--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Stopping_Time.thy Thu Oct 20 18:41:59 2016 +0200
@@ -0,0 +1,262 @@
+(* Author: Johannes Hölzl <hoelzl@in.tum.de> *)
+
+section {* Stopping times *}
+
+theory Stopping_Time
+ imports "../Analysis/Analysis"
+begin
+
+subsection \<open>Stopping Time\<close>
+
+text \<open>This is also called strong stopping time. Then stopping time is T with alternative is
+ \<open>T x < t\<close> measurable.\<close>
+
+definition stopping_time :: "('t::linorder \<Rightarrow> 'a measure) \<Rightarrow> ('a \<Rightarrow> 't) \<Rightarrow> bool"
+where
+ "stopping_time F T = (\<forall>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t))"
+
+lemma stopping_time_cong: "(\<And>t x. x \<in> space (F t) \<Longrightarrow> T x = S x) \<Longrightarrow> stopping_time F T = stopping_time F S"
+ unfolding stopping_time_def by (intro arg_cong[where f=All] ext measurable_cong) simp
+
+lemma stopping_timeD: "stopping_time F T \<Longrightarrow> Measurable.pred (F t) (\<lambda>x. T x \<le> t)"
+ by (auto simp: stopping_time_def)
+
+lemma stopping_timeD2: "stopping_time F T \<Longrightarrow> Measurable.pred (F t) (\<lambda>x. t < T x)"
+ unfolding not_le[symmetric] by (auto intro: stopping_timeD Measurable.pred_intros_logic)
+
+lemma stopping_timeI[intro?]: "(\<And>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t)) \<Longrightarrow> stopping_time F T"
+ by (auto simp: stopping_time_def)
+
+lemma measurable_stopping_time:
+ fixes T :: "'a \<Rightarrow> 't::{linorder_topology, second_countable_topology}"
+ assumes T: "stopping_time F T"
+ and M: "\<And>t. sets (F t) \<subseteq> sets M" "\<And>t. space (F t) = space M"
+ shows "T \<in> M \<rightarrow>\<^sub>M borel"
+proof (rule borel_measurableI_le)
+ show "{x \<in> space M. T x \<le> t} \<in> sets M" for t
+ using stopping_timeD[OF T] M by (auto simp: Measurable.pred_def)
+qed
+
+lemma stopping_time_const: "stopping_time F (\<lambda>x. c)"
+ by (auto simp: stopping_time_def)
+
+lemma stopping_time_min:
+ "stopping_time F T \<Longrightarrow> stopping_time F S \<Longrightarrow> stopping_time F (\<lambda>x. min (T x) (S x))"
+ by (auto simp: stopping_time_def min_le_iff_disj intro!: pred_intros_logic)
+
+lemma stopping_time_max:
+ "stopping_time F T \<Longrightarrow> stopping_time F S \<Longrightarrow> stopping_time F (\<lambda>x. max (T x) (S x))"
+ by (auto simp: stopping_time_def intro!: pred_intros_logic)
+
+section \<open>Filtration\<close>
+
+locale filtration =
+ fixes \<Omega> :: "'a set" and F :: "'t::{linorder_topology, second_countable_topology} \<Rightarrow> 'a measure"
+ assumes space_F: "\<And>i. space (F i) = \<Omega>"
+ assumes sets_F_mono: "\<And>i j. i \<le> j \<Longrightarrow> sets (F i) \<le> sets (F j)"
+begin
+
+subsection \<open>$\sigma$-algebra of a Stopping Time\<close>
+
+definition pre_sigma :: "('a \<Rightarrow> 't) \<Rightarrow> 'a measure"
+where
+ "pre_sigma T = sigma \<Omega> {A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+
+lemma space_pre_sigma: "space (pre_sigma T) = \<Omega>"
+ unfolding pre_sigma_def using sets.space_closed[of "F _"]
+ by (intro space_measure_of) (auto simp: space_F)
+
+lemma measure_pre_sigma[simp]: "emeasure (pre_sigma T) = (\<lambda>_. 0)"
+ by (simp add: pre_sigma_def emeasure_sigma)
+
+lemma sigma_algebra_pre_sigma:
+ assumes T: "stopping_time F T"
+ shows "sigma_algebra \<Omega> {A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+ unfolding sigma_algebra_iff2
+proof (intro sigma_algebra_iff2[THEN iffD2] conjI ballI allI impI CollectI)
+ show "{A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)} \<subseteq> Pow \<Omega>"
+ using sets.space_closed[of "F _"] by (auto simp: space_F)
+next
+ fix A t assume "A \<in> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
+ then have "{\<omega> \<in> space (F t). T \<omega> \<le> t} - {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)"
+ using T stopping_timeD[measurable] by auto
+ also have "{\<omega> \<in> space (F t). T \<omega> \<le> t} - {\<omega> \<in> A. T \<omega> \<le> t} = {\<omega> \<in> \<Omega> - A. T \<omega> \<le> t}"
+ by (auto simp: space_F)
+ finally show "{\<omega> \<in> \<Omega> - A. T \<omega> \<le> t} \<in> sets (F t)" .
