Boualem Djehiche

Boualem Djehiche
KTH Royal Institute of Technology | KTH · Department of Mathematics (SCI-MAT)

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130
Publications
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2,399
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Publications

Publications (130)
Book
Full-text available
Book on mean-field-type games (under preparation)
Article
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 import...
Preprint
We introduce a zero-sum game problem of mean-field type as an extension of the classical zero-sum Dynkin game problem to the case where the payoff processes might depend on the value of the game and its probability law. We establish sufficient conditions under which such a game admits a value and a saddle point. Furthermore, we provide a characteri...
Preprint
Full-text available
The aim of this paper is to address two related estimation problems arising in the setup of hidden state linear time invariant (LTI) state space systems when the dimension of the hidden state is unknown. Namely, the estimation of any finite number of the system's Markov parameters and the estimation of a minimal realization for the system, both fro...
Article
Full-text available
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...
Article
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 in...
Preprint
Full-text available
In this article, we establish the propagation of chaos property for weakly interacting nonlinear Snell envelopes which converge to a class of mean-field reflected backward stochastic differential equations (BSDEs) with jumps, where the mean-field interaction in terms of the distribution of the $Y$-component of the solution enters both the driver an...
Preprint
Full-text available
We study the estimation problem for linear time-invariant (LTI) state-space models with Gaussian excitation of an unknown covariance. We provide non asymptotic lower bounds for the expected estimation error and the mean square estimation risk of the least square estimator, and the minimax mean square estimation risk. These bounds are sharp with exp...
Article
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 different...
Preprint
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 is mapped to quantum states belonging to a quantum feature space, where the a...
Article
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 solut...
Article
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...
Article
Full-text available
In this article, a profit optimization between electricity producers is formulated and solved. The problem is described by a linear jump-diffusion system of conditional mean-field type where the conditioning is with respect to common noise and a quadratic cost functional involving the second moment, the square of the conditional expectation of the...
Preprint
Full-text available
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 in...
Article
Full-text available
We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian mot...
Article
This article presents a class of Stackelberg mean-field-type games with multiple leaders and multiple followers. The decision-makers act in sequential order with informational differences. The state dynamics is driven by jump-diffusion processes and the cost function is non-quadratic and has a polynomial structure. The structures of Stackelberg str...
Article
Full-text available
We show existence of an optimal control and a saddle-point for respectively a control problem and zero-sum differential game associated with payoff functionals of mean field type, under dynamics driven by weak solutions of stochastic differential equations of mean-field type.
Article
Life insurance cash flows become reserve dependent when contract conditions are modified during the contract term on condition that actuarial equivalence is maintained. As a result, insurance cash flows and prospective reserves depend on each other in a circular way, and it is a non-trivial problem to solve that circularity and make cash flows and...
Preprint
We study solutions of 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 we derive a comparison result. Moreover, we show h...
Article
Credit scoring is one of the key problems in financial risk managements. This paper studies the credit scoring problem based on the set-valued identification method, which is used to explain the relation between the individual attribute vectors and classification for the credit worthy and credit worthless lenders. In particular, system parameters a...
Article
In this paper, the credit scoring problem is studied by incorporating networked information, where the advantages of such incorporation are investigated theoretically in two scenarios. Firstly, a Bayesian optimal filter is proposed to provide risk prediction for lenders assuming that published credit scores are estimated merely from structured fina...
Article
Full-text available
We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton-Jacobi equations (HJ), prove the comparison principle and that the value function of the optimal control problem is the unique viscosity solution of the HJ system. This is do...
Article
Full-text available
In this article, we study mean-field-type games with jump-diffusion and regime switching in which the payoffs and the state dynamics depend not only on the state-action profile of the decision-makers but also on a measure of the state-action pair. The state dynamics is a measure-dependent process with jump-diffusion and regime switching. We derive...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Insurance cash flows become reserve dependent whenever contract conditions are modified during the contract term while maintaining actuarial equivalence. As a result, insurance cash flows and prospective reserves depend on each other in a circular way, and it is a non-trivial problem to solve that circularity and make cash flows and reserves well-d...
