Multivalued Backward Stochastic Differential Equations with Time Delayed Generators

Our aim is to study the following new type of multivalued backward stochastic differential equation: \left\{\begin{array} [c]{r}-dY\left(t\right) +\partial\varphi\left(Y\left(t\right)\right) dt\ni F\left(t,Y\left(t\right),Z\left(t\right),Y_{t},Z_{t}\right) dt+Z\left(t\right) dW\left(t\right),\;0\leq t\leq T,\medskip\\ \multicolumn{1}{l}{Y\left(T\right) =\xi,}\end{array} \right. where $\partial\varphi$ is the subdifferential of a convex function and $\left(Y_{t},Z_{t}\right):=(Y(t+\theta),Z(t+\theta))_{\theta\in\lbrack-T,0]}$ represent the past values of the solution over the interval $\left[ 0,t\right]$. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.Comment: 14 page

Multivalued Stochastic Delay Differential Equations and Related Stochastic Control Problems

We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type): \left\{\begin{array} [c]{r} dX(t)+\partial\varphi\left(X(t)\right) dt\ni b\left(t,X(t),Y(t),Z(t)\right) dt+\sigma\left(t,X(t),Y(t),Z(t)\right)dW(t), \medskip\\ t\in(s,T],\medskip\\ \multicolumn{1}{l}{X(t)=\xi\left(t-s\right) ,\;t\in\left[ s-\delta,s\right] .} \end{array} \right. Specify that in this case the coefficients at time $t$ depends also on previous values of $X\left(t\right)$ through $Y(t)$ and $Z(t)$. Also $X$ is constrained with the help of a bounded variation feedback law $K$ to stay in the convex set $\bar{\mathrm{Dom}\left(\varphi\right)}$. Afterwards we consider optimal control problems where the state $X$ is a solution of a controlled delay stochastic system as above. We establish the dynamic programming principle for the value function and finally we prove that the value function is a viscosity solution for a suitable Hamilton-Jacobi-Bellman type equation.Comment: 29 page

Backward stochastic variational inequalities with locally bounded generators

The paper deals with the existence and uniqueness of the solution of the backward stochastic variational inequality: \begin{equation} \left\{\begin{array} {l}-dY_{t}+\partial \varphi(Y_{t})dt \ni F(t,Y_{t},Z_{t})dt-Z_{t}dB_{t},\;0\leq t<T \\ Y_{T}=\eta, \end{array} \right.\end{equation} where $F$ satisfies a local boundedness condition.Comment: Minor edits and a slight change to title have been mad

Backward stochastic variational inequalities on random interval

The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly infinite, time interval: \cases{\displaystyle -\mathrm{d}Y_t+\partial_y\Psi (t,Y_t)\,\mathrm{d}Q_t\ni\Phi (t,Y_t,Z_t)\,\mathrm{d}Q_t-Z_t\,\mathrm{d}W_t,\qquad 0\leq t<\tau,\cr \displaystyle{Y_{\tau}=\eta,}} where $\tau$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and $\partial_y\Psi$ is the subdifferential of the convex lower semicontinuous function $y\longmapsto\Psi (t,y)$. As applications, we obtain from our main results applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.Comment: Published at http://dx.doi.org/10.3150/14-BEJ601 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Stochastic approach for a multivalued Dirichlet-Neumann problem

We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality with a nonlinear multivalued Neumann-Dirichlet boundary condition:% {equation*} \{{array}{r} \dfrac{\partial u(t,x)}{\partial t}-\mathcal{L}_{t}u(t,x) {+}{% \partial \phi}\big(u(t,x)\big)\ni f\big(t,x,u(t,x),(\nabla u\sigma)(t,x)\big), t>0, x\in \mathcal{D},\medskip \multicolumn{1}{l}{\dfrac{\partial u(t,x)}{\partial n}+{\partial \psi}\big(% u(t,x)\big)\ni g\big(t,x,u(t,x)\big), t>0, x\in Bd(\mathcal{D}%),\multicolumn{1}{l}{u(0,x)=h(x), x\in \bar{\mathcal{D}},}% {array}%. {equation*}% where $\partial \phi$ and $\partial \psi$ are subdifferentials operators and $\mathcal{L}_{t}$ is a second differential operator. The result is obtained by a Feynman-Ka\c{c} representation formula starting from the backward stochastic variational inequality:% {equation*} \{{array}{l} dY_{t}{+}F(t,Y_{t},Z_{t}) dt{+}G(t,Y_{t}) dA_{t}\in \partial \phi (Y_{t}) dt{+}\partial \psi (Y_{t}) dA_{t}{+}Z_{t}dW_{t}, 0\leq t\leq T,\medskip \ Y_{T}=\xi .% {array}%. {equation*}Comment: 29 page

THE COMPLEX RELATION BETWEEN CLUSTERS AND INNOVATION IN EUROPEAN UNION

In this paper we proposed to analyze the dynamics of innovation in the European Union countries in order to observe its implications in clusters. Due to the multiple links between cluster members, the innovation transfer is achieved much easier, contributing to the rapid spread of innovative ideas, technologies, labor and know-how. The opportunity for innovation is easier noticeable due to the diversity of the cluster members that operate in a competitive environment, the permanent contact that is created with other companies and institutions allowing just overcome competitive pressure by innovation. Therefore, we analyzed the links between innovation and clusters in the countries of Eastern Europe and those in Western Europe. For this we made a comparison between the most innovative countries and the less innovative ones, identifying the benefits of clusters as a means of enhancing the innovation capacity of each state