{"id":1841,"date":"2014-05-01T11:00:11","date_gmt":"2014-05-01T11:00:11","guid":{"rendered":"http:\/\/casgroups.case.edu\/physics-senior-projects\/?p=1841"},"modified":"2017-01-26T22:38:11","modified_gmt":"2017-01-26T22:38:11","slug":"stochastic-differential-equations-driven-by-compound-poisson-and-levy-processes","status":"publish","type":"post","link":"https:\/\/casgroups.case.edu\/physics-senior-projects\/stochastic-differential-equations-driven-by-compound-poisson-and-levy-processes\/","title":{"rendered":"Stochastic Differential Equations Driven by Compound Poisson and L\u00e9vy Processes"},"content":{"rendered":"<h3 class=\"p4\" style=\"text-align: center\">Thomas Norton with Wojbor A. Woyczy\u0144ski<\/h3>\n<h3 class=\"p5\" style=\"text-align: center\"><i>Stochastic Differential Equations Driven by Compound Poisson and L\u00e9vy Processes<\/i><\/h3>\n<div class=\"panel panel-default\">\n\t\t\t\t<div class=\"panel-heading\">\n\t\t\t\t\t<h4 class=\"panel-title\">\n\t\t\t\t\t\t<a class=\"collapsed\" data-toggle=\"collapse\" data-parent=\"#single-collapse\" href=\"#single-collapse-0\">\u201dPoster\u201d<\/a>\n\t\t\t\t\t<\/h4>\n\t\t\t\t<\/div>\n\t\t\t\t<div id=\"single-collapse-0\" class=\"panel-collapse collapse\">\n\t\t\t\t\t<div class=\"panel-body\">\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-2125\" src=\"https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/176\/2014\/05\/15022448\/Norton_Woyczynski.jpg\" alt=\"Norton_Woyczynski\" width=\"4032\" height=\"2592\" srcset=\"https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/176\/2014\/05\/15022448\/Norton_Woyczynski.jpg 4032w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/176\/2014\/05\/15022448\/Norton_Woyczynski-300x193.jpg 300w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/176\/2014\/05\/15022448\/Norton_Woyczynski-768x494.jpg 768w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/176\/2014\/05\/15022448\/Norton_Woyczynski-1024x658.jpg 1024w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/176\/2014\/05\/15022448\/Norton_Woyczynski-500x321.jpg 500w\" sizes=\"(max-width: 4032px) 100vw, 4032px\" \/><\/p>\n<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n<p class=\"p6\">Stochastic differential equations (SDEs) provide a rigorous setting for investigating the dynamics of systems subject to \u201cnoise,\u201d by which we mean source terms whose behavior is not deterministic, but governed by a probability distribution.<span class=\"Apple-converted-space\">\u00a0 <\/span>The noise can have quite irregular behavior (such as continuous but nowhere differentiable paths), giving rise to a rich theory of stochastic integration that allows for multiple interpretations of the same SDE.<span class=\"Apple-converted-space\">\u00a0 <\/span>Most commonly, the noise is assumed to follow the same probability distribution as Brownian Motion, a pure continuous-time random walk, but the theory is further enriched by considering sources of noise whose amplitude is governed by alternative distributions, such as a compound Poisson process or, more generally, a L\u00e9vy process. The student\u2019s summer research focused on exploring the consequences of differing interpretations of SDEs driven by Brownian Motion noise, with applications to biological networks.<span class=\"Apple-converted-space\">\u00a0 <\/span>This research compared the two most common interpretations of this class of SDEs, the It\u00f4 and Stratonovich interpretations.<span class=\"Apple-converted-space\">\u00a0 <\/span>In this project, the student will explore the theory of SDEs driven by L\u00e9vy processes, with particular attention to stochastic processes on graphs (the dynamics of a population of particles moving on a lattice would be an example) and with the goal of investigating (at least numerically) an analogue of the Stratonovich SDE interpretation for SDEs driven by L\u00e9vy processes.<\/p>\n<p class=\"p6\" style=\"text-align: center\"><a href=\"https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/176\/2014\/05\/15022517\/Norton_Woyczynski.pdf\">Paper<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Thomas Norton with Wojbor A. Woyczy\u0144ski<br \/>\n<i>Stochastic Differential Equations Driven by Compound Poisson and L\u00e9vy Processes<\/i><\/p>\n<p class=\"p6\">Stochastic differential equations (SDEs) provide a rigorous setting for investigating the dynamics of systems subject to \u201cnoise,\u201d by which we mean source terms whose behavior is not deterministic, but governed by a probability distribution.\u00a0 The noise can have quite irregular behavior (such as continuous but nowhere differentiable paths), giving rise to a rich theory of stochastic integration that allows for multiple interpretations of the same SDE.\u00a0 Most commonly, the noise is assumed to follow the same probability distribution as Brownian Motion,<\/p>\n<p><a href=\"https:\/\/casgroups.case.edu\/physics-senior-projects\/stochastic-differential-equations-driven-by-compound-poisson-and-levy-processes\/\" class=\"more-link\">Continue reading&#8230; <span class=\"screen-reader-text\">Stochastic Differential Equations Driven by Compound Poisson and L\u00e9vy Processes<\/span><\/a><\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[44,1,38,54,68,74],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/posts\/1841"}],"collection":[{"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/comments?post=1841"}],"version-history":[{"count":6,"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/posts\/1841\/revisions"}],"predecessor-version":[{"id":3043,"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/posts\/1841\/revisions\/3043"}],"wp:attachment":[{"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/media?parent=1841"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/categories?post=1841"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/casgroups.case.edu\/physics-senior-projects\/wp-json\/wp\/v2\/tags?post=1841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}