The aim of the paper is to introduce a generalization of the Feynman-Kac theorem in Hilbert spaces. Connection between solutions to the abstract stochastic differential equation <svg style="vertical-align:-2.3205pt;width:196.675px;" id="M1" height="15.1125" version="1.1" viewBox="0 0 196.675 15.1125" width="196.675"      xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,0,0,-.017,.062,12.163)"><path id="x1D451" d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5&#xD;&#xA;h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z"/></g><g transform="matrix(.017,0,0,-.017,9.497,12.163)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231&#xD;&#xA;l-110 215q-26 51 -44 63t-72 17l6 28h250l-6 -28l-27 -4q-30 -5 -35 -10t3 -27q17 -43 95 -190q70 78 154 185q15 21 10 29.5t-33 12.5l-30 4l5 28h236z"/></g><g transform="matrix(.017,0,0,-.017,22.96,12.163)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z"/></g><g transform="matrix(.017,0,0,-.017,28.841,12.163)"><path id="x1D461" d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z"/></g><g transform="matrix(.017,0,0,-.017,34.638,12.163)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z"/></g><g transform="matrix(.017,0,0,-.017,45.228,12.163)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z"/></g><g transform="matrix(.017,0,0,-.017,59.932,12.163)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z"/></g><g transform="matrix(.017,0,0,-.017,72.392,12.163)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,0,0,-.017,85.855,12.163)"><use xlink:href="#x28"/></g><g transform="matrix(.017,0,0,-.017,91.737,12.163)"><use xlink:href="#x1D461"/></g><g transform="matrix(.017,0,0,-.017,97.533,12.163)"><use xlink:href="#x29"/></g><g transform="matrix(.017,0,0,-.017,103.415,12.163)"><use xlink:href="#x1D451"/></g><g transform="matrix(.017,0,0,-.017,112.849,12.163)"><use xlink:href="#x1D461"/></g><g transform="matrix(.017,0,0,-.017,125.258,12.163)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z"/></g><g transform="matrix(.017,0,0,-.017,141.849,12.163)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xD;&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z"/></g><g transform="matrix(.017,0,0,-.017,152.337,12.163)"><use xlink:href="#x1D451"/></g><g transform="matrix(.017,0,0,-.017,161.772,12.163)"><path id="x1D44A" d="M1004 650l-6 -29q-54 -6 -71 -19.5t-51 -74.5l-271 -539h-33l-98 506h-3l-258 -506h-30l-78 532q-10 67 -21.5 80t-66.5 21l6 29h241l-8 -29l-26 -5q-34 -6 -41 -16t-3 -47l59 -425h4l251 510h31l102 -510h2q150 299 198 423q14 40 8.5 48.5t-47.5 16.5l-28 5l7 29h231z&#xD;&#xA;"/></g><g transform="matrix(.017,0,0,-.017,179.042,12.163)"><use xlink:href="#x28"/></g><g transform="matrix(.017,0,0,-.017,184.924,12.163)"><use xlink:href="#x1D461"/></g><g transform="matrix(.017,0,0,-.017,190.72,12.163)"><use xlink:href="#x29"/></g>

		
	
</svg> and solutions to the deterministic partial differential (with derivatives in Hilbert spaces) equation for the probability characteristic <svg style="vertical-align:-2.3205pt;width:82.949997px;" id="M2" height="18.2125" version="1.1" viewBox="0 0 82.949997 18.2125" width="82.949997"      xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,0,0,-.017,.062,15.263)"><path id="x1D53C" d="M666 158q-18 -100 -35 -158h-593v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h570q0 -5 8 -144l-29 -4q-11 51 -32 85q-11 17 -35 23t-78 6h-76q-23 0 -28 -5t-5 -27v-228h94q67 0 83 12t25 63h29v-191h-29q-10 54 -26 65.5t-81 11.5h-95v-189q0 -65 24 -81&#xD;&#xA;q19 -13 104 -13q57 0 83 7t43 27q29 34 50 94q5 0 14.5 -2t14.5 -2zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5t7.5 79.5z"/></g>

