{"id":15256,"date":"2021-09-20T17:16:30","date_gmt":"2021-09-20T15:16:30","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=15256"},"modified":"2021-09-21T11:06:42","modified_gmt":"2021-09-21T09:06:42","slug":"completeness-and-right-continuity-of-filtrations","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2021\/09\/20\/completeness-and-right-continuity-of-filtrations\/","title":{"rendered":"Completeness and right-continuity of filtrations"},"content":{"rendered":"<figure id=\"attachment_15257\" aria-describedby=\"caption-attachment-15257\" style=\"width: 236px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" src=\"http:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2021\/09\/Luzin.jpeg\" alt=\"Nikolai Nikolaevich Luzin (1883 - 1950)\" width=\"236\" height=\"326\" class=\"size-full wp-image-15257\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2021\/09\/Luzin.jpeg 236w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2021\/09\/Luzin-217x300.jpeg 217w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><figcaption id=\"caption-attachment-15257\" class=\"wp-caption-text\">Nikolai Nikolaevich Luzin (1883 - 1950)<\/figcaption><\/figure>\n<p style=\"text-align: justify;\">If you believe that the completion and right-continuity of filtrations are typical abstract non sense of the general theory of stochastic processes, useless and obscure, you are maybe missing something interesting. Contrary to discrete time or space stochastic processes, continuous time and space stochastic processes lead naturally to measurability issues, when considering for instance natural objects such as running suprema or stopping times.<\/p>\n<p style=\"text-align: justify;\"><b>Negligible sets and completeness.<\/b> In a probability space \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\), we say that \\( {A\\subset\\Omega} \\) is <i>negligible<\/i> when there exists \\( {A'\\in\\mathcal{F}} \\) with \\( {A\\subset A'} \\) and \\( {\\mathbb{P}(A')=0} \\). We say that the \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) is <i>complete<\/i> when \\( {\\mathcal{F}} \\) contains the negligible subsets of \\( {\\Omega} \\). A filtration \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\) on \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) is <i>complete<\/i> when \\( {\\mathcal{F}_0} \\) contains the negligible elements of \\( {\\mathcal{F}} \\).<\/p>\n<p style=\"text-align: justify;\">Completeness emerges naturally via almost sure events which are complement of negligible sets (as for running supremum below) as well as via projections of measurable sets of product spaces (as for hitting times of Borel sets below) .<\/p>\n<p style=\"text-align: justify;\">We say that a process \\( {{(X_t)}_{t\\in\\mathbb{R}_+}} \\) taking values in a topological space equipped with its Borel \\( {\\sigma} \\)-field is <i>continuous<\/i> when it has almost surely continuous trajectories. This is the case for instance of Brownian motion constructed from random series.<\/p>\n<p style=\"text-align: justify;\"><b>Measurability of running supremum from completeness.<\/b> Let \\( {{(X_t)}_{t\\in\\mathbb{R}_+}} \\) be <i>continuous<\/i>, defined on a probability space \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\), and taking values in a topological space \\( {E} \\) equipped with its Borel \\( {\\sigma} \\)-field \\( {\\mathcal{E}} \\). Let \\( {f:E\\rightarrow\\mathbb{R}} \\) be a <i>measurable<\/i> function.