We propose a copula-based approach to solve the option pricing problem in the risk-neutral setting and with respect to a structured derivative written on several underlying assets. Our analysis generalizes similar results already present in the literature but limited to the trivariate case. The main difficulty of such a generalization consists in selecting the appropriate vine structure which turns to be of D-vine type, contrary to what happens in the trivariate setting where the canonical vine is sufficient. We first define the general procedure for multivariate options and then we will give a concrete example for the case of an option written on four indexes of stocks, namely, the S&#x26;P 500 Index, the Nasdaq 100 Index, the Nasdaq Composite Index, and the Nyse Composite Index. Moreover, we calibrate the proposed model, also providing a comparison analysis between real prices and simulated data to show the goodness of obtained estimates. We underline that our pair-copula decomposition method produces excellent numerical results, without restrictive assumptions on the assets dynamics or on their dependence structure, so that our copula-based approach can be used to model heterogeneous dependence structure existing between market assets of interest in a rigorous and effective way.

Anna Barban

Luca Di Persio

Multivariate Option Pricing with Pair-Copulas

The expected number of 0-1 strings of a limited length is a potentially useful index of the behavior of stochastic processes describing the occurrence of critical events (e.g., records, extremes, and exceedances). Such model sequences might be derived by a Hoppe-Polya or a Polya-Eggenberger urn model interpreting the drawings of white balls as occurrences of critical events. Numerical results, concerning average numbers of constrained length interruptions of records as well as how on the average subsequent exceedances are separated, demonstrate further certain urn models.

Frosso S. Makri

Zaharias M. Psillakis

On the Expected Number of Limited Length Binary Strings Derived by Certain Urn Models

The Inverse Weibull distribution has been applied to a wide range of situations including applications in medicine, reliability, and ecology. It can also be used to describe the degradation phenomenon of mechanical components. We introduce Inverse Generalized Weibull and Generalized Inverse Generalized Weibull (GIGW) distributions. GIGW distribution is a generalization of several distributions in literature. The mathematical properties of this distribution have been studied and the mixture model of two Generalized Inverse Generalized Weibull distributions is investigated. Estimates of parameters using method of maximum likelihood have been computed through simulations for complete and censored data.

Kanchan Jain

Neetu Singla

Suresh Kumar Sharma

The Generalized Inverse Generalized Weibull Distribution and Its Properties

The law <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:35.0625px;" id="M1" height="15.0875" version="1.1" viewBox="0 0 35.0625 15.0875" width="35.0625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z"></path></g><g transform="matrix(.017,-0,0,-.017,9.769,12.138)"><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"></path></g><g transform="matrix(.017,-0,0,-.017,15.65,12.138)"><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&#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"></path></g><g transform="matrix(.017,-0,0,-.017,29.113,12.138)"><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"></path></g>

		
	
</svg> of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.29163pt;width:41.174999px;" id="M2" height="11.85" version="1.1" viewBox="0 0 41.174999 11.85" width="41.174999">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,18.234,11.113)"><path id="x2265" d="M531 285l-474 -214v56l416 183l-416 184v56l474 -215v-50zM531 -40h-474v50h474v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,32.938,11.113)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z"></path></g>

		
	
</svg> has distribution function <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:10.7px;" id="M3" height="11.175" version="1.1" viewBox="0 0 10.7 11.175" width="10.7">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D439" d="M599 650l-18 -160l-30 -2q-2 70 -11 94q-5 16 -26 24t-75 8h-94q-27 0 -33.5 -6.5t-12.5 -32.5l-43 -226h108q52 0 72.5 5t31 18t28.5 52h28l-40 -200h-28q-1 42 -6 56.5t-24.5 21t-70.5 6.5h-108l-32 -177q-13 -65 1 -80.5t88 -22.5l-8 -28h-279l6 28q61 5 77.5 20.5&#xA;t29.5 81.5l72 387l7.5 40.5t1 26.5t-4 18t-15.5 10t-24 6t-37 4l8 28h461z"></path></g>

		
	