			<g transform="matrix(.012,0,0,-.012,11.763,7.113)"><use xlink:href="#x1D461"/></g><g transform="matrix(.012,0,0,-.012,15.82,7.113)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z"/></g><g transform="matrix(.012,0,0,-.012,18.533,7.113)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xD;&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z"/></g>

			<g transform="matrix(.017,0,0,-.017,25.8,15.263)"><path id="x210E" d="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24&#xD;&#xA;q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z"/></g><g transform="matrix(.017,0,0,-.017,34.571,15.263)"><use xlink:href="#x28"/></g><g transform="matrix(.017,0,0,-.017,40.453,15.263)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,0,0,-.017,53.916,15.263)"><use xlink:href="#x28"/></g><g transform="matrix(.017,0,0,-.017,59.798,15.263)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xD;&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z"/></g><g transform="matrix(.017,0,0,-.017,71.119,15.263)"><use xlink:href="#x29"/></g><g transform="matrix(.017,0,0,-.017,77,15.263)"><use xlink:href="#x29"/></g>

		
	
</svg> is proved. Interpretation of objects in the equations is given.

I. V. Melnikova

V. S. Parfenenkova

Relations between Stochastic and Partial Differential Equations in Hilbert Spaces

Let <svg style="vertical-align:-3.21404pt;width:16.512501px;" id="M1" height="14.3375" version="1.1" viewBox="0 0 16.512501 14.3375" width="16.512501"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,14.3375)">
		<g transform="translate(72,-60.53)">
			<text transform="matrix(1,0,0,-1,-71.95,63.79)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-62.07,60.66)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

		</g>
	</g>
</svg> be any <i  >d</i>-dimensional continuous process that takes values in an open connected domain <svg style="vertical-align:-0.1881pt;width:10.9375px;" id="M2" height="10.9875" version="1.1" viewBox="0 0 10.9375 10.9875" width="10.9375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9875)">
		<g transform="translate(72,-63.21)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝒪</tspan>
			</text>

		</g>
	</g>
</svg> in <svg style="vertical-align:-0.0pt;width:18.9px;" id="M3" height="13.6875" version="1.1" viewBox="0 0 18.9 13.6875" width="18.9"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.6875)">
		<g transform="translate(72,-61.05)">
			<text transform="matrix(1,0,0,-1,-71.95,61.1)">
				<tspan style="font-size: 12.50px; " x="0" y="0">ℝ</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-62.61,66.1)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑑</tspan>
			</text>

		</g>
	</g>
</svg>. In this paper, we give equivalent formulations of the conditional full support (CFS) property of <svg style="vertical-align:-3.21404pt;width:16.512501px;" id="M4" height="14.3375" version="1.1" viewBox="0 0 16.512501 14.3375" width="16.512501"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,14.3375)">
		<g transform="translate(72,-60.53)">
			<text transform="matrix(1,0,0,-1,-71.95,63.79)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-62.07,60.66)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

		</g>
	</g>
</svg> in <svg style="vertical-align:-0.1881pt;width:10.9375px;" id="M5" height="10.9875" version="1.1" viewBox="0 0 10.9375 10.9875" width="10.9375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9875)">
		<g transform="translate(72,-63.21)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝒪</tspan>
			</text>

		</g>
	</g>
</svg>. We use them to show that the CFS property of <i  >X</i> in <svg style="vertical-align:-0.1881pt;width:10.9375px;" id="M6" height="10.9875" version="1.1" viewBox="0 0 10.9375 10.9875" width="10.9375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9875)">
		<g transform="translate(72,-63.21)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝒪</tspan>
			</text>