<\/p>\n<ul>\n<li>If \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) is <i>complete<\/i> then \\( {\\sup_{s\\in[0,t]}f(X_s)} \\) is <i>measurable<\/i> for all \\( {t\\in\\mathbb{R}_+} \\).<\/li>\n<li>If \\( {X} \\) is <i>adapted<\/i> for a <i>complete<\/i> \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\) then \\( {{(\\sup_{s\\in[0,t]}f(X_s))}_{t\\in\\mathbb{R}_+}} \\) is <i>adapted<\/i>.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>Proof.<\/b> Let \\( {\\Omega'\\in\\mathcal{F}} \\) be an a.s. event on which \\( {X} \\) is continuous. Set \\( {S_t=\\sup_{s\\in[0,t]}f(X_s)} \\).<\/p>\n<ul>\n<li>We have \\( {\\Omega'\\in\\mathcal{F}} \\). Next, for all \\( {t\\in\\mathbb{R}_+} \\) and \\( {A\\in\\mathcal{E}} \\), we have\n<p style=\"text-align: center;\">\\[ \\Omega'\\cap\\{S_t\\in A\\} =\\Omega'\\cap \\bigr\\{\\sup_{s\\in[0,t]\\cap\\mathbb{Q}}f(X_s)\\in A\\bigr\\}\\in\\mathcal{F}, \\]<\/p>\n<p> while \\( {(\\Omega\\setminus\\Omega')\\cap\\{S_t\\in A\\}\\subset\\Omega\\setminus\\Omega'} \\) is negligible and thus in \\( {\\mathcal{F}} \\) by completeness.<\/li>\n<li>Same argument as before with \\( {\\mathcal{F}_t} \\) instead of \\( {\\mathcal{F}} \\).<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>Universal completeness.<\/b> The notion of completeness is relative to the probability measure \\( {\\mathbb{P}} \\). There is also a notion of universal completeness, that does not depend on the probability measure, see Dellacherie and Meyer 1978. This is not that useful in probability.<\/p>\n<p style=\"text-align: justify;\"><b>Stopping times.<\/b> A map \\( {T:\\Omega\\rightarrow[0,+\\infty]} \\) is a <i>stopping time<\/i> for \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\) when<\/p>\n<p style=\"text-align: center;\">\\[ \\{T\\leq t\\}\\in\\mathcal{F}_t \\]<\/p>\n<p style=\"text-align: justify;\">for all \\( {t\\in\\mathbb{R}_+} \\). Contrary to discrete time filtrations, the notion of stopping times for continuous time filtrations leads naturally to the notions of <i>complete<\/i> filtration and <i>right continuous<\/i> filtration. This is visible notably with <i>hitting times<\/i> as follows.<\/p>\n<p style=\"text-align: justify;\"><b>Hitting times as archetypal examples of stopping times.<\/b> Let \\( {X={(X_t)}_{t\\in\\mathbb{R}_+}} \\) be a <i>continuous<\/i> and <i>adapted<\/i> process defined on \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) with respect to a <i>complete<\/i> filtration \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\), and taking its values in a <i>metric space<\/i> \\( {G} \\) equipped with its Borel \\( {\\sigma} \\)-field. Then, for all <i>closed subset<\/i> \\( {A\\subset G} \\), the <i>hitting time<\/i> \\( {T_A:\\Omega\\rightarrow[0,+\\infty]} \\) of \\( {A} \\), given by<\/p>\n<p style=\"text-align: center;\">\\[ T_A=\\inf\\{t\\in\\mathbb{R}_+:X_t\\in A\\}, \\]<\/p>\n<p style=\"text-align: justify;\">with convention \\( {\\inf\\varnothing=+\\infty} \\), is a <i>stopping time<\/i>.<\/p>\n<p style=\"text-align: justify;\"><b>Proof.<\/b> Let \\( {\\Omega'} \\) be the a.s. event on which \\( {X} \\) is continuous. On \\( {\\Omega'} \\), since \\( {X} \\) is continuous and \\( {A} \\) is closed, we have \\( {\\{t\\in\\mathbb{R}_+:X_t\\in A\\}=\\{t\\in\\mathbb{R}_+:\\mathrm{dist}(X_t,A)=0\\}} \\), the map \\( {t\\in\\mathbb{R}_+\\mapsto\\mathrm{dist}(X_t,A)} \\) is continuous, and the \\( {\\inf} \\) in the definition of \\( {T_A} \\) is a \\( {\\min} \\). Now, since \\( {X} \\) is adapted, we have, for all \\( {t\\in\\mathbb{R}_+} \\),<\/p>\n<p style=\"text-align: center;\">\\[ \\Omega'\\cap\\{T_A\\leq t\\} =\\Omega'\\cap\\bigcap_{s\\in[0,t]\\cap\\mathbb{Q}}\\{X_s\\in A\\} \\in\\mathcal{F}_t, \\]<\/p>\n<p style=\"text-align: justify;\">where we have also used \\( {\\Omega'\\in\\mathcal{F}_t} \\) for all \\( {t\\in\\mathbb{R}_+} \\) since \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\) is complete. Moreover \\( {(\\Omega\\setminus\\Omega')\\cap\\{T_A\\leq t\\}\\subset\\Omega\\setminus\\Omega'} \\) is negligible, and thus in \\( {\\mathcal{F}_t} \\) by completeness.