The criterion and sufficient condition for the existence of moments of one-parameter increasing predictable processes is presented in terms of an associated potential. The estimates of moments of special functional connected with two-parameter increasing predictable processes are given in the 
case when the associated potential is bounded. The application of these 
estimates to the local time for purely discontinuous strong martingales in 
the plane is also presented.

Yu.  Mishura

Ya.  Oltsik

Existence of moments of increasing predictable processes associated with one- and two-parameter potentials

In this paper we study an abstract stochastic equation of second order and 
stochastic boundary problem for the telegraph equation in a strip. We 
prove the existence of solutions, which are <mml:math alttext="$d$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-periodic (periodic in distribution) random processes.

A. Ya. Dorogovtsev

Periodic in distribution solution for a telegraph equation

Statistical analysis on various stocks reveals long range dependence behavior of the stock prices that is not consistent with the classical Black and 
Scholes model. This memory or nondeterministic trend behavior is often 
seen as a reflection of market sentiments and causes that the historical volatility estimator becomes unreliable in practice. We propose an extension 
of the Black and Scholes model by adding a term to the original Wiener 
term involving a <em>smoother</em> process which accounts for these effects. The 
problem of arbitrage will be discussed. Using a generalized stochastic integration theory [8], we show that it is possible to construct a self financing 
replicating portfolio for a European option without any further knowledge 
of the extension and that, as a consequence, the classical concept of volatility needs to be re-interpreted.

John G. M. Schoenmakers

Peter E. Kloeden

Robust option replication for a Black-Scholes model extended with nondeterministic trends

We consider a Gaussian process <mml:math alttext="$\left\{ {X_t ,t \in T} \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math> with an arbitrary index set <mml:math alttext="$T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> 
and study consequences of transformations of the index set on the Skorokhod integral and Skorokhod derivative with respect to <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>. The results applied to Skorokhod SDEs of diffusion type provide uniqueness of the solution 
for the time-reversed equation and, to Ogawa line integral, give an analogue of the fundamental theorem of calculus.

Leszek  Gawarecki

Transformations of index set for Skorokhod integral with respect to Gaussian processes

This is a brief biographical account on late Julian Keilson, his scientific 
contributions, and his collaboration with the author.

Ryszard  Syski

In memoriam: Julian Keilson (November 19, 1924 - March 8, 1999)

An asymptotic method for stability analysis of quasilinear functional differential equations, with small perturbations dependent on phase coordinates 
and an ergodic Markov process, is presented. The proposed method is 
based on an averaging procedure with respect to: 1) time along critical 
solutions of the linear equation; and 2) the invariant measure of the 
Markov process. For asymptotic analysis of the initial random equation 
with delay, it is proved that one can approximate its solutions (which are 
stochastic processes) by corresponding solutions of a specially constructed 
averaged, deterministic ordinary differential equation. Moreover, it is 
proved that exponential stability of the resulting deterministic equation is 
sufficient for exponential <mml:math alttext="$p$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-stability of the initial random system for all 
positive numbers <mml:math alttext="$p$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>, and for sufficiently small perturbation terms.

Lambros Katafygiotis

Yevgeny Tsarkov

Averaging and stability of quasilinear functional differential equations with Markov parameters

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International Journal of Stochastic Analysis - Table of contents 1999

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International Journal of Stochastic Analysis

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International Journal of Stochastic Analysis