arXiv subjects

Lucian Maticiuc

Publications and source records attributed to Lucian Maticiuc.

16 recordsLinked to original sources

arXiv2023

Time-Delayed Generalized BSDEs

We prove the existence and uniqueness of the solution of a BSDE with time-delayed generators in the small delay setting (or equivalently small Lipschitz constant), which employs the Stieltjes integral with respect to an increasing continuous stochastic process. Moreover, we obtain a result of continuity of the solution with regard to the increasing process, assuming only uniform convergence, but not in variation. We also prove the existence in the case of an arbitrary delay by imposing monotonicity and linearity on generators. Lastly, we provide an application of the theoretical framework within an insurance based example.

math.PR↗

arXiv2019

$L^{p}$-Variational Solutions of Multivalued Backward Stochastic Differential Equations

The aim of the paper is to prove the existence and uniqueness of the $L^{p}$--variational solution, with $p>1,$ of the following multivalued backward stochastic differential equation with $p$--integrable data: \begin{equation*} \left\{ \begin{array}[c]{l} -dY_{t}+\partial_{y}Ψ(t,Y_{t})dQ_{t}\ni H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<τ,\\[0.1cm] Y_τ=η, \end{array} \right. \end{equation*} where $τ$ is a stopping time, $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}Ψ$ is the subdifferential of the convex lower semicontinuous function $y\mapstoΨ(t,y).$

math.PR↗

arXiv2019

A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: \[ \begin{cases} \partial_{t}u(t,ϕ)+\mathcal{L}u(t,ϕ)+f(t,ϕ,u(t,ϕ),\partial_{x}u(t,ϕ) σ(t,ϕ),(u(\cdot,ϕ))_{t})=0,\;t\in[0,T),\;ϕ\in\mathbbΛ\, ,u(T,ϕ)=h(ϕ),\;ϕ\in\mathbbΛ, \end{cases} \] where $\mathbbΛ=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot ,ϕ))_{t}:=(u(t+θ,ϕ))_{θ\in[-δ,0]}$ and \[ \mathcal{L}u(t,ϕ):=\langle b(t,ϕ),\partial_{x}u(t,ϕ)\rangle+\dfrac {1}{2}\mathrm{Tr}\big[σ(t,ϕ)σ^{\ast}(t,ϕ)\partial_{xx} ^{2}u(t,ϕ)\big]. \] The result is obtained by a stochastic approach. In particular we prove a new type of nonlinear Feynman-Kac representation formula associated to a backward stochastic differential equation with time-delayed generator which is of non-Markovian type. Applications to the large investor problem and risk measures via $g$-expectations are also provided.

math.PR↗

arXiv2016

Multivalued Monotone Stochastic Differential Equations with Jumps

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.

math.PR↗

arXiv2015

On the continuity of the probabilistic representation of a semilinear Neumann-Dirichlet problem

In this article we prove the continuity of the deterministic function $u:[0,T]\times \mathcal{\bar{D}}\rightarrow \mathbb{R}$, defined by $u(t,x):=Y_{t}^{t,x}$, where the process $(Y_{s}^{t,x})_{s\in[t,T]}$ is given by the generalized multivalued backward stochastic differential equation: \begin{equation*} \left\{ \begin{array}{l} -dY_{s}^{t,x}+\partial φ(Y_{s}^{t,x})ds+\partialψ(Y_{s}^{t,x})dA_{s}^{t,x}\ni f(s,X_{s}^{t,x},Y_{s}^{t,x})ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+g(s,X_{s}^{t,x},Y_{s}^{t,x})dA_{s}^{t,x}-Z_{s}^{t,x}dW_{s}~,\;t\leq s < T, \\ {Y_{T}=h(X_{T}^{t,x}).} \end{array} \right. \end{equation*} The process $(X_{s}^{t,x},A_{s}^{t,x})_{s\geq t}$ is the solution of a stochastic differential equation with reflecting boundary conditions.

math.PR↗

arXiv2015

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: \[\cases{\displaystyle -\mathrm{d}Y_t+\partial_yΨ(t,Y_t)\,\mathrm{d}Q_t\niΦ(t,Y_t,Z_t)\,\mathrm{d}Q_t-Z_t\,\mathrm{d}W_t,\qquad 0\leq t<τ,\cr \displaystyle{Y_τ=η,}}\] where $τ$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and $\partial_yΨ$ is the subdifferential of the convex lower semicontinuous function $y\longmapstoΨ(t,y)$. 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.