+next
+ fix AA :: "nat \<Rightarrow> 'a set" and t assume "range AA \<subseteq> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
+ then have "(\<Union>i. {\<omega> \<in> AA i. T \<omega> \<le> t}) \<in> sets (F t)" for t
+ by auto
+ also have "(\<Union>i. {\<omega> \<in> AA i. T \<omega> \<le> t}) = {\<omega> \<in> UNION UNIV AA. T \<omega> \<le> t}"
+ by auto
+ finally show "{\<omega> \<in> UNION UNIV AA. T \<omega> \<le> t} \<in> sets (F t)" .
+qed auto
+
+lemma sets_pre_sigma: "stopping_time F T \<Longrightarrow> sets (pre_sigma T) = {A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+ unfolding pre_sigma_def by (rule sigma_algebra.sets_measure_of_eq[OF sigma_algebra_pre_sigma])
+
+lemma sets_pre_sigmaI: "stopping_time F T \<Longrightarrow> (\<And>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)) \<Longrightarrow> A \<in> sets (pre_sigma T)"
+ unfolding sets_pre_sigma by auto
+
+lemma pred_pre_sigmaI:
+ assumes T: "stopping_time F T"
+ shows "(\<And>t. Measurable.pred (F t) (\<lambda>\<omega>. P \<omega> \<and> T \<omega> \<le> t)) \<Longrightarrow> Measurable.pred (pre_sigma T) P"
+ unfolding pred_def space_F space_pre_sigma by (intro sets_pre_sigmaI[OF T]) simp
+
+lemma sets_pre_sigmaD: "stopping_time F T \<Longrightarrow> A \<in> sets (pre_sigma T) \<Longrightarrow> {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)"
+ unfolding sets_pre_sigma by auto
+
+lemma stopping_time_le_const: "stopping_time F T \<Longrightarrow> s \<le> t \<Longrightarrow> Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> \<le> s)"
+ using stopping_timeD[of F T] sets_F_mono[of _ t] by (auto simp: pred_def space_F)
+
+lemma measurable_stopping_time_pre_sigma:
+ assumes T: "stopping_time F T" shows "T \<in> pre_sigma T \<rightarrow>\<^sub>M borel"
+proof (intro borel_measurableI_le sets_pre_sigmaI[OF T])
+ fix t t'
+ have "{\<omega>\<in>space (F (min t' t)). T \<omega> \<le> min t' t} \<in> sets (F (min t' t))"
+ using T unfolding pred_def[symmetric] by (rule stopping_timeD)
+ also have "\<dots> \<subseteq> sets (F t)"
+ by (rule sets_F_mono) simp
+ finally show "{\<omega> \<in> {x \<in> space (pre_sigma T). T x \<le> t'}. T \<omega> \<le> t} \<in> sets (F t)"
+ by (simp add: space_pre_sigma space_F)
+qed
+
+lemma mono_pre_sigma:
+ assumes T: "stopping_time F T" and S: "stopping_time F S"
+ and le: "\<And>\<omega>. \<omega> \<in> \<Omega> \<Longrightarrow> T \<omega> \<le> S \<omega>"
+ shows "sets (pre_sigma T) \<subseteq> sets (pre_sigma S)"
+ unfolding sets_pre_sigma[OF S] sets_pre_sigma[OF T]
+proof safe
+ interpret sigma_algebra \<Omega> "{A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+ using T by (rule sigma_algebra_pre_sigma)
+ fix A t assume A: "\<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)"
+ then have "A \<subseteq> \<Omega>"
+ using sets_into_space by auto
+ from A have "{\<omega>\<in>A. T \<omega> \<le> t} \<inter> {\<omega>\<in>space (F t). S \<omega> \<le> t} \<in> sets (F t)"
+ using stopping_timeD[OF S] by (auto simp: pred_def)
