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Viscosity solutions of integro-differential equations for nonruin probabilities
Theory of Probability and Its Applications. 2016. Vol. 60. No. 4. P. 671–679.
Belkina T. A., Kabanov Y.
We consider a model of an insurance company investing its reserve into a risky asset whose price follows a geometric Lévy process. We show that the nonruin probability is a viscosity solution of a second order integro-differential equation and prove a uniqueness theorem for the latter.
Averboukh Y., / Cornell University. Серия arXiv "math". 2025. № 2506.21373.
This work extends weak KAM theory to the case of a nonsmooth Lagrangian satisfying a superlinear growth condition. Using the solution of a weak KAM equation that is a stationary Hamilton-Jacobi equation and the proximal aiming method, we construct a family of discontinuous feedback strategies that are nearly optimal for every time interval. This result ...
Added: November 28, 2025
Kabanov Y., Лёгенький Д. В., Промыслов П. В., / Series arXiv "math". 2025.
This note is an addendum to the work initiated by Eberlein, Kabanov, and Schmidt and developed further by Kabanov and Promyslov on the asymptotics of the ruin probabilities in the Sparre Andersen model with investments in a risky asset. Using more advanced methods of the implicit renewal theory, we provide complements to some results of ...
Added: June 30, 2025
Kabanov Y., Промыслов П. В., / Series arXiv "math". 2023.
Added: June 30, 2025
Kabanov Y., Промыслов П. В., Finance and Stochastics 2023 Vol. 27 No. 4 P. 887–902
This note is a complement to the paper (Stoch. Process. Appl. 144:72–84, 2022) by Eberlein, Kabanov and Schmidt on the asymptotics of the ruin probability in a Sparre Andersen non-life insurance model with investments into a risky asset whose price follows a geometric Lévy process. Using techniques from the theory of semi-Markov processes, we extend ...
Added: June 30, 2025
Ascione G., Mehrdoust F., Orlando G. et al., Applied Mathematics and Computation 2023 No. 446 Article 127851
In this paper, we consider the Heston-CIR model with Lévy process for pricing in the foreign exchange (FX) market by providing a new formula that better fits the distribution of prices. To do that, first, we study the existence and uniqueness of the solution to this model. Second, we examine the strong convergence of the ...
Added: February 16, 2024
Averboukh Y., Nonlinear Differential Equations and Applications 2021 Vol. 28 No. 6 Article 65
The theory of first-order mean field type differential games examines the systems of infinitely many identical agents interacting via some external media under assumption that each agent is controlled by two players. We study the approximations of the value function of the first-order mean field type differential game using solutions of model finite-dimensional differential games. ...
Added: October 20, 2021
Alexander Gushchin, Pavlyukevich I., Ritsch M., Statistical Inference for Stochastic Processes 2020 Vol. 23 No. 3 P. 553–570
We consider the problem of estimation of the drift parameter of an ergodic Ornstein–
Uhlenbeck type process driven by a Lévy process with heavy tails. The process is observed
continuously on a long time interval [0, T ], T →∞. We prove that the statistical model is
locally asymptotic mixed normal and the maximum likelihood estimator is asymptotically
efficient. ...
Added: October 27, 2020
Галкин Е. Г., Nikitin A. A., Вестник Московского университета. Серия 15: Вычислительная математика и кибернетика 2020 № 2 С. 11–18
The article presents the main approaches to the study of the stochastic process of population dynamics with continuous time and space and with fixed individuals, a countable system of integro-differential equations corresponding to the dynamics of spatial moments of this process is derived, and a method for finding an approximate solution using the method of ...
Added: February 18, 2020
Галкин Е. Г., Зеленков В. К., Nikitin A. A., International Journal of Open Information Technologies 2019 Т. 7 № 12 С. 18–23
This publication begins a series of works devoted to the comparison of the results of numerical methods for solving integral equations and the results of computer simulations using Poisson processes. The pros and cons of these two approaches are highlighted. The main subject of study is the model of two-species communities proposed in the works ...
Added: December 8, 2019
Moreno-Franco H. A., Applied Mathematics and Optimization 2018 Vol. 78 No. 1 P. 25–60
The main goal of this paper is to establish existence, regularity and uniqueness results for the solution of a Hamilton–Jacobi–Bellman (HJB) equation, whose operator is an elliptic integro-differential operator. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a controlled d-dimensional Lévy process. ...
Added: October 12, 2016
Khanin K., Sobolevski A., Archive for Rational Mechanics and Analysis 2016 Vol. 219 No. 2 P. 861–885
Characteristic curves of a Hamilton–Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth “viscosity” solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian ...
Added: August 18, 2015
Gushchin A. A., Küchler U., Stochastic Processes and their Applications 2000 Vol. 88 No. 2 P. 195–211
Let a be a finite signed measure on [-r,0], Z a Lévy process (that is a real process with independent stationary increments and càdlàg paths). A linear stochastic delay differential equation
X(t)=X(0)+∫ 0 t ∫ [-r,0] X(s+u)da(u)ds+Z(t),t≥0,(1)
driven by Z is studied, only càdlàg solutions to (1) such that Z and (X(t),-r≤t≤0) are independent being considered. Set ...
Added: October 8, 2013