<\/p>\n<p style=\"text-align: justify;\"><b>Right-continuity.<\/b> A filtration \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\) is <i>right-continuous<\/i> if for all \\( {t\\in\\mathbb{R}_+} \\) we have<\/p>\n<p style=\"text-align: center;\">\\[ \\mathcal{F}_t=\\mathcal{F}_{t^+} \\quad\\text{where}\\quad \\mathcal{F}_{t+} =\\bigcap_{\\varepsilon&gt;0}\\mathcal{F}_{t+\\varepsilon} =\\bigcap_{s&gt;t}\\mathcal{F}_s. \\]<\/p>\n<p style=\"text-align: justify;\"><b>Alternative definition of stopping times.<\/b> If \\( {T:\\Omega\\rightarrow[0,+\\infty]} \\) is a stopping time with respect to a filtration \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\) then<\/p>\n<p style=\"text-align: center;\">\\[ \\{T&lt;t\\}\\in\\mathcal{F}_t \\]<\/p>\n<p style=\"text-align: justify;\">for all \\( {t\\in\\mathbb{R}_+} \\). Conversely this property implies that \\( {T} \\) is a stopping time when the filtration is <i>right-continuous<\/i>. Indeed, if \\( {T} \\) is a stopping time then for all \\( {t\\in\\mathbb{R}_+} \\) we have<\/p>\n<p style=\"text-align: center;\">\\[ \\{T&lt;t\\} =\\bigcup_{n=1}^\\infty\\{T\\leq t-{\\textstyle\\frac{1}{n}}\\} \\in\\mathcal{F}_t, \\]<\/p>\n<p style=\"text-align: justify;\">Conversely \\( {\\{T\\leq t\\}\\in\\cap_{s&gt;t}\\mathcal{F}_s=\\mathcal{F}_{t+}} \\) since for all \\( {s&gt;t} \\),<\/p>\n<p style=\"text-align: center;\">\\[ \\{T\\leq t\\} =\\bigcap_{n=1}^\\infty\\{T&lt;(t+{\\textstyle\\frac{1}{n}})\\wedge s\\} \\in\\mathcal{F}_{s}. \\]<\/p>\n<p style=\"text-align: justify;\">Note that if \\( {T} \\) is a stopping time then \\( {\\{T=t\\}=\\{T\\leq t\\}\\cap\\{T&lt;t\\}^c\\in\\mathcal{F}_t} \\).<\/p>\n<p style=\"text-align: justify;\"><b>Progressively measurable processes.<\/b> Recall that a process \\( {{(X_t)}_{t\\in\\mathbb{R}_+}} \\) defined on a probability space \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) is <i>progressively measurable<\/i> for \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\) when for all \\( {t\\in\\mathbb{R}_+} \\) the map \\( {(\\omega,s)\\in\\Omega\\times[0,t]\\mapsto X_s(\\omega)} \\) is measurable for \\( {\\mathcal{F}_t\\otimes\\mathcal{B}_{[0,t]}} \\). Example of <i>progressively measurable<\/i> processes include <i>adapted<\/i> <i>right-continuous<\/i> processes.<\/p>\n<p style=\"text-align: justify;\"><b>Hitting time of Borel sets.<\/b> Let \\( {X={(X_t)}_{t\\in\\mathbb{R}_+}} \\) be a <i>progressively measurable<\/i> process defined on a probability space \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) equipped with a <i>right continuous<\/i> and <i>complete<\/i> filtration \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\), and taking its values in a <i>measurable space<\/i> \\( {G} \\). Then for all <i>measurable subset<\/i> \\( {A\\subset G} \\), the <i>hitting time<\/i> \\( {T_A:\\Omega\\rightarrow[0,+\\infty]} \\) defined by<\/p>\n<p style=\"text-align: center;\">\\[ T_A=\\inf\\{t\\in\\mathbb{R}_+:X_t\\in A\\}, \\]<\/p>\n<p style=\"text-align: justify;\">with convention \\( {\\inf\\varnothing=+\\infty} \\), is a <i>stopping time<\/i>.