Preprint
This paper introduces a system of stochastic differential equations (SDE) of mean-field type that by means of sticky boundaries and boundary diffusion accounts for the possibility of pedestrians to spend time at, and to move along, walls. As an alternative to Neumann-type boundary conditions, sticky boundaries and boundary diffusion have a 'smoothi...
Preprint
Full-text available
In this paper, the credit scoring problem is studied by incorporating network information, where the advantages of such incorporation are investigated in two scenarios. Firstly, a Bayesian optimal filter is proposed to provide a prediction for lenders assuming that published credit scores are estimated merely from structured individual data. Such p...
Preprint
We study a class of infinite horizon impulse control problems with execution delay when the dynamics of the system is described by a general adapted stochastic process. 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...
Preprint
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...
Preprint
Full-text available
We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton-Jacobi equations (HJ), prove the comparison principle and that the value function of the optimal control problem is the unique viscosity solution of the HJ system. This is do...
Chapter
Full-text available
In this paper we examine mean-field-type games in blockchain-based distributed power networks with several different entities: investors, consumers, prosumers, producers and miners. Under a simple model of jump-diffusion and regime switching processes, we identify risk-aware mean-field-type optimal strategies for the decision-makers.
Article
This paper suggests a mean-field model for the movement of tagged pedestrians, distinguishable from a surrounding crowd, with a targeted final destination. The tagged pedestrians move through a dynamic crowd, interacting with it while optimizing their path. The model includes distribution-dependent effects like congestion and crowd aversion. The be...
Article
We establish existence of controlled Markov chain of mean-field type with unbounded jump intensities by means of a fixed point argument using the Wasserstein distance. Using a Markov chain entropic backward SDE approach, we further suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zer...
Article
We extend the class of pedestrian crowd models introduced by Lachapelle and Wolfram [Transp. Res. B: Methodol., 45 (2011), pp. 1572–1589] to allow for nonlocal crowd aversion and arbitrarily but finitely many interacting crowds. The new crowd aversion feature grants pedestrians a “personal space” where crowding is undesirable. We derive the model f...
Data
We study risk-sensitive optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state and control processes. Moreover the risk-sensitive cost functional is also of mean-field type. We derive optimality equations in infinite dimensions c...
Article
We study risk-sensitive optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state and control processes. Moreover the risk-sensitive cost functional is also of mean-field type. We derive optimality equations in infinite dimensions c...
Article
Full-text available
With the ever increasing amounts of data becoming available, strategic data analysis and decision-making will become more pervasive as a necessary ingredient for societal infrastructures. In many network engineering games, the performance metrics depend only on some aggregate of the parameters/choices. One typical example is the congestion field in...
Article
This work puts forward a simple mean-field network game that captures some of the key dynamic features of crowd and pedestrian flows in multi-level building evacuations. It considers both microscopic and macroscopic route choice by strategic agents. To achieve this, we use mean-field differential game with local congestion measure based on the loca...
Article
We extend the class of pedestrian crowd models introduced by Lachapelle and Wolfram (2011) to allow for nonlocal congestion and arbitrarily but finitely many interacting crowds. The new congestion feature grants pedestrians a personal space where crowding is undesirable. We treat the model as a mean-field type game which we derive from a particle p...
Article
Full-text available
We show existence of an optimal control and a saddle-point for the zero-sum games associated with payoff functionals of mean-field type, under a dynamics driven by a class of Markov chains of mean-field type.
Article
This paper considers importance sampling for estimation of rare-event probabilities in a Markovian intensity model commonly used in the context of credit risk. The main contribution is the design of efficient importance sampling algorithms using subsolutions of a certain Hamilton-Jacobi equation. We provide theoretical results that quantify the per...
Article
We consider large insurance portfolios consisting of life or disability insurance policies that are assumed independent, conditional on a stochastic process representing the economic–demographic environment. Using the conditional law of large numbers, we show that when the portfolio of liabilities becomes large enough, its value on a δ -year horizo...