+ also have "{\<omega>\<in>A. T \<omega> \<le> t} \<inter> {\<omega>\<in>space (F t). S \<omega> \<le> t} = {\<omega>\<in>A. S \<omega> \<le> t}"
+ using \<open>A \<subseteq> \<Omega>\<close> sets_into_space[of A] le by (auto simp: space_F intro: order_trans)
+ finally show "{\<omega>\<in>A. S \<omega> \<le> t} \<in> sets (F t)"
+ by auto
+qed
+
+lemma stopping_time_less_const:
+ assumes T: "stopping_time F T" shows "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> < t)"
+proof -
+ guess D :: "'t set" by (rule countable_dense_setE)
+ note D = this
+ show ?thesis
+ proof cases
+ assume *: "\<forall>t'<t. \<exists>t''. t' < t'' \<and> t'' < t"
+ { fix t' assume "t' < t"
+ with * have "{t' <..< t} \<noteq> {}"
+ by fastforce
+ with D(2)[OF _ this]
+ have "\<exists>d\<in>D. t'< d \<and> d < t"
+ by auto }
+ note ** = this
+
+ show ?thesis
+ proof (rule measurable_cong[THEN iffD2])
+ show "T \<omega> < t \<longleftrightarrow> (\<exists>r\<in>{r\<in>D. r < t}. T \<omega> \<le> r)" for \<omega>
+ by (auto dest: ** intro: less_imp_le)
+ show "Measurable.pred (F t) (\<lambda>w. \<exists>r\<in>{r \<in> D. r < t}. T w \<le> r)"
+ by (intro measurable_pred_countable stopping_time_le_const[OF T] countable_Collect D) auto
+ qed
+ next
+ assume "\<not> (\<forall>t'<t. \<exists>t''. t' < t'' \<and> t'' < t)"
+ then obtain t' where t': "t' < t" "\<And>t''. t'' < t \<Longrightarrow> t'' \<le> t'"
+ by (auto simp: not_less[symmetric])
+ show ?thesis
+ proof (rule measurable_cong[THEN iffD2])
+ show "T \<omega> < t \<longleftrightarrow> T \<omega> \<le> t'" for \<omega>
+ using t' by auto
+ show "Measurable.pred (F t) (\<lambda>w. T w \<le> t')"
+ using \<open>t'<t\<close> by (intro stopping_time_le_const[OF T]) auto
+ qed
+ qed
+qed
+
+lemma stopping_time_eq_const: "stopping_time F T \<Longrightarrow> Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> = t)"
+ unfolding eq_iff using stopping_time_less_const[of T t]
+ by (intro pred_intros_logic stopping_time_le_const) (auto simp: not_less[symmetric] )
+
+lemma stopping_time_less:
+ assumes T: "stopping_time F T" and S: "stopping_time F S"
+ shows "Measurable.pred (pre_sigma T) (\<lambda>\<omega>. T \<omega> < S \<omega>)"
+proof (rule pred_pre_sigmaI[OF T])
+ fix t
+ obtain D :: "'t set"
+ where [simp]: "countable D" and semidense_D: "\<And>x y. x < y \<Longrightarrow> (\<exists>b\<in>D. x \<le> b \<and> b < y)"
+ using countable_separating_set_linorder2 by auto
+ show "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> < S \<omega> \<and> T \<omega> \<le> t)"
+ proof (rule measurable_cong[THEN iffD2])
+ let ?f = "\<lambda>\<omega>. if T \<omega> = t then \<not> S \<omega> \<le> t else \<exists>s\<in>{s\<in>D. s \<le> t}. T \<omega> \<le> s \<and> \<not> (S \<omega> \<le> s)"
+ { fix \<omega> assume "T \<omega> \<le> t" "T \<omega> \<noteq> t" "T \<omega> < S \<omega>"
+ then have "T \<omega> < min t (S \<omega>)"
+ by auto
+ then obtain r where "r \<in> D" "T \<omega> \<le> r" "r < min t (S \<omega>)"
