Affiliations: International Center for Decision and Risk Analysis, School of Management, University of Texas at Dallas, Richardson, TX, USA | Graduate School of Business, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China | Graduate Department of Financial Engineering, Ajou University, Suwon, South Korea. E-mail: alain.bensoussan@utdallas.edu | Departamento de Matemáticas, Cinvestav-IPN, México, D.F. México. E-mail: hjasso@math.cinvestav.mx | Université Denis Diderot-Paris 7, Paris, France. E-mail: menozzi@math.univ-paris-diderot.fr | Department of Statistics, The Chinese University of Hong Kong, Hong Kong. E-mail: mertz@sta.cuhk.edu.hk
Note: [] Corresponding author: Stéphane Menozzi, Université Denis Diderot-Paris 7, LPMA, 175 rue du Chevaleret, Paris 75013, France. E-mail: menozzi@math.univ-paris-diderot.fr.
Abstract: In a previous work by the first author with J. Turi [Appl. Math. Optim. 58(1) (2008), 1–27], a stochastic variational inequality has been introduced to model an elasto-plastic oscillator with noise. A major advantage of the stochastic variational inequality is to overcome the need to describe the trajectory by phases (elastic or plastic). This is useful, since the sequence of phases cannot be characterized easily. In particular, when a change of regime occurs, there are numerous small elastic phases which may appear as an artefact of the Wiener process. However, it remains important to have informations on both the elastic and plastic phases. In order to reconcile these contradictory issues, we introduce an approximation of stochastic variational inequalities by imposing artificial small jumps between phases allowing a clear separation of the elastic and plastic regimes. In this work, we prove that the approximate solution converges on any finite time interval, when the size of jumps tends to 0.