Conference Paper
This is a short introduction to some basic aspects of statistical estimation techniques known as graduation technique in life and disability insurance.
Chapter
We establish a stochastic maximum principle (SMP) for control problems of partially observed diffusions of mean-field type with risk-sensitive performance functionals.
Article
We suggest a unified approach to claims reserving for life insurance policies with reserve-dependent payments driven by multi-state Markov chains. The associated prospective reserve is formulated as a recursive utility function using the framework of backward stochastic differential equations (BSDE). We show that the prospective reserve satisfies a...
Article
Full-text available
In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the stat...
Article
Full-text available
In this paper we study a generalization of the continuous time Principal-Agent problem allowing for time inconsistent utility functions, for instance of mean-variance type. Using recent results on the Pontryagin maximum principle for FBSDEs we suggest a method of characterizing optimal contracts for such models. To illustrate this we consider a ful...
Article
We propose a stochastic semi-Markovian framework for disability modelling in a multi-period discrete-time setting. The logistic transforms of disability inception and recovery probabilities are modelled by means of stochastic risk factors and basis functions, using counting processes and generalized linear models. The model for disability inception...
Article
Full-text available
Point and interval estimation of future disability inception and recovery rates are predominantly carried out by combining generalized linear models (GLM) with time series forecasting techniques into a two-step method involving parameter estimation from historical data and subsequent calibration of a time series model. This approach may in fact lea...
Article
Full-text available
In this paper we formulate and solve a mean-field game described by a linear stochastic dynamics and a quadratic or exponential-quadratic cost functional for each generic player. The optimal strategies for the players are given explicitly using a simple and direct method based on square completion and a Girsanov-type change of measure. This approac...
Article
Full-text available
We formulate and solve a finite horizon full balance sheet two-modes optimal switching problem related to trade-off strategies between expected profit and cost yields. Given the current mode, this model allows for either a switch to the other mode or termination of the project, and this happens for both sides of the balance sheet. A novelty in this...
Article
Full-text available
We study a general class of Principal-Agent problems in continuous time under hidden action. By formulating the model as a coupled stochastic optimal control problem we are able to find a set of necessary conditions characterizing optimal contracts, using the stochastic maximum principle. An example is carried out to illustrate the proposed approac...
Article
In this paper, we deal with the solutions of systems of PDEs with bilateral interconnected obstacles of min–max and max–min types. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are regular, the solutions of the min–max and max–min systems coincide. Then, this common v...
Article
Full-text available
This article examines mean-field games for marriage. The results support the argument that optimizing the long-term well-being through effort and social feeling state distribution (mean-field) will help to stabilize marriage. However , if the cost of effort is very high, the couple fluctuates in a bad feeling state or the marriage breaks down. We t...
Article
Full-text available
In this paper a duality relation between the Ma\~{n}\'e potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain min-max representations of viscosity solutions of first order Hamilton-Jacobi equations. These min-max representations naturally suggest class\-...
Chapter
Historical notes on the Scandinavian Actuarial Journal are presented.
Article
Full-text available
We consider the problem of switching a large number of production lines between two modes, high production and low production. The switching is based on the optimal expected profit and cost yields of the respective production lines and considers both sides of the balance sheet. Furthermore, the production lines are all assumed to be interconnected...
Data
Full-text available
In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the stat...
Article
Full-text available
We study a class of dynamic decision problems of mean field type with time inconsistent cost functionals, and derive a stochastic maximum principle to characterize subgame perfect Nash equilibrium points. Subsequently, this approach is extended to a mean field game to construct decentralized strategies and obtain an estimate of their performance.
Article
We consider a large, homogeneous portfolio of life or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic-demographic environment. Using a conditional law of large numbers, we establish the connection between claims reserving and risk aggregation for large p...
Article
Full-text available
We propose a functional version of the Hodrick-Prescott fi?lter for functional data which take values in an in?finite dimensional separable Hilbert space. We further characterize the associated optimal smoothing parameter when the associated linear operator is compact and the underlying distribution of the data is Gaussian.
