arXiv subjects

Adrian Zălinescu

Publications and source records attributed to Adrian Zălinescu.

4 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↗

arXiv2022

Feynman-Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations

We consider a system of Forward Backward Stochastic Differential Equations (FBSDEs), with time delayed generator and driven by Lèvy-type noise. We establish a non linear Feynman Kac representation formula associating the solution given by the FBSDEs-system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization of the so-called Large Investor Problem where the stock price evolves according to a jump-diffusion dynamic.

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↗

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↗