This paper considers the pricing of a European
option using a 
<svg style="vertical-align:-2.21957pt;width:38.087502px;" id="M1" height="13.45" version="1.1" viewBox="0 0 38.087502 13.45" width="38.087502"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
		<g transform="translate(72,-61.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝐵</tspan>
				<tspan style="font-size: 12.50px; " x="12.865587" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="18.066835" y="0">𝑆</tspan>
				<tspan style="font-size: 12.50px; " x="26.193785" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>-market 
in which the stock price and the asset in the riskless bank 
account both have hereditary price structures described by 
the authors of this paper (1999). Under the smoothness assumption of the 
payoff function, it is shown that the infinite dimensional 
Black-Scholes equation possesses a unique classical solution. A 
spectral approximation scheme is developed using the Fourier 
series expansion in the space <svg style="vertical-align:-1.95624pt;width:55.075001px;" id="M2" height="13.2375" version="1.1" viewBox="0 0 55.075001 13.2375" width="55.075001"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.2375)">
		<g transform="translate(72,-61.41)">
			<text transform="matrix(1,0,0,-1,-71.95,63.41)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝐶</tspan>
				<tspan style="font-size: 12.50px; " x="8.389513" y="0">[</tspan>
				<tspan style="font-size: 12.50px; " x="12.553012" y="0">−</tspan>
				<tspan style="font-size: 12.50px; " x="21.117567" y="0">ℎ</tspan>
				<tspan style="font-size: 12.50px; " x="28.331799" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="33.533047" y="0">0</tspan>
				<tspan style="font-size: 12.50px; " x="39.784546" y="0">]</tspan>
			</text>

		</g>
	</g>
</svg> for the Black-Scholes equation. It is also shown that the <svg style="vertical-align:-0.10033pt;width:7.8874998px;" id="M3" height="7.1374998" version="1.1" viewBox="0 0 7.8874998 7.1374998" width="7.8874998"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,7.1375)">
		<g transform="translate(72,-66.29)">
			<text transform="matrix(1,0,0,-1,-71.95,66.44)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg>th approximant resembles the classical Black-Scholes equation in finite
dimensions.

Mou-Hsiung Chang

Roger K. Youree

Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory

We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.

A. L. Marhoune

F. Lakhal

A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations

We investigate the arbitrage-free property of stock price models where the local
martingale component is based on an ergodic diffusion with a specified stationary distribution. These models are particularly useful for long horizon asset-liability management as they allow the modelling of long term stock returns with heavy tail ergodic diffusions, with tractable, time homogeneous dynamics, and which moreover admit a complete financial market, leading to unique pricing and hedging strategies. Unfortunately the standard specifications of these models in literature admit arbitrage opportunities. We investigate in detail the features of the existing model specifications which create these arbitrage opportunities and consequently construct a modification that is arbitrage free.

Bernard Wong

On Modelling Long Term Stock Returns with Ergodic Diffusion Processes: Arbitrage and Arbitrage-Free Specifications

The valuation and hedging of defaultable game options is studied in a hazard process model of credit risk. A convenient pricing formula with respect to a reference filteration is derived. A connection of arbitrage prices with a suitable notion of hedging is obtained. The main result shows that the arbitrage prices are the minimal superhedging prices with sigma martingale cost under a risk neutral measure.

Tomasz R. Bielecki

Stéphane Crépey

Monique Jeanblanc

Marek Rutkowski

Defaultable Game Options in a Hazard Process Model

We consider the interloss times in the <svg style="vertical-align:-0.17555pt;width:60.200001px;" id="M2" height="10.9125" version="1.1" viewBox="0 0 60.200001 10.9125" width="60.200001"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
		<g transform="translate(72,-63.27)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
				<tspan style="font-size: 12.50px; " x="12.565515" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="16.041349" y="0">𝑀</tspan>
				<tspan style="font-size: 12.50px; " x="28.606865" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="32.082699" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="38.334198" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="41.810032" y="0">1</tspan>
			</text>