		</g>
	</g>
</svg> implies the existence of a martingale <i  >M</i> under an equivalent probability measure such that <i  >M</i> lies in the <svg style="vertical-align:-0.30096pt;width:34.674999px;" id="M7" height="11.0625" version="1.1" viewBox="0 0 34.674999 11.0625" width="34.674999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,11.0625)">
		<g transform="translate(72,-63.15)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜖</tspan>
				<tspan style="font-size: 12.50px; " x="9.3397408" y="0">&gt;</tspan>
				<tspan style="font-size: 12.50px; " x="21.380131" y="0">0</tspan>
			</text>

		</g>
	</g>
</svg> neighborhood of <svg style="vertical-align:-3.21404pt;width:16.512501px;" id="M8" height="14.3375" version="1.1" viewBox="0 0 16.512501 14.3375" width="16.512501"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,14.3375)">
		<g transform="translate(72,-60.53)">
			<text transform="matrix(1,0,0,-1,-71.95,63.79)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-62.07,60.66)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

		</g>
	</g>
</svg> for any given <svg style="vertical-align:-0.13794pt;width:7.4749999px;" id="M9" height="7.1875" version="1.1" viewBox="0 0 7.4749999 7.1875" width="7.4749999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,7.1875)">
		<g transform="translate(72,-66.25)">
			<text transform="matrix(1,0,0,-1,-71.95,66.44)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜖</tspan>
			</text>

		</g>
	</g>
</svg> under the supremum norm. The existence of such martingales, which are called consistent price systems (CPSs), has relevance with absence of arbitrage and hedging problems in
markets with proportional transaction costs as discussed in the recent paper by Guasoni et al. (2008), where the CFS property is introduced and shown sufficient for CPSs for processes with certain state space. The current paper extends the results in  the work of Guasoni et al. (2008), to processes with more general state space.

Florian Maris

Hasanjan Sayit

Consistent Price Systems in Multiasset Markets

We consider the class of semi-Markov modulated jump diffusions (sMMJDs) whose operator turns out to be an integro-partial differential operator. We find conditions under which the solutions of this class of switching jump-diffusion processes are almost surely exponentially stable and moment exponentially stable. We also provide conditions that imply almost sure convergence of the trivial solution when the moment exponential stability of the trivial solution is guaranteed. We further investigate and determine the conditions under which the trivial solution of the sMMJD-perturbed nonlinear system of differential equations <svg style="vertical-align:-3.21404pt;width:97.25px;" id="M1" height="14.7" version="1.1" viewBox="0 0 97.25 14.7" width="97.25"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,14.7)">
		<g transform="translate(72,-60.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan>
				<tspan style="font-size: 12.50px; " x="6.6515961" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-55.42,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-52.18,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="3.4758339" y="0">𝑑</tspan>
				<tspan style="font-size: 12.50px; " x="10.12743" y="0">𝑡</tspan>
				<tspan style="font-size: 12.50px; " x="17.5042" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="29.544588" y="0">𝑓</tspan>
				<tspan style="font-size: 12.50px; " x="36.483753" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="40.647251" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-1.65,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,1.59,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg> is almost surely exponentially stable. It is observed that for a one-dimensional state space, a linear unstable system of differential equations when stabilized just by the addition of the jump part of an sMMJD process does not get destabilized by any addition of a Brownian motion. However, in a state space of dimension at least two, we show that a corresponding nonlinear system of differential equations stabilized by jumps gets destabilized by addition of Brownian motion.

Amogh Deshpande

Asymptotic Stability of Semi-Markov Modulated Jump Diffusions

We propose a dependent hidden Markov model of credit quality. We suppose that the "true" credit quality is not observed directly but only through noisy observations given by posted credit ratings. The model is formulated in discrete time with a Markov chain observed in martingale noise, where "noise" terms of the state and observation processes are possibly dependent. The model provides estimates for the state of the Markov chain governing the evolution of the credit rating process and the parameters of the model, where the latter are estimated using the EM algorithm. The dependent dynamics allow for the so-called "rating momentum" discussed in the credit literature and also provide a convenient test of independence between the state and observation dynamics.