<\/p>\n<p style=\"text-align: justify;\"><b>Proof.<\/b> The <i>debut<\/i> \\( {D_B} \\) of any \\( {B\\in\\mathcal{F}\\otimes\\mathcal{B}(\\mathbb{R}_+)} \\) is defined for all \\( {\\omega\\in\\Omega} \\) by<\/p>\n<p style=\"text-align: center;\">\\[ D_B(\\omega)=\\inf\\{t\\in\\mathbb{R}_+:(\\omega,t)\\in B\\}\\in[0,+\\infty]. \\]<\/p>\n<p style=\"text-align: justify;\">If \\( {B} \\) is progressive, then \\( {D_B} \\) is a stopping time (this is known as the <i>debut theorem<\/i>). Indeed, for all \\( {t\\in\\mathbb{R}_+} \\) the set \\( {\\{D_B&lt;t\\}} \\) is then the projection on \\( {\\Omega} \\) of<\/p>\n<p style=\"text-align: center;\">\\[ C=\\{s\\in[0,t):(\\omega,s)\\in B\\}, \\]<\/p>\n<p style=\"text-align: justify;\">which belongs to \\( {\\mathcal{B}(\\mathbb{R}_+)\\otimes\\mathcal{F}_t} \\) since \\( {B} \\) is progressive. Since the filtration is complete, this projection belongs to \\( {\\mathcal{F}_t} \\), see for instance Dellacherie and Meyer Theorem IV.50 page 116. Now \\( {\\{D_B&lt;t\\}\\in\\mathcal{F}_t} \\) for all \\( {t\\in\\mathbb{R}_+} \\) implies that \\( {D_B} \\) is a stopping time since the filtration is right continuous. Finally it remains to note that<\/p>\n<p style=\"text-align: center;\">\\[ T_A=D_B\\quad\\text{with}\\quad B=\\{(\\omega,t):X_t\\in A\\}, \\]<\/p>\n<p style=\"text-align: justify;\">which is progressive as pre-image of \\( {\\mathbb{R}_+\\times A} \\) by \\( {(\\omega,t)\\mapsto X_t(\\omega)} \\) (\\( {X} \\) is progressive).<\/p>\n<p style=\"text-align: justify;\"><b>A famous mistake.<\/b> This is related to a famous mistake made by <a href= \"https:\/\/en.wikipedia.org\/wiki\/Henri_Lebesgue\">Henri Lebesgue<\/a> (1875 -- 1941) on the measurability of projections of measurable sets in product spaces, that motivated <a href= \"https:\/\/en.wikipedia.org\/wiki\/Nikolai_Luzin\">Nikolai Luzin<\/a> (1883 -- 1950) and his student <a href= \"https:\/\/en.wikipedia.org\/wiki\/Mikhail_Suslin\">Mikhail Suslin<\/a> (1894 -- 1919) to forge the concept of <a href= \"https:\/\/en.wikipedia.org\/wiki\/Analytic_set\">analytic set<\/a> and <a href= \"https:\/\/en.wikipedia.org\/wiki\/Descriptive_set_theory\">descriptive set theory<\/a>.<\/p>\n<p style=\"text-align: justify;\">``<em>[\u2026] Sans que le terme de tribu soit utilis\u00e9 \u00e0 l\u2019\u00e9poque, il semblait \u00e0 Borel qu\u2019aucune op\u00e9ration de l\u2019analyse ne ferait jamais sortir de la tribu bor\u00e9lienne. C\u2019\u00e9tait aussi l\u2019avis de Lebesgue, et il avait cru le d\u00e9montrer en 1905. Rarement erreur a \u00e9t\u00e9 plus fructueuse. Au d\u00e9but de l\u2019ann\u00e9e 1917, les Comptes rendus publient deux notes des Russes Nicolas Lusin et M. Ya. Souslin. [\u2026] La projection d\u2019un bor\u00e9lien n\u2019est pas n\u00e9cessairement un bor\u00e9lien. L\u2019analyse classique force donc \u00e0 sortir de la tribu bor\u00e9lienne. Entre la tribu de Borel et celle de Lebesgue se trouve la tribu de Lusin, constitu\u00e9e par les ensembles que Lusin appelle analytiques et qui sont des images continues de bor\u00e9liens. Au cours des ann\u00e9es 1920 s\u2019est d\u00e9velopp\u00e9e \u00e0 Moscou une \u00e9cole math\u00e9matique extr\u00eamement brillante, dont Lusin a \u00e9t\u00e9 le fondateur. Ainsi Lebesgue et son \u0153uvre ont \u00e9t\u00e9 beaucoup mieux connus \u00e0 Moscou qu\u2019ils ne l\u2019\u00e9taient en France. Hongrie, Pologne et Russie ont \u00e9t\u00e9 les foyers de rayonnement de la pens\u00e9e de Lebesgue et de son h\u00e9ritage. [\u2026]<\/em>''<\/p>\n<p style=\"text-align: justify;\">Jean-Pierre Kahane (2001)<\/p>\n<p style=\"text-align: justify;\"><b>Canonical filtration.