		</g>
	</g>
</svg> Erlang Loss System. Here we present the explicit form of the probability density
function of the time spent between two consecutive losses in the <svg style="vertical-align:-0.17555pt;width:60.200001px;" id="M3" height="10.9125" version="1.1" viewBox="0 0 60.200001 10.9125" width="60.200001"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
		<g transform="translate(72,-63.27)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
				<tspan style="font-size: 12.50px; " x="12.565515" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="16.041349" y="0">𝑀</tspan>
				<tspan style="font-size: 12.50px; " x="28.606865" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="32.082699" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="38.334198" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="41.810032" y="0">1</tspan>
			</text>

		</g>
	</g>
</svg>
model. This density function solves a Cauchy problem for the second-order
differential equations, which was used to evaluate the corresponding laplace
transform. Finally the connection between the Erlang&#x27;s loss rate and the evaluated probability density function is showed.

Pierpaolo Ferrante

Interloss Time in <svg style="vertical-align:-2.22495pt" height="14.85" version="1.1" viewBox="0 0 72.162498 14.85" width="72.162498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g><g transform="matrix(.017,-0,0,-.017,17.418,12.012)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,24.422,12.012)"><use xlink:href="#x1D440"/></g><g transform="matrix(.017,-0,0,-.017,41.777,12.012)"><use xlink:href="#x2F"/></g><g transform="matrix(.017,-0,0,-.017,48.781,12.012)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,56.94,12.012)"><use xlink:href="#x2F"/></g><g transform="matrix(.017,-0,0,-.017,63.944,12.012)"><use xlink:href="#x31"/></g></svg> Loss System

We study a new type of reflected backward stochastic differential
equations (RBSDEs), where the reflecting process enters the drift in a nonlinear manner. This
type of the reflected BSDEs is based on a variance of the Skorohod problem studied recently by
Bank and El Karoui (2004), and is hence named the “Variant Reflected BSDEs” (VRBSDE) in this paper. The special nature of the Variant Skorohod problem leads to a hidden forward-backward feature of the BSDE, and as a consequence this type of BSDE cannot be treated in a usual way. We shall prove that in a small-time duration most of the well-posedness, comparison, and stability results are still valid, although some extra conditions on the boundary process are needed. We will also provide some possible applications where the VRBSDE can be potentially useful. These applications show that the VRBSDE could become a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable.