</svg> and first moment <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.30096pt;width:77.875px;" id="M4" height="11.125" version="1.1" viewBox="0 0 77.875 11.125" width="77.875">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,12.931,10.862)"><path id="x3C" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z"></path></g><g transform="matrix(.017,-0,0,-.017,27.634,10.862)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z"></path></g><g transform="matrix(.017,-0,0,-.017,45.772,10.862)"><use xlink:href="#x3C"></use></g><g transform="matrix(.017,-0,0,-.017,60.476,10.862)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z"></path></g>

		
	
</svg>. The law <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:35.0625px;" id="M5" height="18.075001" version="1.1" viewBox="0 0 35.0625 18.075001" width="35.0625">
	
		
			<g transform="matrix(.017,-0,0,-.017,18.187,11.5)"><path id="xF5C02" d="M658 557l-9 -25q-129 23 -320 70q-210 -50 -320 -70l-9 25q232 77 318 116h22q89 -40 318 -116z"></path></g><g transform="matrix(.017,-0,0,-.017,.062,15.125)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,9.769,15.125)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,15.125)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,29.113,15.125)"><use xlink:href="#x29"></use></g>

		
	
</svg> of the length-biased version of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:13.5875px;" id="M6" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"></use></g>

		
	
</svg> has by definition the distribution function <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.59964pt;width:100.9375px;" id="M7" height="23.825001" version="1.1" viewBox="0 0 100.9375 23.825001" width="100.9375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.4)"><use xlink:href="#x1D45A"></use></g>

			<g transform="matrix(.012,-0,0,-.012,13.475,8.238)"><path id="x2212" d="M535 230h-483v50h483v-50z"></path></g><g transform="matrix(.012,-0,0,-.012,20.46,8.238)"><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"></path></g>

			<g transform="matrix(.017,-0,0,-.017,29.625,16.412)"><path id="x222B" d="M480 853q0 -20 -13.5 -32.5t-34.5 -12.5q-19 0 -34.5 18t-27.5 18q-13 0 -21 -11.5t-13 -50.5t-4 -109l6 -574q6 -488 -183 -488q-28 0 -46.5 16t-18.5 38q0 19 14 32.5t33 13.5t40.5 -19t30.5 -19q14 0 21.5 12.5t11.5 48.5t4 103l-6 654q-1 201 46.5 305t138.5 104&#xA;q24 0 40 -13.5t16 -33.5z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,39.312,5.387)"><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&#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"></path></g><g transform="matrix(.012,-0,0,-.012,36.25,23.612)"><use xlink:href="#x30"></use></g>

			<g transform="matrix(.017,-0,0,-.017,49.425,16.4)"><path id="x1D466" d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26&#xA;q13 28 43 28q18 0 30 -15.5t12 -39.5z"></path></g><g transform="matrix(.017,-0,0,-.017,59.267,16.4)"><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&#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"></path></g><g transform="matrix(.017,-0,0,-.017,68.702,16.4)"><use xlink:href="#x1D439"></use></g><g transform="matrix(.017,-0,0,-.017,79.275,16.4)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,85.156,16.4)"><use xlink:href="#x1D466"></use></g><g transform="matrix(.017,-0,0,-.017,94.999,16.4)"><use xlink:href="#x29"></use></g>

		
	
</svg>. It is known that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:35.0625px;" id="M8" height="15.0875" version="1.1" viewBox="0 0 35.0625 15.0875" width="35.0625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,9.769,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,12.138)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,29.113,12.138)"><use xlink:href="#x29"></use></g>

		
	
</svg> is infinitely divisible if and only if <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.51414pt;width:76.262497px;" id="M9" height="20.9375" version="1.1" viewBox="0 0 76.262497 20.9375" width="76.262497">
	
		
			<g transform="matrix(.017,-0,0,-.017,2.6,17.2)"><use xlink:href="#xF5C02"></use></g><g transform="matrix(.017,-0,0,-.017,.062,20.825)"><use xlink:href="#x1D44B"></use></g>

			<g transform="matrix(.012,-0,0,-.012,19.938,11.087)"><use xlink:href="#x1D451"></use></g>

			<g transform="matrix(.017,-0,0,-.017,18.25,20.825)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,32.937,20.825)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,50.174,20.825)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,63.943,20.825)"><path id="x1D44D" d="M698 636l-541 -596q60 -5 176 -5q91 0 138.5 5t69.5 24q44 36 85 124l29 -15q-38 -125 -64 -173h-559l-9 16l545 598h-182q-81 0 -109 -8t-48 -31q-26 -29 -55 -103l-29 3q23 86 42 200h22q11 -16 21 -20.5t34 -4.5h428z"></path></g>