<\/b> It is customary to assume that the underlying filtration is right-continuous and complete. For a given filtration \\( {{(\\mathcal{F}_t)}_{t\\in\\mathbb{R}_+}} \\), it is always possible to consider its <i>completion<\/i> \\( {(\\sigma_t)}_{t\\in\\mathbb{R}_+}={(\\sigma(\\mathcal{N}\\cup\\mathcal{F}_t))}_{t\\in\\mathbb{R}_+} \\) where \\( {\\mathcal{N}} \\) is the collection of negligible elements of \\( {\\mathcal{F}} \\). It is also customary to consider the <i>right-continuous version<\/i> \\( {{(\\sigma_{t+})}_{t\\in\\mathbb{R}_+}} \\), called the <i>canonical filtration<\/i>. A process is always adapted with respect to the canonical filtration constructed from its completed natural filtration.<\/p>\n<p style=\"text-align: justify;\"><b>Subtleties about righ-continuity of filtrations.<\/b> The natural filtration of a right-continuous process is <i>not right-continuous in general<\/i>, indeed a counter example is given by \\( {X_t=tZ} \\) for all \\( {t\\in\\mathbb{R}_+} \\) where \\( {Z} \\) is a non-constant random variable. Indeed, we have \\( {\\sigma(X_0)=\\{\\varnothing,\\Omega\\}} \\) while \\( {\\sigma(X_{0+\\varepsilon}:\\varepsilon&gt;0)=\\sigma(Z)\\neq\\sigma(X_0)} \\). However it can be shown that the completion of the natural filtration of a Feller Markov process, including all L\u00e9vy processes and in particular Brownian motion, is always right-continuous.<\/p>\n<p style=\"text-align: justify;\"><b>Further reading.<\/b><\/p>\n<ul>\n<li><a href= \"https:\/\/en.wikipedia.org\/wiki\/Claude_Dellacherie\">Claude Dellacherie<\/a> and <a href= \"https:\/\/en.wikipedia.org\/wiki\/Paul-Andr%C3%A9_Meyer\">Paul-Andr\u00e9 Meyer<\/a><br \/> <a href=\"https:\/\/zbmath.org\/?q=an:0494.60001\">Probabilities and potential<\/a><br \/> North-Holland Mathematics Studies 29 (1978)<\/li>\n<li><a href= \"https:\/\/en.wikipedia.org\/wiki\/Jean-Pierre_Kahane\">Jean-Pierre Kahane<\/a><br \/> <a href= \"http:\/\/smf4.emath.fr\/Publications\/Gazette\/2001\/89\/smf_gazette_89_5-20.pdf\"> Naissance et post\u00e9rit\u00e9 de l'int\u00e9grale de Lebesgue<\/a><br \/> Gazette des math\u00e9maticiens 89 (2001)<\/li>\n<li><a href= \"https:\/\/djalil.chafai.net\/blog\/2016\/03\/21\/integration-alpha-et-omega\/\"> Int\u00e9gration \u2013 alpha et omega<\/a><br \/> On this blog (2016)<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>If you believe that the completion and right-continuity of filtrations are typical abstract non sense of the general theory of stochastic processes, useless and obscure,&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2021\/09\/20\/completeness-and-right-continuity-of-filtrations\/\">Continue reading<span class=\"screen-reader-text\">Completeness and right-continuity of filtrations<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1962},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/15256"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=15256"}],"version-history":[{"count":19,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/15256\/revisions"}],"predecessor-version":[{"id":15278,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/15256\/revisions\/15278"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=15256"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=15256"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=15256"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}