Jin Ma

Yusun Wang

On Variant Reflected Backward SDEs, with Applications

Let <svg style="vertical-align:-0.0pt;width:15.175px;" id="M1" height="10.325" version="1.1" viewBox="0 0 15.175 10.325" width="15.175" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,10.325)"><g transform="translate(72,-63.74)"><text transform="matrix(1,0,0,-1,-71.95,63.79)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐻</tspan></text></g></g></svg> be a real Hilbert space, let <svg style="vertical-align:-0.1254pt;width:11.375px;" id="M2" height="10.725" version="1.1" viewBox="0 0 11.375 10.725" width="11.375" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,10.725)"><g transform="translate(72,-63.42)"><text transform="matrix(1,0,0,-1,-71.95,63.6)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑆</tspan></text></g></g></svg>, <svg style="vertical-align:-0.0pt;width:11.225px;" id="M3" height="10.325" version="1.1" viewBox="0 0 11.225 10.325" width="11.225" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,10.325)"><g transform="translate(72,-63.74)"><text transform="matrix(1,0,0,-1,-71.95,63.79)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑇</tspan></text></g></g></svg> be two nonexpansive mappings such that <svg style="vertical-align:-2.21957pt;width:103.475px;" id="M4" height="13.45" version="1.1" viewBox="0 0 103.475 13.45" width="103.475" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,13.45)"><g transform="translate(72,-61.24)"><text transform="matrix(1,0,0,-1,-71.95,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐹</tspan><tspan style="font-size: 12.50px; " x="7.7268538" y="0">(</tspan><tspan style="font-size: 12.50px; " x="11.890353" y="0">𝑆</tspan><tspan style="font-size: 12.50px; " x="20.017303" y="0">)</tspan><tspan style="font-size: 12.50px; " x="26.956469" y="0">∩</tspan><tspan style="font-size: 12.50px; " x="37.483994" y="0">𝐹</tspan><tspan style="font-size: 12.50px; " x="45.21085" y="0">(</tspan><tspan style="font-size: 12.50px; " x="49.374348" y="0">𝑇</tspan><tspan style="font-size: 12.50px; " x="56.250996" y="0">)</tspan><tspan style="font-size: 12.50px; " x="62.502499" y="0">≠</tspan><tspan style="font-size: 12.50px; " x="73.142548" y="0">∅</tspan></text></g></g></svg>, let <svg style="vertical-align:-2.34499pt;width:10.675px;" id="M5" height="13.4875" version="1.1" viewBox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,13.4875)"><g transform="translate(72,-61.21)"><text transform="matrix(1,0,0,-1,-71.95,63.6)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑓</tspan></text></g></g></svg> be a contractive mapping, and let <svg style="vertical-align:-0.0pt;width:11.325px;" id="M6" height="10.55" version="1.1" viewBox="0 0 11.325 10.55" width="11.325" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,10.55)"><g transform="translate(72,-63.56)"><text transform="matrix(1,0,0,-1,-71.95,63.61)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐴</tspan></text></g></g></svg> be a strongly positive linear bounded operator on <svg style="vertical-align:-0.0pt;width:15.175px;" id="M7" height="10.325" version="1.1" viewBox="0 0 15.175 10.325" width="15.175" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,10.325)"><g transform="translate(72,-63.74)"><text transform="matrix(1,0,0,-1,-71.95,63.79)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐻</tspan></text></g></g></svg>. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as <svg style="vertical-align:-3.49493pt;width:168.85001px;" id="M8" height="15.05" version="1.1" viewBox="0 0 168.85001 15.05" width="168.85001" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,15.05)"><g transform="translate(72,-59.96)"><text transform="matrix(1,0,0,-1,-71.95,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑥</tspan></text><text transform="matrix(1,0,0,-1,-65.07,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan><tspan style="font-size: 8.75px; " x="4.3497438" y="0">+</tspan><tspan style="font-size: 8.75px; " x="10.344864" y="0">1</tspan></text><text transform="matrix(1,0,0,-1,-46.38,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">=</tspan><tspan style="font-size: 12.50px; " x="12.027886" y="0">𝛽</tspan></text><text transform="matrix(1,0,0,-1,-27.37,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-22.52,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑥</tspan></text><text transform="matrix(1,0,0,-1,-15.64,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-8.02,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">+</tspan><tspan style="font-size: 12.50px; " x="11.340221" y="0">(</tspan><tspan style="font-size: 12.50px; " x="15.50372" y="0">1</tspan><tspan style="font-size: 12.50px; " x="24.530886" y="0">−</tspan><tspan style="font-size: 12.50px; " x="35.88361" y="0">𝛽</tspan></text><text transform="matrix(1,0,0,-1,34.84,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,39.69,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">)</tspan><tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑆</tspan><tspan style="font-size: 12.50px; " x="12.290449" y="0">𝑦</tspan></text><text transform="matrix(1,0,0,-1,58.18,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text></g></g></svg>, <svg style="vertical-align:-3.20526pt;width:192.425px;" id="M9" height="14.6875" version="1.1" viewBox="0 0 192.425 14.6875" width="192.425" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.6875)"><g transform="translate(72,-60.25)"><text transform="matrix(1,0,0,-1,-71.95,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑦</tspan></text><text transform="matrix(1,0,0,-1,-65.75,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-57.43,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">=</tspan><tspan style="font-size: 12.50px; " x="12.027886" y="0">𝛼</tspan></text><text transform="matrix(1,0,0,-1,-38.34,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-33.49,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝛾</tspan><tspan style="font-size: 12.50px; " x="6.3390212" y="0">𝑓</tspan><tspan style="font-size: 12.50px; " x="13.278186" y="0">(</tspan><tspan style="font-size: 12.50px; " x="17.441685" y="0">𝑥</tspan></text><text transform="matrix(1,0,0,-1,-9.16,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-4.31,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">)</tspan><tspan style="font-size: 12.50px; " x="6.9391651" y="0">+</tspan><tspan style="font-size: 12.50px; " x="18.279387" y="0">(</tspan><tspan style="font-size: 12.50px; " x="22.442884" y="0">𝐼</tspan><tspan style="font-size: 12.50px; " x="31.219992" y="0">−</tspan><tspan style="font-size: 12.50px; " x="42.560211" y="0">𝛼</tspan></text><text transform="matrix(1,0,0,-1,45.31,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,50.16,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐴</tspan><tspan style="font-size: 12.50px; " x="8.9646511" y="0">)</tspan><tspan style="font-size: 12.50px; " x="13.12815" y="0">𝑇</tspan><tspan style="font-size: 12.50px; " x="20.004801" y="0">𝑥</tspan></text><text transform="matrix(1,0,0,-1,77.04,60.37)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text></g></g></svg>, <svg style="vertical-align:-1.29163pt;width:35.099998px;" id="M10" height="12.3" version="1.1" viewBox="0 0 35.099998 12.3" width="35.099998" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,12.3)"><g transform="translate(72,-62.16)"><text transform="matrix(1,0,0,-1,-71.95,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan><tspan style="font-size: 12.50px; " x="9.6773224" y="0">≥</tspan><tspan style="font-size: 12.50px; " x="21.71771" y="0">0</tspan></text></g></g></svg> is a real number and <svg style="vertical-align:-3.20526pt;width:30px;" id="M11" height="14.75" version="1.1" viewBox="0 0 30 14.75" width="30" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.75)"><g transform="translate(72,-60.2)"><text transform="matrix(1,0,0,-1,-71.95,63.45)"><tspan style="font-size: 12.50px; " x="0" y="0">{</tspan><tspan style="font-size: 12.50px; " x="6.00144" y="0">𝛼</tspan></text><text transform="matrix(1,0,0,-1,-58.9,60.32)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-54.05,63.45)"><tspan style="font-size: 12.50px; " x="0" y="0">}</tspan></text></g></g></svg>, <svg style="vertical-align:-3.20526pt;width:29.9125px;" id="M12" height="14.75" version="1.1" viewBox="0 0 29.9125 14.75" width="29.9125" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.75)"><g transform="translate(72,-60.2)"><text transform="matrix(1,0,0,-1,-71.95,63.45)"><tspan style="font-size: 12.50px; " x="0" y="0">{</tspan><tspan style="font-size: 12.50px; " x="6.00144" y="0">𝛽</tspan></text><text transform="matrix(1,0,0,-1,-58.97,60.32)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-54.12,63.45)"><tspan style="font-size: 12.50px; " x="0" y="0">}</tspan></text></g></g></svg> are two sequences in <svg style="vertical-align:-2.21957pt;width:32.674999px;" id="M13" height="13.45" version="1.1" viewBox="0 0 32.674999 13.45" width="32.674999" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,13.45)"><g transform="translate(72,-61.24)"><text transform="matrix(1,0,0,-1,-71.95,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">(</tspan><tspan style="font-size: 12.50px; " x="4.1634989" y="0">0</tspan><tspan style="font-size: 12.50px; " x="10.414999" y="0">,</tspan><tspan style="font-size: 12.50px; " x="15.616247" y="0">1</tspan><tspan style="font-size: 12.50px; " x="21.867746" y="0">)</tspan></text></g></g></svg> satisfying the following
control conditions:
(C1) <svg style="vertical-align:-3.20526pt;width:100.8375px;" id="M14" height="14.8" version="1.1" viewBox="0 0 100.8375 14.8" width="100.8375" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.8)"><g transform="translate(72,-60.16)"><text transform="matrix(1,0,0,-1,-71.95,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">l</tspan><tspan style="font-size: 12.50px; " x="3.4758339" y="0">i</tspan><tspan style="font-size: 12.50px; " x="6.9516678" y="0">m</tspan></text><text transform="matrix(1,0,0,-1,-51.1,60.28)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan><tspan style="font-size: 8.75px; " x="5.8025761" y="0">→</tspan><tspan style="font-size: 8.75px; " x="15.35976" y="0">∞</tspan></text><text transform="matrix(1,0,0,-1,-25.04,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan></text><text transform="matrix(1,0,0,-1,-17.99,60.28)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-9.67,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">=</tspan><tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan></text></g></g></svg>, (C3) <svg style="vertical-align:-3.2303pt;width:275.27499px;" id="M15" height="14.825" version="1.1" viewBox="0 0 275.27499 14.825" width="275.27499" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.825)"><g transform="translate(72,-60.14)"><text transform="matrix(1,0,0,-1,-71.95,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">0</tspan><tspan style="font-size: 12.50px; " x="9.7148314" y="0">&lt;</tspan><tspan style="font-size: 12.50px; " x="21.75522" y="0">l</tspan><tspan style="font-size: 12.50px; " x="25.231054" y="0">i</tspan><tspan style="font-size: 12.50px; " x="28.706888" y="0">m</tspan><tspan style="font-size: 12.50px; " x="44.685722" y="0">i</tspan><tspan style="font-size: 12.50px; " x="48.161556" y="0">n</tspan><tspan style="font-size: 12.50px; " x="54.413055" y="0">f</tspan></text><text transform="matrix(1,0,0,-1,-7.95,60.28)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan><tspan style="font-size: 8.75px; " x="5.8025761" y="0">→</tspan><tspan style="font-size: 8.75px; " x="15.368512" y="0">∞</tspan></text><text transform="matrix(1,0,0,-1,18.11,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan></text><text transform="matrix(1,0,0,-1,25.09,60.28)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,29.94,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">≤</tspan><tspan style="font-size: 12.50px; " x="14.115887" y="0">l</tspan><tspan style="font-size: 12.50px; " x="17.591721" y="0">i</tspan><tspan style="font-size: 12.50px; " x="21.067554" y="0">m</tspan><tspan style="font-size: 12.50px; " x="37.046391" y="0">s</tspan><tspan style="font-size: 12.50px; " x="41.910057" y="0">u</tspan><tspan style="font-size: 12.50px; " x="48.161556" y="0">p</tspan></text><text transform="matrix(1,0,0,-1,88.52,60.26)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan><tspan style="font-size: 8.75px; " x="5.8025761" y="0">→</tspan><tspan style="font-size: 8.75px; " x="15.368512" y="0">∞</tspan></text><text transform="matrix(1,0,0,-1,114.58,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan></text><text transform="matrix(1,0,0,-1,121.56,60.28)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,129.88,63.41)"><tspan style="font-size: 12.50px; " x="0" y="0">&lt;</tspan><tspan style="font-size: 12.50px; " x="12.027886" y="0">1</tspan></text></g></g></svg>, then <svg style="vertical-align:-3.49493pt;width:112.575px;" id="M16" height="15.2625" version="1.1" viewBox="0 0 112.575 15.2625" width="112.575" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,15.2625)"><g transform="translate(72,-59.79)"><text transform="matrix(1,0,0,-1,-71.95,63.32)"><tspan style="font-size: 12.50px; " x="0" y="0">‖</tspan><tspan style="font-size: 12.50px; " x="6.5390692" y="0">𝑥</tspan></text><text transform="matrix(1,0,0,-1,-58.53,60.2)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan><tspan style="font-size: 8.75px; " x="4.3497438" y="0">+</tspan><tspan style="font-size: 8.75px; " x="10.344864" y="0">1</tspan></text><text transform="matrix(1,0,0,-1,-40.54,63.32)"><tspan style="font-size: 12.50px; " x="0" y="0">−</tspan><tspan style="font-size: 12.50px; " x="11.340221" y="0">𝑥</tspan></text><text transform="matrix(1,0,0,-1,-22.32,60.2)"><tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan></text><text transform="matrix(1,0,0,-1,-17.47,63.32)"><tspan style="font-size: 12.50px; " x="0" y="0">‖</tspan><tspan style="font-size: 12.50px; " x="12.090401" y="0">→</tspan><tspan style="font-size: 12.50px; " x="29.219511" y="0">0</tspan></text></g></g></svg>. We also discuss several special cases of this iterative algorithm.

Yonghong Yao

Muhammad Aslam Noor

Syed Tauseef Mohyud-Din

Modified Iterative Algorithms for Nonexpansive Mappings

The expected number of real zeros of an algebraic polynomial
<svg style="vertical-align:-3.2316pt;width:181.0125px;" id="M1" height="17.799999" version="1.1" viewBox="0 0 181.0125 17.799999" width="181.0125"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,17.8)">
		<g transform="translate(72,-57.76)">
			<text transform="matrix(1,0,0,-1,-71.95,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-65.67,57.91)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑜</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-58.39,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="11.340221" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-40.77,57.91)">
				<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-35.89,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑥</tspan>
				<tspan style="font-size: 12.50px; " x="9.6398134" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="20.992537" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-8.62,57.91)">
				<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-3.75,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑥</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,3.13,66.03)">
				<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,10.78,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="11.340221" y="0">⋯</tspan>
				<tspan style="font-size: 12.50px; " x="27.781666" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="39.121887" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,56.19,57.91)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,61.03,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑥</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,67.91,66.03)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg> with random coefficient
<svg style="vertical-align:-4.77652pt;width:120.975px;" id="M2" height="16.637501" version="1.1" viewBox="0 0 120.975 16.637501" width="120.975"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,16.6375)">
		<g transform="translate(72,-58.69)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-65.67,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑗</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-61.25,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="5.2137508" y="0">𝑗</tspan>
				<tspan style="font-size: 12.50px; " x="13.540749" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="25.581139" y="0">0</tspan>
				<tspan style="font-size: 12.50px; " x="31.832638" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="37.046391" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="43.29789" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="48.499138" y="0">2</tspan>
				<tspan style="font-size: 12.50px; " x="54.750637" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="59.96439" y="0">…</tspan>
				<tspan style="font-size: 12.50px; " x="74.555389" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="79.756638" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg> is known. The distribution of the
coefficients is often assumed to be identical albeit allowed to
have different classes of distributions. For the nonidentical case,
there has been much interest where the variance of the <svg style="vertical-align:-2.34499pt;width:7.1374998px;" id="M3" height="12.675" version="1.1" viewBox="0 0 7.1374998 12.675" width="7.1374998"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,12.675)">
		<g transform="translate(72,-61.86)">
			<text transform="matrix(1,0,0,-1,-71.95,64.25)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑗</tspan>
			</text>

		</g>
	</g>
</svg>th coefficient
is  <svg style="vertical-align:-4.77652pt;width:93.4375px;" id="M4" height="19.6" version="1.1" viewBox="0 0 93.4375 19.6" width="93.4375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,19.6)">
		<g transform="translate(72,-56.32)">
			<text transform="matrix(1,0,0,-1,-71.95,61.13)">
				<tspan style="font-size: 12.50px; " x="0" y="0">v</tspan>
				<tspan style="font-size: 12.50px; " x="6.00144" y="0">a</tspan>
				<tspan style="font-size: 12.50px; " x="11.552772" y="0">r</tspan>
				<tspan style="font-size: 12.50px; " x="19.87977" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="24.043268" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-41.63,58.01)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑗</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-37.2,61.13)">
				<tspan style="font-size: 12.50px; " x="19.68" y="2.51"></tspan>
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="7.6268301" y="0">=</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-9.59,66.75)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 8.75px; " x="0.20999999" y="7.1999998">𝑗</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-3.15,58.62)">
				<tspan style="font-size: 12.50px; " x="0" y="0"></tspan>
			</text>

