arXiv subjects
Publications and source records attributed to Aurel Rascanu.
We study multivalued stochastic differential equations (MSDEs) with maximal monotone operators driven by semimartingales with jumps. We discuss in detail some methods of approximation of solutions of MSDEs based on discretization of processes and Yosida approximation of the monotone operator. We also study the general problem of stability of solutions of MSDEs with respect to the convergence of driving semimartingales.
It is well-known since the work of Pardoux and Peng [12] that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman-Kac formula to semilinear PDEs, see also Pardoux and Rascanu [14]. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang [15]. However, the proof of continuity of the extended Feynman-Kac formula with respect to x (resp. to (t,x)) is not correct in that paper. Here we consider a more general situation, where both the equation and the boundary condition involve the (possibly multivalued) gradient of a convex function. We prove the required continuity. The result for the class of equations studied in [15] is a Corollary of our main results.
This work deals with a Skorokhod problem driven by a maximal operator: \begin{aligned} &du(t)+Au(t)(dt)\ni f(t)dt+dM(t), \; 0<t<T,\\ &u(0)=u_{0}, \end{aligned} which is a multivalued deterministic differential equation with a singular inputs $dM(t)$, where $t\rightarrow M(t)$ is a continuous function. The existence and uniqueness result is used to study an Itô's stochastic differential equation \begin{aligned} &du(t)+Au(t)(dt)\ni f(t,u(t))dt+B(t,u(t))dW(t),\; 0<t<T,\\ &u(0)=u_{0}, \end{aligned} in a real Hilbert space $H$, where $A$ is a multivalued ($α$-)maximal monotone operator on $H$, and $f(t,u)$ and $B(t,u)$ are Lipschitz continuous with respect to $u$. Some asymptotic properties in the stochastic case are also found.
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 φ(Y_{t})dt \ni F(t,Y_{t},Z_{t})dt-Z_{t}dB_{t},\;0\leq t<T \\ Y_{T}=η, \end{array} \right.\end{equation} where $F$ satisfies a local boundedness condition.
We study the existence and uniqueness of the solution for the following backward stochastic variational inequality with oblique reflection (for short, $BSVI\left(H(t,y),φ,F\right)$), written under differential form \[ \left\{\begin{array} [c]{l}% -dY_{t}+H\left(t,Y_{t}\right) \partialφ\left(Y_{t}\right) \left(dt\right) \ni F\left(t,Y_{t},Z_{t}\right) dt-Z_{t}dB_{t},\quad t\in\left[ 0,T\right] ,\smallskip\\ Y_{T}=η, \end{array} \right. \] where $H$ is a bounded symmetric smooth matrix and $φ$ is a proper convex lower semicontinuous function, with $\partialφ$ being its subdifferential operator. The presence of the product $H\partialφ$ does not permit the use of standard techniques because it does conserve neither the Lipschitz property of the matrix nor the monotonicity property of the subdifferential operator. We prove that, if we consider the dependence of $H$ only on the time, the equation admits a unique strong solution and, allowing the dependence also on the state of the system, the above $BSVI\left(H(t,y),φ,F\right)$ admits a weak solution in the sense of the Meyer-Zheng topology. However, for that purpose we must renounce at the dependence on $Z$ for the generator function and we situate our problem in a Markovian framework.
In this paper we will study the existence and uniqueness of the solution for the stochastic variational inequality with oblique subgradients of the following form:{l} dX_{t}+H(X_{t}) \partial ϕ(X_{t}) (dt) \ni f(t,X_{t}) dt+g(t,X_{t}) dB_{t},\quad t>0,\smallskip \ X_{0}=x\in \bar{\emph{Dom}(ϕ)}.% This problem is the generalization of the stochastic differential equation with oblique reflection considered by Lions and Sznitman in `84. The existence result is based on a deterministic approach; first, we prove the existence and uniqueness of the solution of a differential system with singular input.
In this paper, we study direct and inverse images for fractional stochastic tangent sets and we establish the deterministic necessary and sufficient conditions that guarantee that the solution of a given stochastic differential equation driven by the fractional Brownian motion evolves in some particular sets $K$. A comparison theorem is derived.
Using the Fitzpatrick function, we characterize the solutions for different classes of deterministic and stochastic differential equations driven by maximal monotone operators (or in particular subdifferential operators) as the minimum point of a suitably chosen convex lower semicontinuous function. Such technique provides a new approach for the existence of the solutions for the considered equations.
In this paper we prove a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion with Hurst parameter1/2 < H < 1, using pathwise approach. The sufficient condition is also an alternative global existence result for the fractional differential equations with restrictions on the state.
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 ϕ}\big(u(t,x)\big)\ni f\big(t,x,u(t,x),(\nabla uσ)(t,x)\big), t>0, x\in \mathcal{D},\medskip \multicolumn{1}{l}{\dfrac{\partial u(t,x)}{\partial n}+{\partial ψ}\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 $\partial ϕ$ and $\partial ψ$ are subdifferentials operators and $\mathcal{L}_{t}$ is a second differential operator. The result is obtained by a Feynman-Kaç 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 ϕ(Y_{t}) dt{+}\partial ψ(Y_{t}) dA_{t}{+}Z_{t}dW_{t}, 0\leq t\leq T,\medskip \ Y_{T}=ξ.% {array}%. {equation*}