Article
Full-text available
We study a version of the functional Hodrick-Prescott filter where the associated operator is not necessarily compact, but merely closed and densely defined with closed range. We show that the associated optimal smoothing operator preserves the structure obtained in the compact case, when the underlying distribution of the data is Gaussian.
Article
We suggest an explicit data-driven consistent estimator of the optimal smooth trend in a multivariate Hodrick-Prescott filter, when the associated disturbances (i.e., signal and cycle components) follow a moving average, and a vector autoregressive process, respectively. This is done through deriving consistent estimators of the covariance matrices...
Article
Full-text available
We study a general class of nonlinear second-order variational inequalities with interconnected bilateral obstacles, related to a multiple modes switching game. Under rather weak assumptions, using systems of penalized unilateral backward SDEs, we construct a continuous viscosity solution of polynomial growth. Moreover, we establish a comparison re...
Article
In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger...
Article
We study the optimal control for stochastic differential equations (SDEs) of mean-field type, in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman optimality prin...
Article
We suggest an explicit solution to the graduation problem that aims at designing an optimal curve fit which simultaneously behaves as a smoothing B-spline in each time interval and as the Hodrick-Prescott filter in the knots We also outline its numerical performance and compare it to the existing estimates of the standard Hodrick-Prescott filter.
Article
Full-text available
We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimali...
Article
We suggest an optimality criterion, for choosing the best smoothing parameters for an extension of the so-called Hodrick-Prescott Multivariate (HPMV) �lter. We show that this criterion admits a whole set of optimal smoothing parameters, to which belong the widely used noise-to-signal ratios. We also propose explicit consistent estimators of these n...
Article
Stocks are generally used to provide higher returns in the long run. But the dramatic fall in equity prices at the beginning of this century, triggering large underfundings in pension plans, raised the question as to whether stocks can really help mend the asset and liability mismatch. To understand some aspects of this topical issue, we examine wh...
Article
Full-text available
We consider a finite horizon optimal stopping problem related to trade-off strategies between expected profit and cost cash-flows of an investment under uncertainty. The optimal problem is first formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We then construct both a minimal an...
Article
We study optimal 2-switching and nn-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the nn-switching problem we obtain regularity results for the solutions of the...
Article
The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control probl...
Article
Full-text available
We address the issue of finding a strategy to sustain structural profitability of an investment project, whose production activity depends on the market price of a number of underlying commodities. Depending on the fluctuating prices of these commodities, the activity will either continue until the project's profitability reaches a critical low lev...
Article
Full-text available
For a controlled stochastic di¤erential equation with a …nite horizon cost functional, a necessary conditions for optimal control of degenerate di¤usions with non smooth coe¢ cients is derived. The main idea is to show that the SDE S admit a unique linearized version interpreted as its distributional derivative with respect to the initial condition...
Article
We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. Depending on the regions considered, probabilities are determined by different parts of the model.
Article
Full-text available
The univariate Hodrick-Prescott filter depends on the noise-to-signal ratio that acts as a smoothing parameter. We first propose an optimality criterion for choosing the best smoothing parameters. We show that the noise-to-signal ratio is the unique minimizer of this criterion, when we use an orthogonal parametrization of the trend, whereas it is n...
Article
Full-text available
We study singular stochastic control of a two dimensional stochastic differential equation, where the first component is linear with random and unbounded coefficients. We derive existence of an optimal relaxed control and necessary conditions for optimality in the form of a mixed relaxed-singular maximum principle in a global form. A motivating exa...
Article
Full-text available
We suggest an optimality criterium for choosing the best smoothing parameters for the so-called Hodrick-Prescott Multivariate (HPMV) filter. We show that this criterium admits a whole set of optimal smoothing parameters, to which belong the widely used noise-to-signal ratios. We also propose explicit consistent estimators of these noise-to-signal r...
Article
Full-text available
We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients....