{"id":17417,"date":"2022-08-18T13:24:55","date_gmt":"2022-08-18T05:24:55","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=17417"},"modified":"2026-06-08T16:36:14","modified_gmt":"2026-06-08T08:36:14","slug":"8-5-d-dimensional-brownian-motions","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-science\/department-of-applied-mathematics\/introduction-to-financial-mathematics-ii-department-of-applied-mathematics-prof-ching-tang-wu\/8-5-d-dimensional-brownian-motions","title":{"rendered":"8.5 d-dimensional Brownian motions"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"17417\" class=\"elementor elementor-17417\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section elementor-top-section elementor-element elementor-element-16c525bd elementor-section-boxed elementor-section-height-default 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href=\"https:\/\/ocw.nycu.edu.tw\/\">Homepage<\/a><\/span><\/span><\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-top-section elementor-element elementor-element-22fce837 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"22fce837\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-382be233\" data-id=\"382be233\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-63b47ed3 elementor-widget elementor-widget-penci-info-box\" data-id=\"63b47ed3\" data-element_type=\"widget\" data-widget_type=\"penci-info-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"penci_info_box_80679\" class=\"penci-block-vc penci-info-box penci-ibox-float-left penci-view-default penci-shape-circle\">\r\n\t\t\t<div class=\"penci-ibox-inner\">\r\n\t\t\t\t\t\t\t\t<div class=\"penci-ibox-icon penci-ibox-icon--icon penci-icon penci-tibox-text\"><span class=\"penci-ibox-icon-fa\"><\/span><\/div>\t\t\t\t<div class=\"penci-ibox-content-wrap\">\r\n\t\t\t\t\t\t\t\t\t\t<h3 class=\"penci-ibox-title\">8.5 d-dimensional Brownian motions\n<\/h3>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-64c004d7 elementor-widget elementor-widget-video\" data-id=\"64c004d7\" data-element_type=\"widget\" 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penci-disSticky penci-structure-20 elementor-section elementor-inner-section elementor-element elementor-element-238a4f4a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"238a4f4a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-fe8f973\" data-id=\"fe8f973\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-29d56571 elementor-widget elementor-widget-penci-button\" data-id=\"29d56571\" data-element_type=\"widget\" data-widget_type=\"penci-button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pcbtn-wrapper pcbtn-align-left\">\r\n\t\t\t<a class=\"pcbtn 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class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<div class=\"elementor-element elementor-element-3349365d elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"3349365d\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-divider{--divider-border-style:none;--divider-border-width:1px;--divider-color:#0c0d0e;--divider-icon-size:20px;--divider-element-spacing:10px;--divider-pattern-height:24px;--divider-pattern-size:20px;--divider-pattern-url:none;--divider-pattern-repeat:repeat-x}.elementor-widget-divider .elementor-divider{display:flex}.elementor-widget-divider .elementor-divider__text{font-size:15px;line-height:1;max-width:95%}.elementor-widget-divider .elementor-divider__element{margin:0 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100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5ea39f19 elementor-widget elementor-widget-shortcode\" data-id=\"5ea39f19\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-31\" class=\"tablepress tablepress-id-31\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">Week<\/th><th class=\"column-2\">Course Content<\/th><th class=\"column-3\">Course Video<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u55ae\u5143\u4e03 Coutinuous-time Martingales<br \/>\n7.1 Stochastic process (1\/2)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17352&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7.1 Stochastic process (2\/2)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17357&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7.2 Uniform integrability<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17363&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7.3 Martingale theory in continuous-time<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17369&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7.4 Local martingales<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17375&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7.5 Doob-Meyer decomposition<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17381&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7.6 Semimartingales<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17387&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u55ae\u5143\u516b Brownian Motions<br \/>\n8.1 Scaled random walk<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17393&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">8.2 Brownian motions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17399&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">8.3 The Brownian sample paths<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17405&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">8.4 Exponential martingales<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17411&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">8.5 d-dimensional Brownian motions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17417&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u55ae\u5143\u4e5d Stochastic Integrals<br \/>\n9.1 Construction of stochastic integrals with respect to martingales (1\/3)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17423&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.1 Construction of stochastic integrals with respect to martingales (2\/3)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17429&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.1 Construction of stochastic integrals with respect to martingales (3\/3)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17435&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-17 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.2 Stochastic integrals with respect to semimartingales<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17441&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-18 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.3 Stochastic integrals with respect to local martingales<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17447&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-19 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.4 It\u00f4 formula  (1\/2)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17453&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-20 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.4 It\u00f4 formula  (2\/2)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17459&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-21 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.5 Integration by parts<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17465&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-22 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.6 Martingale representation theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17471&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-23 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.7 Change of Measures<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17477&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-24 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.8 Girsanov theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17483&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-25 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">9.9 Local times<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17489&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-26 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u55ae\u5143\u5341 Stochastic Differential Equations<br \/>\n10.1 Examples and some solution methods (1\/2)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17495&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-27 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">10.1 Examples and some solution methods (2\/2)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17501&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-28 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">10.2 An existence and uniqueness result<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17507&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-29 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">10.3 Weak and strong solutions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17513&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-30 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">10.4 Feynman-Kac theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17519&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-31 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u55ae\u5143\u5341\u4e00 Continuous-Time Models<br \/>\n11.1 Market portfolios and