		</g>
	</g>
</svg>. It is shown that this class
of polynomials has significantly more zeros than the classical
algebraic polynomials with identical coefficients. However,
in the case of nonidentically distributed coefficients it is
analytically necessary to assume that the means
of coefficients are zero. In this work we
study a case when the moments of the coefficients have both
binomial and geometric progression elements. That is we assume <svg style="vertical-align:-4.77652pt;width:107.1px;" id="M5" height="19.725" version="1.1" viewBox="0 0 107.1 19.725" width="107.1"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,19.725)">
		<g transform="translate(72,-56.22)">
			<text transform="matrix(1,0,0,-1,-71.95,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan>
				<tspan style="font-size: 12.50px; " x="8.9271421" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="13.090641" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-52.58,57.91)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑗</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-48.16,61.03)">
				<tspan style="font-size: 12.50px; " x="19.68" y="2.51"></tspan>
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="7.6393328" y="0">=</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-20.55,66.65)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 8.75px; " x="0.20999999" y="7.21">𝑗</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-14.11,58.52)">
				<tspan style="font-size: 12.50px; " x="0" y="0"></tspan>
				<tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51">𝜇</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-0.65,66.03)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑗</tspan>
				<tspan style="font-size: 8.75px; " x="3.4045279" y="0">+</tspan>
				<tspan style="font-size: 8.75px; " x="9.3996477" y="0">1</tspan>
			</text>

		</g>
	</g>
</svg> and <svg style="vertical-align:-4.77652pt;width:113.6125px;" id="M6" height="19.725" version="1.1" viewBox="0 0 113.6125 19.725" width="113.6125"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,19.725)">
		<g transform="translate(72,-56.22)">
			<text transform="matrix(1,0,0,-1,-71.95,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">v</tspan>
				<tspan style="font-size: 12.50px; " x="6.00144" y="0">a</tspan>
				<tspan style="font-size: 12.50px; " x="11.552772" y="0">r</tspan>
				<tspan style="font-size: 12.50px; " x="19.87977" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="24.043268" y="0">𝑎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-41.63,57.91)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑗</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-37.2,61.03)">
				<tspan style="font-size: 12.50px; " x="19.68" y="2.51"></tspan>
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="7.6268301" y="0">=</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-9.59,66.65)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 8.75px; " x="0.20999999" y="7.21">𝑗</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-3.15,58.52)">
				<tspan style="font-size: 12.50px; " x="0" y="0"></tspan>
				<tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51">𝜎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,10.04,66.03)">
				<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
				<tspan style="font-size: 8.75px; " x="4.3759999" y="0">𝑗</tspan>
			</text>

		</g>
	</g>
</svg>. We show how the above expected number of real zeros is
dependent on values of <svg style="vertical-align:-0.13794pt;width:15.3875px;" id="M7" height="13.9375" version="1.1" viewBox="0 0 15.3875 13.9375" width="15.3875"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.9375)">
		<g transform="translate(72,-60.85)">
			<text transform="matrix(1,0,0,-1,-71.95,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜎</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-64.61,66.03)">
				<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
			</text>

		</g>
	</g>
</svg> and <svg style="vertical-align:-2.29482pt;width:9.6374998px;" id="M8" height="9.6750002" version="1.1" viewBox="0 0 9.6374998 9.6750002" width="9.6374998"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,9.675)">
		<g transform="translate(72,-64.26)">
			<text transform="matrix(1,0,0,-1,-71.95,66.6)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
			</text>

		</g>
	</g>
</svg> in various cases.

K. Farahmand

M. Sambandham

Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

We give a theorem on implicit difference functional inequalities generated
by mixed problems for nonlinear systems of first-order partial differential functional equations. We apply this result in the investigations of the stability of difference methods. Classical solutions of mixed problems are approximated in the paper by solutions of suitable implicit difference schemes. The proof of the convergence of difference method
is based on comparison technique, and the result on difference functional inequalities is used. Numerical examples are presented.

Z. Kamont

K. Kropielnicka

Implicit Difference Inequalities Corresponding to First-Order Partial Differential  Functional Equations

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International Journal of Stochastic Analysis - Table of contents 2009

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International Journal of Stochastic Analysis

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International Journal of Stochastic Analysis