Ma&#x142;gorzata Wiktoria Korolkiewicz

A Dependent Hidden Markov Model of Credit Quality

This paper analyzes an <svg style="vertical-align:-2.22495pt;width:57px;" id="M1" height="14.85" version="1.1" viewBox="0 0 57 14.85" width="57" 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="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg> queueing system with two heterogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The vacationing server, however, returns to serve at a low rate as an arrival finds the other server busy. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime computed. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided.

A. Krishnamoorthy

C. Sreenivasan

An <i>M/M/2</i> Queueing System with Heterogeneous Servers Including One with Working Vacation

As a new formulation in structural analysis, Integrated Force Method has been successfully applied to many structures for civil, mechanical, and aerospace engineering due to the accurate estimate of forces in computation. Right now, it is being further extended to the probabilistic domain. For the assessment of uncertainty effect in system optimization and identification, the probabilistic sensitivity analysis of IFM was further investigated in this study. A set of stochastic sensitivity analysis formulation of Integrated Force Method was developed using the perturbation method. Numerical examples are presented to illustrate its application. Its efficiency and accuracy were also substantiated with direct Monte Carlo simulations and the reliability-based sensitivity method. The numerical algorithm was shown to be readily adaptable to the existing program since the models of stochastic finite element and stochastic design sensitivity are almost identical.

X. F. Wei

S. N. Patnaik

Application of Stochastic Sensitivity Analysis to Integrated Force Method

This paper deals with the general optimal control problem for fully coupled forward-backward
stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Necessary conditions of Pontraygin's type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.

Jingtao Shi

Necessary Conditions for Optimal Control of Forward-Backward Stochastic Systems with Random Jumps

We consider a one-dimensional stochastic equation <svg style="vertical-align:-3.21404pt;width:198.60001px;" id="M1" height="14.7" version="1.1" viewBox="0 0 198.60001 14.7" width="198.60001"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,14.7)">
		<g transform="translate(72,-60.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan>
				<tspan style="font-size: 12.50px; " x="6.6515961" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-55.42,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-48.71,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="12.027886" y="0">𝑏</tspan>
				<tspan style="font-size: 12.50px; " x="17.904297" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="22.067795" y="0">𝑡</tspan>
				<tspan style="font-size: 12.50px; " x="25.981234" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="31.194984" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-7.63,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
				<tspan style="font-size: 8.75px; " x="2.7393761" y="0">−</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,1.6,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑑</tspan>
				<tspan style="font-size: 12.50px; " x="10.815095" y="0">𝑍</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,22.07,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,28.08,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="11.340221" y="0">𝑎</tspan>
				<tspan style="font-size: 12.50px; " x="17.616728" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="21.780226" y="0">𝑡</tspan>
				<tspan style="font-size: 12.50px; " x="25.693665" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="30.907415" y="0">𝑋</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,68.87,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,72.11,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑑</tspan>
				<tspan style="font-size: 12.50px; " x="10.815095" y="0">𝑡</tspan>
			</text>

		</g>
	</g>
</svg>, <svg style="vertical-align:-1.29163pt;width:32.224998px;" id="M2" height="12.3" version="1.1" viewBox="0 0 32.224998 12.3" width="32.224998"  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">𝑡</tspan>
				<tspan style="font-size: 12.50px; " x="7.37677" y="0">≥</tspan>
				<tspan style="font-size: 12.50px; " x="19.417158" y="0">0</tspan>
			</text>

		</g>
	</g>
</svg>, with respect to a symmetric stable process <svg style="vertical-align:-0.0pt;width:13.2875px;" id="M3" height="10.325" version="1.1" viewBox="0 0 13.2875 10.325" width="13.2875"  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">𝑍</tspan>
			</text>

		</g>
	</g>
</svg> of index <svg style="vertical-align:-1.29163pt;width:63.349998px;" id="M4" height="12.3" version="1.1" viewBox="0 0 63.349998 12.3" width="63.349998"  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">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">𝛼</tspan>
				<tspan style="font-size: 12.50px; " x="32.282745" y="0">≤</tspan>
				<tspan style="font-size: 12.50px; " x="44.323135" y="0">2</tspan>
			</text>

