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	<title>Almost Sure</title>
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	<description>A random mathematical blog</description>
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		<title>Almost Sure</title>
		<link>http://almostsure.wordpress.com</link>
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		<item>
		<title>The Doob-Meyer Decomposition</title>
		<link>http://almostsure.wordpress.com/2011/12/30/the-doob-meyer-decomposition/</link>
		<comments>http://almostsure.wordpress.com/2011/12/30/the-doob-meyer-decomposition/#comments</comments>
		<pubDate>Fri, 30 Dec 2011 12:30:40 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[Doob-Meyer decomposition]]></category>
		<category><![CDATA[Martingale]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[predictable process]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Submartingale]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1443</guid>
		<description><![CDATA[The Doob-Meyer decomposition was a very important result, historically, in the development of stochastic calculus. This theorem states that every cadlag submartingale uniquely decomposes as the sum of a local martingale and an increasing predictable process. For one thing, if X is a square-integrable martingale then Jensen&#8217;s inequality implies that is a submartingale, so the [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1443&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>3</slash:comments>
	
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			<media:title type="html">George</media:title>
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		<title>Compensators of Counting Processes</title>
		<link>http://almostsure.wordpress.com/2011/12/27/compensators-of-counting-processes/</link>
		<comments>http://almostsure.wordpress.com/2011/12/27/compensators-of-counting-processes/#comments</comments>
		<pubDate>Tue, 27 Dec 2011 12:30:13 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[compensators]]></category>
		<category><![CDATA[counting process]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[Poisson process]]></category>
		<category><![CDATA[predictable process]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Stopping Time]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1418</guid>
		<description><![CDATA[A counting process, X, is defined to be an adapted stochastic process starting from zero which is piecewise constant and right-continuous with jumps of size 1. That is, letting be the first time at which , then By the debut theorem, are stopping times. So, X is an increasing integer valued process counting the arrivals [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1418&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">George</media:title>
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		<title>Compensators of Stopping Times</title>
		<link>http://almostsure.wordpress.com/2011/12/20/compensators-of-stopping-times/</link>
		<comments>http://almostsure.wordpress.com/2011/12/20/compensators-of-stopping-times/#comments</comments>
		<pubDate>Tue, 20 Dec 2011 12:30:37 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[compensators]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[predictable process]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Stopping Time]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1405</guid>
		<description><![CDATA[The previous post introduced the concept of the compensator of a process, which is known to exist for all locally integrable semimartingales. In this post, I&#8217;ll just look at the very special case of compensators of processes consisting of a single jump of unit size. Definition 1 Let be a stopping time. The compensator of [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1405&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">George</media:title>
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		<item>
		<title>Compensators</title>
		<link>http://almostsure.wordpress.com/2011/11/22/compensators/</link>
		<comments>http://almostsure.wordpress.com/2011/11/22/compensators/#comments</comments>
		<pubDate>Tue, 22 Nov 2011 12:30:17 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[compensators]]></category>
		<category><![CDATA[FV process]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[predictable process]]></category>
		<category><![CDATA[Semimartingale]]></category>
		<category><![CDATA[Special Semimartingale]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1316</guid>
		<description><![CDATA[A very common technique when looking at general stochastic processes is to break them down into separate martingale and drift terms. This is easiest to describe in the discrete time situation. So, suppose that is a stochastic process adapted to the discrete-time filtered probability space . If X is integrable, then it is possible to [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1316&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://almostsure.wordpress.com/2011/11/22/compensators/feed/</wfw:commentRss>
		<slash:comments>13</slash:comments>
	
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			<media:title type="html">George</media:title>
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		<item>
		<title>Special Semimartingales</title>
		<link>http://almostsure.wordpress.com/2011/10/03/special-semimartingales/</link>
		<comments>http://almostsure.wordpress.com/2011/10/03/special-semimartingales/#comments</comments>
		<pubDate>Mon, 03 Oct 2011 12:00:32 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[compensators]]></category>
		<category><![CDATA[FV process]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[predictable process]]></category>
		<category><![CDATA[Semimartingale]]></category>
		<category><![CDATA[Special Semimartingale]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Stochastic Integration]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1280</guid>
		<description><![CDATA[For stochastic processes in discrete time, the Doob decomposition uniquely decomposes any integrable process into the sum of a martingale and a predictable process. If is an integrable process adapted to a filtration then we write . Here, M is a martingale, so that , and A is predictable with . By saying that A [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1280&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>14</slash:comments>
	
