arXiv subjects

L. Vostrikova

Publications and source records attributed to L. Vostrikova.

9 recordsLinked to original sources

arXiv2018

On distributions of exponential functionals of the processes with independent increments

The aim of this paper is to study the laws of the exponential functionals of the processes $X$ with independent increments, namely $$I_t= \int _0^t\exp(-X_s)ds, \,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ Under suitable conditions we derive the integro-differential equations for the density of $I_t$ and $I_{\infty}$. We give sufficient conditions for the existence of smooth density of the laws of these functionals. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.

math.PR↗

arXiv2018

On exponential functionals of processes with independent increments

In this paper we study the exponential functionals of the processes $X$ with independent increments , namely $$I_t= \int _0^t\exp(-X_s)ds, _,\,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ When $X$ is a semi-martingale with absolutely continuous characteristics, we derive recurrent integral equations for Mellin transform ${\bf E}( I_t^α)$, $α\in\mathbb{R}$, of the integral functional $I_t$. Then we apply these recurrent formulas to calculate the moments. We present also the corresponding results for the exponential functionals of Levy processes, which hold under less restrictive conditions then in the paper of Bertoin, Yor (2005). In particular, we obtain an explicit formula for the moments of $I_t$ and $I_{\infty}$, and we precise the exact number of finite moments of $I_{\infty}$.

math.PR↗

arXiv2011

$F$-divergence minimal equivalent martingale measures and optimal portfolios for exponential Levy models with a change-point

We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-point model and we give the conditions for the existence of f-divergence minimal equivalent martingale measure. Using the connection between utility maximisation and $f$-divergence minimisation, we obtain a general formula for optimal strategy in change-point case for initially enlarged filtration and also for progressively enlarged filtration in the case of exponential utility. We illustrate our results considering the Black-Scholes model with change-point.

q-fin.PM↗

arXiv2011

Levy preservation and associated properties for $f$-divergence minimal equivalent martingale measures

We study such important properties of $f$-divergence minimal martingale measure as Levy preservation property, scaling property, invariance in time property for exponential Levy models. We give some useful decomposition for $f$-divergence minimal martingale measures and we answer on the question which form should have $f$ to ensure mentioned properties. We show that $f$ is not necessarily common $f$-divergence. For common $f$-divergences, i.e. functions verifying $f"(x) = ax^ γ,\, a>0,\, γ\in \mathbb R$, we give necessary and sufficient conditions for existence of $f$-minimal martingale measure.

math.PR↗

arXiv2010

An $f$-divergence approach for optimal portfolios in exponential Levy models

We present a unified approach to get explicit formulas for utility maximising strategies in Exponential Levy models. This approach is related to $f$-divergence minimal martingale measures and based on a new concept of preservation of the Levy property by $f$-divergence minimal martingale measures. For common $f$-divergences, i.e. functions which satisfy $f"(x)= ax^ γ,\, a>0, \, γ\in \mathbb R$, we give the conditions for the existence of corresponding $u_f$- maximising strategies, as well as explicit formulas.

math.PR↗

arXiv2009

On continuity properties for option prices in exponential Lévy models

For a converging sequence of exponential Lévy models, we give conditions under which the associated sequence of option prices converges. We also study the behaviour of the prices when no such convergence holds. We then consider two special cases, first when the martingale measure is chosen by minimisation of entropy and then when it minimises Hellinger integrals.

math.PR↗

arXiv2009

On the stability of call/put option's prices in incomplete models under statistical estimations

In exponential semi-martingale setting for risky asset we estimate the difference of prices of options when initial physical measure $P$ and corresponding martingale measure $Q$ change to $\tilde{P}$ and $\tilde{Q}$ respectively. Then, we estimate $L_1$-distance of option's prices for corresponding parametric models with known and estimated parameters. The results are applied to exponential Levy models with special choice of martingale measure as Esscher measure, minimal entropy measure and $f^q$-minimal martingale measure. We illustrate our results by considering GMY and CGMY models.

math.PR↗

arXiv2009

On regularity properties of Bessel flow

We study the differentiability of Bessel flow $ρ: x \to ρ^x_t$, where $(ρ^x_t)_{t\geq 0}$ is BES $^x(δ$) process of dimension $δ>1$ starting from $x$. For $δ\geq 2$ we prove the existence of bicontinuous derivatives in P-a.s. sense at $x\geq 0$ and we study the asymptotic behaviour of the derivatives at $x=0$. For $1< δ<2$ we prove the existence of a modification of Bessel flow having derivatives in probability sense at $x\geq 0$. We study the asymptotic behaviour of the derivatives at $t=τ_0(x)$ where $τ_0(x)$ is the first zero of $(ρ^x_t)_{t\geq 0}$.

math.PR↗

arXiv2008

Reflection principle and Ocone martingales

Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $σ$-field of the invariant events by all reflections at levels $a_n$, $n\ge1$ is trivial. We establish similar results for skip free $\mathbb{Z}$-valued processes and use them for the proof in continuous time, via a discretisation in space.

math.PR↗