+ by (metis semidense_D)
+ then have "\<exists>s\<in>{s\<in>D. s \<le> t}. T \<omega> \<le> s \<and> s < S \<omega>"
+ by auto }
+ then show "(T \<omega> < S \<omega> \<and> T \<omega> \<le> t) = ?f \<omega>" for \<omega>
+ by (auto simp: not_le)
+ show "Measurable.pred (F t) ?f"
+ by (intro pred_intros_logic measurable_If measurable_pred_countable countable_Collect
+ stopping_time_le_const predE stopping_time_eq_const T S)
+ auto
+ qed
+qed
+
+end
+
+lemma stopping_time_SUP_enat:
+ fixes T :: "nat \<Rightarrow> ('a \<Rightarrow> enat)"
+ shows "(\<And>i. stopping_time F (T i)) \<Longrightarrow> stopping_time F (SUP i. T i)"
+ unfolding stopping_time_def SUP_apply SUP_le_iff by (auto intro!: pred_intros_countable)
+
+lemma less_eSuc_iff: "a < eSuc b \<longleftrightarrow> (a \<le> b \<and> a \<noteq> \<infinity>)"
+ by (cases a) auto
+
+lemma stopping_time_Inf_enat:
+ fixes F :: "enat \<Rightarrow> 'a measure"
+ assumes F: "filtration \<Omega> F"
+ assumes P: "\<And>i. Measurable.pred (F i) (P i)"
+ shows "stopping_time F (\<lambda>\<omega>. Inf {i. P i \<omega>})"
+proof (rule stopping_timeI, cases)
+ fix t :: enat assume "t = \<infinity>" then show "Measurable.pred (F t) (\<lambda>\<omega>. Inf {i. P i \<omega>} \<le> t)"
+ by auto
+next
+ fix t :: enat assume "t \<noteq> \<infinity>"
+ moreover
+ { fix i \<omega> assume "Inf {i. P i \<omega>} \<le> t"
+ with \<open>t \<noteq> \<infinity>\<close> have "(\<exists>i\<le>t. P i \<omega>)"
+ unfolding Inf_le_iff by (cases t) (auto elim!: allE[of _ "eSuc t"] simp: less_eSuc_iff) }
+ ultimately have *: "\<And>\<omega>. Inf {i. P i \<omega>} \<le> t \<longleftrightarrow> (\<exists>i\<in>{..t}. P i \<omega>)"
+ by (auto intro!: Inf_lower2)
+ show "Measurable.pred (F t) (\<lambda>\<omega>. Inf {i. P i \<omega>} \<le> t)"
+ unfolding * using filtration.sets_F_mono[OF F, of _ t] P
+ by (intro pred_intros_countable_bounded) (auto simp: pred_def filtration.space_F[OF F])
+qed
+
+lemma stopping_time_Inf_nat:
+ fixes F :: "nat \<Rightarrow> 'a measure"
+ assumes F: "filtration \<Omega> F"
+ assumes P: "\<And>i. Measurable.pred (F i) (P i)" and wf: "\<And>i \<omega>. \<omega> \<in> \<Omega> \<Longrightarrow> \<exists>n. P n \<omega>"
+ shows "stopping_time F (\<lambda>\<omega>. Inf {i. P i \<omega>})"
+ unfolding stopping_time_def
+proof (intro allI, subst measurable_cong)
+ fix t \<omega> assume "\<omega> \<in> space (F t)"
+ then have "\<omega> \<in> \<Omega>"
+ using filtration.space_F[OF F] by auto
+ from wf[OF this] have "((LEAST n. P n \<omega>) \<le> t) = (\<exists>i\<le>t. P i \<omega>)"
+ by (rule LeastI2_wellorder_ex) auto
+ then show "(Inf {i. P i \<omega>} \<le> t) = (\<exists>i\<in>{..t}. P i \<omega>)"
+ by (simp add: Inf_nat_def Bex_def)
+next
+ fix t from P show "Measurable.pred (F t) (\<lambda>w. \<exists>i\<in>{..t}. P i w)"
+ using filtration.sets_F_mono[OF F, of _ t]
+ by (intro pred_intros_countable_bounded) (auto simp: pred_def filtration.space_F[OF F])
+qed
+
+end