math.PR↗

arXiv2015

Stochastic Variational Inequalities on Non-Convex Domains

The objective of this work is to prove, in a first step, the existence and the uniqueness of a solution of the following multivalued deterministic differential equation: $dx(t)+\partial ^-φ(x(t))(dt)\ni dm(t),\ t>0$, $x(0)=x_0$, where $m:\mathbb{R}_+\rightarrow\mathbb{R}^d$ is a continuous function and $\partial^-φ$ is the Fréchet subdifferential of a semiconvex function $φ$; the domain of $φ$ can be non-convex, but some regularities of the boundary are required. The continuity of the map $m\mapsto x:C([0,T];\mathbb{R}^{d})\rightarrow C([0,T] ;\mathbb{R}^{d})$, which associate the input function $m$ with the solution $x$ of the above equation, as well as tightness criteria allow to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion: $X_t+K_t = ξ+\int_0^t F(s,X_{s})ds + \int_0^t G(s,X_s) dB_s,\; t\geq0$, $\;$ with $dK_{t}(ω)\in\partial^-φ( X_t (ω))(dt)$.

math.DS↗

arXiv2015

Cadlag Skorokhod problem driven by a maximal monotone operator

The article deals with existence and uniqueness of the solution of the following differential equation (a càdlàg Skorokhod problem) driven by a maximal monotone operator and with singular input generated by the càdlàg function $m$: \[ \left\{ \begin{array} [c]{l} dx_{t}+A\left( x_{t}\right) \left( dt\right) +dk_{t}^{d}\ni dm_{t} \,,~t\geq0,\\ x_{0}=m_{0}, \end{array} \right. \] where $k^{d}$ is a pure jump function. The jumps outside of the constrained domain $\overline{\mathrm{D}(A)}$ are counteracted through the generalized projection $Π$, by taking $x_{t}=Π(x_{t-}+Δm_{t})$, whenever $x_{t-}+Δm_{t}\notin\overline {\mathrm{D}(A)}\,$. Approximations of the solution based on discretization and Yosida penalization are considered.

math.DS↗

arXiv2013

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 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.

math.PR↗

arXiv2013

Fractional backward stochastic differential euqations and fractional backward variational inequalities

In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: [{[c]{l}% -dY(t)= f(t,η(t),Y(t),Z(t))dt-Z(t)δB^{H}(t), \quad t\in[0,T], Y(T)=ξ,.] where $η$ is a stochastic process given by $η(t)=η(0) +\int_{0}^{t}σ(s) δB^{H}(s)$, $t\in[0,T]$, and $B^{H}$ is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng's paper, \textit{BDSEs driven by fBm}, SIAM J Control Optim. (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equation [{[c]{l} -dY(t)+\partialφ(Y(t))dt\ni f(t,η(t),Y(t),Z(t))dt-Z(t)δB^{H}(t),\quad t\in[0,T], Y(T)=ξ,.] where $\partialφ$ is a multivalued operator of subdifferential type associated with the convex function $φ$.

math.PR↗

arXiv2013

Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs

In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by solutions of penalized partial differential equations.

math.PR↗

arXiv2013

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): \[ \left\{\begin{array} [c]{r} dX(t)+\partialφ\left(X(t)\right) dt\ni b\left(t,X(t),Y(t),Z(t)\right) dt+σ\left(t,X(t),Y(t),Z(t)\right)dW(t), \medskip\\ t\in(s,T],\medskip\\ \multicolumn{1}{l}{X(t)=ξ\left(t-s\right) ,\;t\in\left[ s-δ,s\right] .} \end{array} \right. \] 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 $\bar{\mathrm{Dom}\left(φ\right)}$. 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.

math.PR↗

arXiv2013

Multivalued Backward Stochastic Differential Equations with Time Delayed Generators

Our aim is to study the following new type of multivalued backward stochastic differential equation: \[ \left\{\begin{array} [c]{r}-dY\left(t\right) +\partialφ\left(Y\left(t\right)\right) dt\ni F\left(t,Y\left(t\right),Z\left(t\right),Y_{t},Z_{t}\right) dt+Z\left(t\right) dW\left(t\right),\;0\leq t\leq T,\medskip\\ \multicolumn{1}{l}{Y\left(T\right) =ξ,}\end{array} \right. \] where $\partialφ$ is the subdifferential of a convex function and $\left(Y_{t},Z_{t}\right):=(Y(t+θ),Z(t+θ))_{θ\in\lbrack-T,0]}$ 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.

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↗

arXiv2010

Viscosity solutions for systems of parabolic variational inequalities

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\[\{[c]{l}\dfrac{\partial u}{\partial t}(t,x)+\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\in\partialϕ(u(t,x)),\quad t\in[0,T),x\in\mathbb{R}^d, u(T,x)=h(x),\quad x\in\mathbb{R}^d,\] where $\partialϕ$ is the subdifferential operator of the proper convex lower semicontinuous function $ϕ:\mathbb{R}^k\to (-\infty,+\infty]$ and $\mathcal{L}_t$ is a second differential operator given by $\mathcal{L}_tv_i(x)={1/2}\operatorname {Tr}[σ(t,x)σ^*(t,x)\mathrm{D}^2v_i(x)]+< b(t,x),\nabla v_i(x)>$, $i\in\bar{1,k}$. We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u:[0,T]\times\mathbb{R}^d\to\mathbb{R}^k$ of the above parabolic variational inequality.

math.DS↗

arXiv2008

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 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*}

math.DS↗