
World Scientific Pub Co Pte Ltd, 2021
Keywords
backward SDEs, Mean-field, nonlinear diffusion process, nonzero-sum game, open loop Nash equilibrium, optimal control
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-291688 (URN)10.1142/S0219493721500362 (DOI)000694654300001 ()2-s2.0-85097436518 (Scopus ID)
Note

QC 20210323. QC 20211010

Available from: 2021-03-23 Created: 2021-03-23 Last updated: 2023-06-26Bibliographically approved
Agram, N. & Djehiche, B. (2021). On a class of reflected backward stochastic Volterra integral equations and related time-inconsistent optimal stopping problems. Systems & control letters (Print), 155, 104989, Article ID 104989.
Open this publication in new window or tab >>On a class of reflected backward stochastic Volterra integral equations and related time-inconsistent optimal stopping problems
2021 (English)In: Systems & control letters (Print), ISSN 0167-6911, E-ISSN 1872-7956, Vol. 155, p. 104989-, article id 104989Article in journal (Refereed) Published
Abstract [en]

We introduce a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We prove existence and uniqueness by a fixed point argument and derive a comparison result. Moreover, we show how the solution of our problem is related to a time-inconsistent optimal stopping problem and derive an optimal strategy.

Place, publisher, year, edition, pages
Elsevier BV, 2021
Keywords
Backward stochastic differential equation, Snell envelope, Volterra integral equation, Time-inconsistent optimal stopping problem
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-309318 (URN)10.1016/j.sysconle.2021.104989 (DOI)000754889700003 ()2-s2.0-85110267391 (Scopus ID)
Note

QC 20220302

Available from: 2022-03-02 Created: 2022-03-02 Last updated: 2022-06-25Bibliographically approved
Djehiche, B. & Löfdahl, B. (2021). Quantum Support Vector Regression for Disability Insurance. Risks, 9(12), 216, Article ID 216.
Open this publication in new window or tab >>Quantum Support Vector Regression for Disability Insurance
2021 (English)In: Risks, E-ISSN 2227-9091, Vol. 9, no 12, p. 216-, article id 216Article in journal (Refereed) Published
Abstract [en]

We propose a hybrid classical-quantum approach for modeling transition probabilities in health and disability insurance. The modeling of logistic disability inception probabilities is formulated as a support vector regression problem. Using a quantum feature map, the data are mapped to quantum states belonging to a quantum feature space, where the associated kernel is determined by the inner product between the quantum states. This quantum kernel can be efficiently estimated on a quantum computer. We conduct experiments on the IBM Yorktown quantum computer, fitting the model to disability inception data from a Swedish insurance company.

Place, publisher, year, edition, pages
MDPI AG, 2021
Keywords
disability insurance, machine learning, support vector machines, quantum computing
National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-307159 (URN)10.3390/risks9120216 (DOI)000738306400001 ()2-s2.0-85121817427 (Scopus ID)
Note

QC 20220127

Available from: 2022-01-27 Created: 2022-01-27 Last updated: 2024-03-05Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-6608-0715

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