		
	
</svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:12.375px;" id="M10" height="11.6" version="1.1" viewBox="0 0 12.375 11.6" width="12.375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.537)"><use xlink:href="#x1D44D"></use></g>

		
	
</svg> is independent of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:13.5875px;" id="M11" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"></use></g>

		
	
</svg>. Here we assume this relation and ask whether <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:33.849998px;" id="M12" height="15.0875" version="1.1" viewBox="0 0 33.849998 15.0875" width="33.849998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,9.769,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,12.138)"><use xlink:href="#x1D44D"></use></g><g transform="matrix(.017,-0,0,-.017,27.906,12.138)"><use xlink:href="#x29"></use></g>

		
	
</svg> or <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:35.0625px;" id="M13" height="18.075001" version="1.1" viewBox="0 0 35.0625 18.075001" width="35.0625">
	
		
			<g transform="matrix(.017,-0,0,-.017,18.187,11.5)"><use xlink:href="#xF5C02"></use></g><g transform="matrix(.017,-0,0,-.017,.062,15.125)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,9.769,15.125)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,15.125)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,29.113,15.125)"><use xlink:href="#x29"></use></g>

		
	
</svg> is infinitely divisible. Examples show that both, neither, or exactly one of the components of the pair <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:88.462502px;" id="M14" height="18.075001" version="1.1" viewBox="0 0 88.462502 18.075001" width="88.462502">
	
		
			<g transform="matrix(.017,-0,0,-.017,65.7,11.5)"><use xlink:href="#xF5C02"></use></g><g transform="matrix(.017,-0,0,-.017,.062,15.125)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,5.944,15.125)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,15.125)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,21.532,15.125)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,34.995,15.125)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,40.876,15.125)"><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"></path></g><g transform="matrix(.017,-0,0,-.017,47.574,15.125)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,57.28,15.125)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,63.175,15.125)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,76.638,15.125)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,82.52,15.125)"><use xlink:href="#x29"></use></g>

		
	
</svg> can be infinitely divisible. Some general algorithms facilitate exploring the general question. It is shown that length-biasing up to the fourth order preserves infinite divisibility when <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:35.0625px;" id="M15" height="15.0875" version="1.1" viewBox="0 0 35.0625 15.0875" width="35.0625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,9.769,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,12.138)"><use xlink:href="#x1D44B"></use></g><g transform="matrix(.017,-0,0,-.017,29.113,12.138)"><use xlink:href="#x29"></use></g>

		
	
</svg> has a certain compound Poisson law or the Lambert law. It is conjectured for these examples that this extends to all orders of length-biasing.

Anthony G. Pakes

On the Preservation of Infinite Divisibility under Length-Biasing

We derive formulas for the expected hitting times of general random walks on graphs, in terms of voltages, with very elementary electric means. Under this new light we revise bounds and hitting times for birth-and-death Markov chains and for walks on graphs with cutpoints, and give some exact computations on the necklace graph. We also prove Tetali&#x2019;s formula for hitting times without making use of the reciprocity principle. In fact this principle follows as a corollary of our argument that also yields as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour.

Jos&#xe9; Luis Palacios

Eduardo G&#xf3;mez

Miguel Del R&#xED;o

Hitting Times of Walks on Graphs through Voltages

The notion of <svg     style="vertical-align:-0.10033pt;width:8.6625004px;" id="M1" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg>-fold iterated It&#xf4; integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-It&#xf4; chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

M. A. Alshanskiy

Wiener-It&#xF4;  Chaos Expansion of Hilbert Space Valued Random Variables

We introduce and characterize a new family of distributions, Marshall-Olkin discrete uniform distribution. The natures of hazard rate, entropy, and distribution of minimum of sequence of i.i.d. random variables are derived. First order autoregressive (AR (1)) model with this distribution for marginals is considered. The maximum likelihood estimates for the parameters are found out. Also, the goodness of the distribution is tested with real data.

E. Sandhya

C. B. Prasanth

Marshall-Olkin Discrete Uniform Distribution

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Journal of Probability