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Yuri Kabanov

Publications and source records attributed to Yuri Kabanov.

16 recordsLinked to original sources
arXiv2026

On convergence of the Mayer problems arising in the theory of financial markets with transaction cost

The geometric approach to financial markets with proportional transaction cost prescribes to imbed a specific model (of stock market, of currency market etc.), usually given in a parametric form, into a natural framework defined by the two random processes, S and K. The first one, d-dimensional, models the price evolution of basic securities while the second one, cone-valued, describes the evolution of the solvency set. It happened that the fundamental questions -- no-arbitrage criteria, hedging problems, portfolio optimization -- can be studied in this general setting opening the door to set-valued techniques. In this note we explore, in such a general framework, the stochastic Mayer control problem, consisting in the maximization of the expected utility of the portfolio terminal wealth. We get results on continuity of the optimal value and the optimal control under price approximations in a general multi-asset framework described by the geometric formalism.

q-fin.MF↗
arXiv2024

Ruin problems with investments on a finite interval: PIDEs and their viscosity solutions

The study deals with the ruin problem when an insurance company invests its reserve in a risky asset whose the price dynamics is given by a geometric Lévy process. Considering the ruin probability as a of the capital reserve we obtain for it a partial integro-differential equation understood in a viscosity sense and prove a result on the uniqueness of the viscosity solution for a corresponding boundary value problem.

math.PR↗
arXiv2023

Ruin probabilities with investments in random environment: smoothness

The paper deals with the ruin problem of an insurance company investing its capital reserve in a risky asset with the price dynamics given by a conditional geometric Brownian motion whose parameters depend on a Markov process describing a random variations in the economic and financial environments. We prove smoothness of the ruin probability as a function of the initial capital and obtain for it an integro-differential equation.

math.PR↗
arXiv2023

Ruin Probabilities for a Sparre Andersen Model with Investments: the Case of Annuity Payments

This note is a complement to the paper by Eberlein, Kabanov, and Schmidt on the asymptotic of the ruin probability in a Sparre Andersen non-life insurance model with investments a risky asset whose price follows a geometric Lévy process. Using the techniques of semi-Markov processes we extend the result of the mentioned paper to the case of annuities and models with two-sided jumps.

math.PR↗
arXiv2022

An Axiomatic Viewpoint on the Rogers--Veraart and Suzuki--Elsinger Models of Systemic Risk

We study a model of clearing in an interbank network with crossholdings and default charges. Following the Eisenberg--Noe approach, we define the model via a set of natural financial regulations including those related with eventual default charges and derive a finite family of fixpoint problems. These problems are parameterized by vectors of binary variables. Our model combines features of the Ararat--Meimanjanov, Rogers--Veraart, and Suzuki--Elsinger networks. We develop methods of computing the maximal and minimal clearing pairs using the mixed integer-linear programming and a Gaussian elimination algorithm.

math.OC↗
arXiv2021

On ruin probabilities with investments in a risky asset with a switching regime price

We investigate the asymptotic of ruin probabilities when the company invests its reserve in a risky asset with a switching regime price. We assume that the asset price is a conditional geometric Brownian motion with parameters modulated by a Markov process with a finite number of states. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities as the initial capital tends to infinity.

math.PR↗
arXiv2020

Ruin Probabilities for a Sparre Andersen Model with Investments

We study a Sparre Andersen model in which the business activity of the company is described by a compound renewal process with drift assuming that the capital reserves are invested in a risky asset. The price of the latter is assumed to evolve according to a geometric Lévy process. We prove that the asymptotic behavior of the ruin probability depends to a large extent only on the properties of the price process.

math.PR↗
arXiv2020

On ruin probabilities with risky investments

We investigate the asymptotic of ruin probabilities when the company combines the life- and non-life insurance businesses and invests its reserve into a risky asset with stochastic volatility and drift driven by a two-state Markov process. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities.

math.PR↗
arXiv2020

Ruin probabilities with investments: smoothness, IDE and ODE, asymptotic behavior

The study deals with the ruin problem when an insurance company having two business branches, life insurance and non-life insurance, invests its reserve into a risky asset with the price dynamics given by a geometric Brownian motion. We prove a result on smoothness of the ruin probability as a function of the initial capital and obtain for it an integro-differential equation understood in the classical sense. For the case of exponentially distributed jumps we show that the survival probability is a solution of an ordinary differential equation of the 4th order. Asymptotic analysis of the latter leads to the conclusion that the ruin probability decays to zero in the same way as in the already studied cases of models with one-side jumps.

math.PR↗
arXiv2018

The ruin problem for Lévy-driven linear stochastic equations with applications to actuarial models with negative risk sums

We study the asymptotic of the ruin probability for a process which is the solution of linear SDE defined by a pair of independent Lévy processes. Our main interest is the model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let $β>0$ be the root of the cumulant-generating function $H$ of the increment of the log price process $V$. We show that the ruin probability admits the exact asymptotic $Cu^{-β}$ as the initial capital $u\to\infty$ assuming only that the law of $V_T$ is non-arithmetic without any further assumptions on the price process.

math.PR↗
arXiv2015

No arbitrage and local martingale deflators

A supermartingale deflator (resp., local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp., local martingales). The supermartingale numeraire (resp., local martingale numeraire) is the wealth processes whose reciprocal is a supermartingale deflator (resp., local martingale deflator). It has been established in previous literature that absence of arbitrage of the first kind (NA1) is equivalent to existence of the supermartingale numeraire, and further equivalent to existence of a strictly positive local martingale deflator; however, under NA1, the local martingale numeraire may fail to exist. In this work, we establish that, under NA1, any total-variation neighbourhood of the original probability has an equivalent probability under which the local martingale numeraire exists. This result, available previously only for single risky-asset models, is in striking resemblance with the fact that any total-variation neighbourhood of a separating measure contains an equivalent $σ$-martingale measure. The presentation of our main result is relatively self-contained, including a proof of existence of the supermartingale numeraire under NA1. We further show that, if the Levy measures of the asset-price process have finite support, NA1 is equivalent to existence of the local martingale numeraire with respect to the original probability.

math.PR↗
arXiv2015

In the Life Insurance Business Risky Investments are Dangerous

We investigate models of the life annuity insurance when the company invests its reserve into a risky asset with price following a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to unit identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome: the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.

math.PR↗
arXiv2015

Consumption-Investment Problem with Transaction Costs for Lévy-Driven Price Processes

We consider an optimal control problem for a linear stochastic integro-diffe\-rential equation with conic constraints on the phase variable and the control of singular-regular type. Our setting includes consumption-investment problems for models of financial markets in the presence of proportional transaction costs where the price of the assets are given by a geometric Lévy process and the investor is allowed to take short positions. We prove that the Bellman function of the problem is a viscosity solution of the HJB equation. A uniqueness theorem for the solution of the latter is established. Special attention is paid to the Dynamic Programming Principle.

math.OC↗