		</g>
	</g>
</svg>. It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation <svg style="vertical-align:-3.21404pt;width:114.35px;" id="M5" height="14.7" version="1.1" viewBox="0 0 114.35 14.7" width="114.35"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,14.7)">
		<g transform="translate(72,-60.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan>
				<tspan style="font-size: 12.50px; " x="6.6515961" y="0">𝐿</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-56.45,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-49.74,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="12.040389" y="0">𝐵</tspan>
				<tspan style="font-size: 12.50px; " x="20.742477" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="24.905975" y="0">𝐿</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-15.98,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan>
				<tspan style="font-size: 8.75px; " x="2.7393761" y="0">−</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-6.75,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑑</tspan>
				<tspan style="font-size: 12.50px; " x="10.815095" y="0">𝑊</tspan>
				<tspan style="font-size: 12.50px; " x="22.267843" y="0">𝑡</tspan>
			</text>

		</g>
	</g>
</svg> with respect to the semimartingale <svg style="vertical-align:-2.21957pt;width:67.712502px;" id="M6" height="13.45" version="1.1" viewBox="0 0 67.712502 13.45" width="67.712502"  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">𝑊</tspan>
				<tspan style="font-size: 12.50px; " x="14.916079" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="26.956469" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="31.119967" y="0">𝑍</tspan>
				<tspan style="font-size: 12.50px; " x="40.772282" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="45.986034" y="0">𝑡</tspan>
				<tspan style="font-size: 12.50px; " x="49.899471" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg> and corresponding matrix <svg style="vertical-align:-0.0pt;width:11.625px;" id="M7" height="10.325" version="1.1" viewBox="0 0 11.625 10.325" width="11.625"  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">𝐵</tspan>
			</text>

		</g>
	</g>
</svg>. In the case of <svg style="vertical-align:-1.29163pt;width:63.349998px;" id="M8" height="12.3" version="1.1" viewBox="0 0 63.349998 12.3" width="63.349998"  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">1</tspan>
				<tspan style="font-size: 12.50px; " x="9.7148314" y="0">≤</tspan>
				<tspan style="font-size: 12.50px; " x="21.75522" y="0">𝛼</tspan>
				<tspan style="font-size: 12.50px; " x="32.282745" y="0">&lt;</tspan>
				<tspan style="font-size: 12.50px; " x="44.323135" y="0">2</tspan>
			</text>

		</g>
	</g>
</svg> we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The
existence proofs are established using the method of Krylov's estimates
for processes satisfying the 2-dimensional equation. On another hand,
the Krylov's estimates are based on some analytical facts of independent
interest that are also proved in the paper.

V. P. Kurenok

On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

We consider the best-choice problem with disorder
and imperfect observation. The decision-maker observes sequentially
a known number of i.i.d random variables from a known distribution
with the object of choosing the largest. At the random time the distribution
law of observations is changed. The random variables cannot
be perfectly observed. Each time a random variable is sampled the
decision-maker is informed only whether it is greater than or less than
some level specified by him. The decision-maker can choose at most
one of the observation. The optimal rule is derived in the class of
Bayes&#39; strategies.

Vladimir Mazalov

Evgeny Ivashko

Bayes&#39; Model of the Best-Choice Problem with Disorder

About this journal

Journal profile

Meet the editorial board

Editor spotlight

Submit your proposal

Do you think there is an emerging area of research that really needs to be highlighted? Or an existing research area that has been overlooked or would benefit from deeper investigation? Raise the profile of a research area by leading a Special Issue.

Special Issues

International Journal of Stochastic Analysis - Table of contents 2012

International Journal of Stochastic Analysis | Hindawi

title

International Journal of Stochastic Analysis

https://images.hindawi.com/journals/ijsa/ijsa.banner.jpg

International Journal of Stochastic Analysis