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			<media:title type="html">George</media:title>
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	</item>
		<item>
		<title>Predictable FV Processes</title>
		<link>http://almostsure.wordpress.com/2011/07/18/predictable-fv-processes/</link>
		<comments>http://almostsure.wordpress.com/2011/07/18/predictable-fv-processes/#comments</comments>
		<pubDate>Mon, 18 Jul 2011 11:00:53 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[FV process]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[predictable process]]></category>
		<category><![CDATA[Semimartingale]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Stochastic Integration]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1247</guid>
		<description><![CDATA[By definition, an FV process is a cadlag adapted stochastic process which almost surely has finite variation over finite time intervals. These are always semimartingales, because the stochastic integral for bounded integrands can be constructed by taking the Lebesgue-Stieltjes integral along sample paths. Also, from the previous post on continuous semimartingales, we know that the [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1247&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">George</media:title>
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	</item>
		<item>
		<title>Predictable Stopping Times</title>
		<link>http://almostsure.wordpress.com/2011/05/26/predictable-stopping-times-2/</link>
		<comments>http://almostsure.wordpress.com/2011/05/26/predictable-stopping-times-2/#comments</comments>
		<pubDate>Thu, 26 May 2011 11:30:51 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[Feller process]]></category>
		<category><![CDATA[Martingale]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[predictable process]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Stopping Time]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1115</guid>
		<description><![CDATA[Although this post is under the heading of `the general theory of semimartingales&#8217; it is not, strictly speaking, about semimartingales at all. Instead, I will be concerned with a characterization of predictable stopping times. The reason for including this now is twofold. First, the results are too advanced to have been proven in the earlier [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1115&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>9</slash:comments>
	
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			<media:title type="html">George</media:title>
		</media:content>
	</item>
		<item>
		<title>Continuous Semimartingales</title>
		<link>http://almostsure.wordpress.com/2011/05/03/continuous-semimartingales/</link>
		<comments>http://almostsure.wordpress.com/2011/05/03/continuous-semimartingales/#comments</comments>
		<pubDate>Tue, 03 May 2011 11:00:34 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[Brownian Motion]]></category>
		<category><![CDATA[FV process]]></category>
		<category><![CDATA[Ito Process]]></category>
		<category><![CDATA[Local Martingale]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[Semimartingale]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Stochastic Integration]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1210</guid>
		<description><![CDATA[A stochastic process is a semimartingale if and only if it can be decomposed as the sum of a local martingale and an FV process. This is stated by the Bichteler-Dellacherie theorem or, alternatively, is often taken as the definition of a semimartingale. For continuous semimartingales, which are the subject of this post, things simplify [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1210&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">George</media:title>
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		<item>
		<title>The Bichteler-Dellacherie Theorem</title>
		<link>http://almostsure.wordpress.com/2011/03/28/the-bichteler-dellacherie-theorem/</link>
		<comments>http://almostsure.wordpress.com/2011/03/28/the-bichteler-dellacherie-theorem/#comments</comments>
		<pubDate>Mon, 28 Mar 2011 11:30:35 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[Bichteler-Dellacherie Theorem]]></category>
		<category><![CDATA[FV process]]></category>
		<category><![CDATA[Martingale]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[Semimartingale]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>
		<category><![CDATA[Stochastic Integration]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1162</guid>
		<description><![CDATA[In this post, I will give a statement and proof of the Bichteler-Dellacherie theorem describing the space of semimartingales. A semimartingale, as defined in these notes, is a cadlag adapted stochastic process X such that the stochastic integral is well-defined for all bounded predictable integrands . More precisely, an integral should exist which agrees with [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1162&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>11</slash:comments>
	
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			<media:title type="html">George</media:title>
		</media:content>
	</item>
		<item>
		<title>The General Theory of Semimartingales</title>
		<link>http://almostsure.wordpress.com/2011/03/11/the-general-theory-of-semimartingales/</link>
		<comments>http://almostsure.wordpress.com/2011/03/11/the-general-theory-of-semimartingales/#comments</comments>
		<pubDate>Fri, 11 Mar 2011 12:30:19 +0000</pubDate>
		<dc:creator>George Lowther</dc:creator>
				<category><![CDATA[Stochastic Calculus Notes]]></category>
		<category><![CDATA[The General Theory of Semimartingales]]></category>
		<category><![CDATA[Martingale]]></category>
		<category><![CDATA[math.PR]]></category>
		<category><![CDATA[Semimartingale]]></category>
		<category><![CDATA[Stochastic Calculus]]></category>

		<guid isPermaLink="false">http://almostsure.wordpress.com/?p=1142</guid>
		<description><![CDATA[Having completed the series of posts applying the methods of stochastic calculus to various special types of processes, I now return to the development of the theory. The next few posts of these notes will be grouped under the heading `The General Theory of Semimartingales&#8217;. Subjects which will be covered include the classification of predictable [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=almostsure.wordpress.com&amp;blog=9609325&amp;post=1142&amp;subd=almostsure&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>7</slash:comments>
	
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			<media:title type="html">George</media:title>
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