arXiv subjects

Eduard Rotenstein

Publications and source records attributed to Eduard Rotenstein.

9 recordsLinked to original sources

arXiv2026

C\`adl\`ag Solutions to Backward Stochastic Dynamics featuring Oblique Subgradients and driven by Martingale Noise

The present study improves the qualitative analysis of backward stochastic variational dynamics on a general complete filtered probability space, considered in the spirit of Liang, Lyons and Qian (2011). Our primary objective is to overcome a substantial limitation in the study of Bensoussan, Li and Yam (2018), where the boundedness condition imposed on the multivalued subdifferential operator excludes standard obstacle-type constraints and indicator functions of convex sets. We prove the existence and uniqueness of a strong c\`adl\`ag solution under the natural assumption that the driving proper lower semicontinuous convex function is merely bounded from below by an affine/quadratic function. Furthermore, we incorporate an oblique reflection governed by a time-dependent, uniformly positive definite symmetric matrix, in the spirit of the pioneering results of Gassous, R\u{a}\c{s}canu and Rotenstein (2012, 2015).

math.PR↗

arXiv2026

Planar Obliquely Reflected BSVIs on General Filtered Spaces: Non-Symmetric Rotation Fields and Associated Control Problems

We prove existence and uniqueness of a c\`adl\`ag solution to a planar backward stochastic variational inequality on a general complete filtered probability space, driven by a square integrable martingale, which may have jumps. The multivalued term is the exterior normal cone operator of a bounded uniformly convex planar domain, and the reflection direction is generated by a time-dependent non-symmetric rotation field. The non-symmetry creates a first-order tangential boundary term that destroys the standard monotonicity and quadratic contraction estimates, used in the symmetric oblique-reflection theory. We overcome this obstruction by constructing an explicit symmetric two-point kernel with state dependent coefficients, whose boundary derivative cancels the leading tangential contribution. A weighted martingale-exponential estimate then controls the second order defect and the jump terms. Under suitable geometric and quantitative compatibility constraints, we obtain the unique strong c\`adl\`ag solution. We also formulate associated control problems for the rotation angle and prove existence of an optimal control in a compact class of bounded-rate angle paths.

math.PR↗

arXiv2016

Approximate and Approximate Null-Controllability of a Class of Piecewise Linear Markov Switch Systems

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in [13]. The proof relies on recent results in [4] allowing to reduce the dual stochastic backward system to a family of ordinary differential equations. Second, we prove by examples that the notion of approximate controllability is strictly stronger than approximate null-controllability. A sufficient criterion for this stronger notion is also provided. The results are illustrated on a model derived from repressed bacterium operon (given in [19] and reduced in [5]).

math.OC↗

arXiv2013

Pricing financial derivatives by a minimizing method

We shall study backward stochastic differential equations and we will present a new approach for the existence of the solution. This type of equation appears very often in the valuation of financial derivatives in complete markets. Therefore, the identification of the solution as the unique element in a certain Banach space where a suitably chosen functional attains its minimum becomes interesting for numerical computations.

math.OC↗

arXiv2013

Multivalued backward stochastic differential equations with oblique subgradients

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.

math.PR↗

arXiv2011

Stochastic variational inequalities with oblique subgradients

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.

math.PR↗

arXiv2011

Numerical Schemes for Multivalued Backward Stochastic Differential Systems

We define some approximation schemes for different kinds of generalized backward stochastic differential systems, considered in the Markovian framework. We propose a mixed approximation scheme for a decoupled system of forward reflected SDE and backward stochastic variational inequality. We use an Euler scheme type, combined with Yosida approximation techniques.

math.PR↗

arXiv2009

The Fitzpatrick function - a bridge between convex analysis and multivalued stochastic differential equations

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.

math.OC↗

arXiv2009

A Generalized Mixed Zero-sum Stochastic Differential Game and Double Barrier Reflected BSDEs with Quadratic Growth Coefficient

This article is dedicated to the study of mixed zero-sum two-player stochastic differential games in the situation when the player's cost functionals are modeled by doubly controlled reflected backward stochastic equations with two barriers whose coefficients have quadratic growth in Z. This is a generalization of the risk-sensitive payoffs. We show that the lower and the upper value function associated with this stochastic differential game with reflection are deterministic and they are also the unique viscosity solutions for two Isaacs equations with obstacles.

math.OC↗