arbitrage<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17525&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-32 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11.2 Equivalent local martingale measures<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17531&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-33 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11.3 Completeness<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17537&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-34 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11.4 Pricing for attainable contingent claim<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17543&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-35 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11.5 Black-Scholes-Merton formula<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17549&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-36 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11.6 The Greeks<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17555&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-37 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11.7 Parity rrelations<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17561&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-38 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u55ae\u5143\u5341\u4e8c Hedging<br \/>\n12.1 Hedging strategy for the simple contingent claim <br \/>\n12.2 Delta and gamma hedging<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17567&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-39 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Appendix F\u3001Characteristic Functions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17573&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-40 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Appendix G\u3001Differntial Equations<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17579&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-41 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Appendix H\u3001Convex Analysis<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=17585&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-31 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-62610854\" data-id=\"62610854\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2f861ee6 elementor-widget-divider--view-line_text 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href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-life-sciences-en\" class=\"elementor-item\">College of Life Sciences<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-141797\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/colege-of-medicine-en\" class=\"elementor-item\">College of Medicine<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118072\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-chinese-center\" class=\"elementor-item\">Course of Chinese Center<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120431\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/liberal-arts-college\" class=\"elementor-item\">Forum Of Liberal Arts<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123746\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/speech-of-general-education-center\" class=\"elementor-item\">Contemporary Civilization<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-128240\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/reading-of-classic-masterpiece\" class=\"elementor-item\">Reading of Classic Masterpiece<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123745\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-general-education-center-2\" class=\"elementor-item\">Course of General Education Center<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"\u9078\u55ae\u5207\u63db\" aria-expanded=\"false\">\n\t\t\t<svg aria-hidden=\"true\" 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200 833 171 804 137 775 137 742 167Z\"><\/path><\/svg>\t\t\t<span class=\"elementor-screen-only\">\u9078\u55ae<\/span>\n\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-23e9544f\" class=\"elementor-nav-menu sm-vertical\"><li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-121577\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/self-learning-courses-for-high-school-students\" class=\"elementor-item\" tabindex=\"-1\">Self-learning courses for high school students<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-121578\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/pre-university-courses\" class=\"elementor-item\" tabindex=\"-1\">Pre-University Courses<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5381\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-science\" class=\"elementor-item\" tabindex=\"-1\">College of Science<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5377\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-engineering\" class=\"elementor-item\" tabindex=\"-1\">College of Engineering<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5376\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-electrical-and-computer-engineering\" class=\"elementor-item\" tabindex=\"-1\">College of Electrical and Computer Engineering<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5380\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-management\" class=\"elementor-item\" tabindex=\"-1\">College of Management<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5375\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-computer-science\" class=\"elementor-item\" tabindex=\"-1\">College of Computer Science<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5374\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-biological-science-and-technology\" class=\"elementor-item\" tabindex=\"-1\">College of Engineering Bioscience<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5379\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-humanities-and-social-sciences\" class=\"elementor-item\" tabindex=\"-1\">College of Humanities Arts and Social Sciences<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5378\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-hakka-affair-culture\" class=\"elementor-item\" tabindex=\"-1\">College of Hakka Affair Culture<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118070\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/school-of-law\" class=\"elementor-item\" tabindex=\"-1\">School of Law<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118069\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/school-of-nursing-2\" class=\"elementor-item\" tabindex=\"-1\">School of Nursing<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118071\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-photonics\" class=\"elementor-item\" tabindex=\"-1\">College of Photonics<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-125874\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-biomedical-science-and-engineering\" class=\"elementor-item\" tabindex=\"-1\">College of Biomedical Science and Engineering<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-132738\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-life-sciences-en\" class=\"elementor-item\" tabindex=\"-1\">College of Life Sciences<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-141797\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/colege-of-medicine-en\" class=\"elementor-item\" tabindex=\"-1\">College of Medicine<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118072\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-chinese-center\" class=\"elementor-item\" tabindex=\"-1\">Course of Chinese Center<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120431\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/liberal-arts-college\" class=\"elementor-item\" tabindex=\"-1\">Forum Of Liberal Arts<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123746\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/speech-of-general-education-center\" class=\"elementor-item\" tabindex=\"-1\">Contemporary Civilization<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-128240\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/reading-of-classic-masterpiece\" class=\"elementor-item\" tabindex=\"-1\">Reading of Classic Masterpiece<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123745\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-general-education-center-2\" class=\"elementor-item\" tabindex=\"-1\">Course of General Education Center<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>8.5 d-dimensio&hellip;<\/p>\n","protected":false},"featured_media":0,"parent":15692,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","format":"standard","meta":{"footnotes":""},"categories":[969],"tags":[],"class_list":["post-17417","course_page","type-course_page","status-publish","format-standard","hentry","category-introduction-to-financial-mathematics-ii-99-en"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.8 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>8.5 d-dimensional Brownian motions - NYCU OCW<\/title>\n<meta name=\"description\" content=\"8.5 d-dimensional Brownian motions            https:\/\/youtu.be\/1Gwrw5AcxTM      Back to Course Homepage       COURSE\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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