Optimal Multi-Modes Switching Problem in Infinite Horizon

This paper studies the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solutions approach is employed to carry out a finne analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. This problem is in relation with the valuation of firms in a financial market

A general comparison theorem for 1-dimensional anticipated BSDEs

Anticipated backward stochastic differential equation (ABSDE) studied the first time in 2007 is a new type of stochastic differential equations. In this paper, we establish a general comparison theorem for 1-dimensional ABSDEs with the generators depending on the anticipated term of $Z$.Comment: 8 page

Machine Learning and Portfolio Optimization

The portfolio optimization model has limited impact in practice due to estimation issues when applied with real data. To address this, we adapt two machine learning methods, regularization and cross-validation, for portfolio optimization. First, we introduce performance-based regularization (PBR), where the idea is to constrain the sample variances of the estimated portfolio risk and return, which steers the solution towards one associated with less estimation error in the performance. We consider PBR for both mean-variance and mean-CVaR problems. For the mean-variance problem, PBR introduces a quartic polynomial constraint, for which we make two convex approximations: one based on rank-1 approximation and another based on a convex quadratic approximation. The rank-1 approximation PBR adds a bias to the optimal allocation, and the convex quadratic approximation PBR shrinks the sample covariance matrix. For the mean-CVaR problem, the PBR model is a combinatorial optimization problem, but we prove its convex relaxation, a QCQP, is essentially tight. We show that the PBR models can be cast as robust optimization problems with novel uncertainty sets and establish asymptotic optimality of both Sample Average Approximation (SAA) and PBR solutions and the corresponding efficient frontiers. To calibrate the right hand sides of the PBR constraints, we develop new, performance-based k-fold cross-validation algorithms. Using these algorithms, we carry out an extensive empirical investigation of PBR against SAA, as well as L1 and L2 regularizations and the equally-weighted portfolio. We find that PBR dominates all other benchmarks for two out of three of Fama-French data sets

Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators

The aim of the present paper is to study the regularity properties of the solution of a backward stochastic differential equation with a monotone generator in infinite dimension. We show some applications to the nonlinear Kolmogorov equation and to stochastic optimal control

On Markovian solutions to Markov Chain BSDEs

We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows us to connect these equations to coupled systems of ODEs, and hence to give fast numerical methods for the evaluation of Markov-Chain BSDEs

Paleoenvironmental and ecological changes during the Eocene-Oligocene transition based on foraminifera from the Cap Bon Peninsula in North East Tunisia

Biostratigraphic analysis of the Eocene-Oligocene transition (E-O) at the Menzel Bou Zelfa and Jhaff composite section in the Cap Bon Peninsula (North East Tunisia) allowed us to recognize a continuous planktic foraminiferal biozonation: E14 Globigerinatheka semiinvoluta Zone, E15 Globigerinatheka index Zone, E16 Hantkenina alabamensis Zone and O1 Pseudohastigerina naguewichiensis Zone. A quantitative study of benthic and planktic foraminifera assemblages was carried out and the richness and diversity of foraminifera allowed us to reconstruct the paleoenvironmental evolution from marine to terrestrial environments. From the Eocene E14 Zone, the foraminiferal association characterizes a relatively warm climate with considerable oxygen content and a dominance of keeled and spinose planktic foraminifera, which became extinct at the E/O boundary, possibly due to cooling of the planktic environment. Nevertheless, the small benthic foraminifera do not show an extinction event at the Eocene/Oligocene (E/O) boundary, indicating that the benthic environment was not significantly affected. In the basal Oligocene O1 Zone, the benthic environment changes to a shallower setting due to cooling of the climate. These changes generated a remarkable dominance of globular forms in the planktic environment. Small benthic foraminifera apparently have a gradual extinction event, or more likely a gradual pattern of local disappearances, that could have been caused by the Oi1 glaciation

An overview of Viscosity Solutions of Path-Dependent PDEs

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12]

Planktic foraminiferal biostratigraphy, paleoecology and chronostratigraphy across the Eocene/Oligocene boundary in northern Tunisia

The biostratigraphic analysis of the Eocene-Oligocene transition of the Menzel Bou Zelfa and Jhaff sections in northeastern Tunisia (Cap Bon peninsula) allows us to identify a continuous planktic foraminiferal biozonation. The following biozones were recognized: Globigerinatheka semiinvoluta Zone (E14), Globigerinatheka index Zone (E15), (Hantkenina alabamensis Zone (E16) of the upper Eocene and Pseudohastigerina naguewichiensis Zone (O1) of the lower Oligocene. A rapid mass extinction event in planktic foraminifera occurred at the Eocene-Oligocene transition, including the extinction of the turborotalids (Turborotalia cerroazulensis, Turborotalia cocoaensis and Turborotalia cunialensis) followed by a significant size reduction of the genus Pseudohastigerina and the extinction of the hantkeninids (Hantkenina alabamensis, Hantkenina brevispina, Hantkenina nanggulanensis and Cribrohantkenina lazzarii), which mark the Eocene/Oligocene boundary. These species were tropical and subtropical surface and intermediate dwellers, with distinctive morphologies (carinate turborotalids and spinose hantkeninids), which were well adapted species of k-strategy. The surviving planktic foraminifera species were quite similar in morphology with globular chambers (globigerinids) and small planispiral (pseudohastigerinids), which were opportunistic species of r-strategy. The recognition of a 4 m thick interval, between the extinction of turborotalids and hantkeninids, indicates that the section is continuous and one of the most expanded throughout the Eocene-Oligocene transition. This section could serve as an auxiliary section (hypostratotype) for the complete definition of the Global Stratotype Section and Point for the Eocene/Oligocene boundary, which mark the base of the Rupelian Stage

Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems

This paper deals with existence and uniqueness, in viscosity sense, of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case of this system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is connected with the valuation of a power plant in the energy market. The main tool is the notion of systems of reflected BSDEs with oblique reflection.Comment: 36 page

Fractional smoothness and applications in finance

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages