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[EQUATION] Proof. The necessity. Let [MATH] be a local regular super-martingale. Then [EQUATION] From here we obtain [MATH] Due to the uniform integrability of [MATH] and [MATH] we obtain |
[EQUATION] where [MATH] [MATH] since [MATH] But [MATH] From ( 33 ) we have [MATH] The last equality gives [MATH] or [EQUATION] The sufficiency. If the conditions of Theorem are satisfied, then [MATH] is a martingale, where [MATH] The last implies the local regularity of [MATH] |
Theorem is proved. Description of local regular super-martingales relative to an arbitrary convex set of equivalent measures. Below, in the paper we assume that an arbitrary convex set of equivalent measures [MATH] on a measurable space [MATH] and a filtration [MATH] on it satisfies the conditions: the density [MATH] i... |
Introduce into consideration a set [MATH] of all integrable nonnegative random values [MATH] relative to a convex set of equivalent measures [MATH] satisfying conditions |
[EQUATION] It is evident that the set [MATH] is not empty, since contains the random value [MATH] More interesting case is as [MATH] contains more then one element. |
Lemma 9 On a measurable space [MATH] and a filtration [MATH] on it, let [MATH] be an arbitrary convex set of equivalent measures. If the nonnegative random value [MATH] is such that [MATH] then |
[MATH] is a super-martingale relative to the convex set of equivalent measures [MATH] Proof. From the definition of [MATH] , for every [MATH] there exists a countable set [MATH] such that |
[EQUATION] The set [MATH] is also countable one and the equality [EQUATION] is true. Really, since [EQUATION] From the other side, |
[EQUATION] The last gives [EQUATION] The inequalities ( 38 ), ( 40 ) prove the needed statement. So, for all [MATH] we can choose the common set [MATH] Let [MATH] Due to Lemma |
, for every [MATH] we have [EQUATION] where [EQUATION] It is evident that [MATH] tends to [MATH] monotonously increasing, as [MATH] Fixing [MATH] and tending [MATH] to the infinity in the inequalities ( 41 ), we obtain |
[EQUATION] The last inequalities implies that for every measure [MATH] belonging to the convex span, constructed on the set [MATH] |
[MATH] is a super-martingale relative to the convex set of equivalent measures, [MATH] Now, if a measure [MATH] does not belong to the convex span, constructed on the set [MATH] then we can add it to the set [MATH] and repeat the proof made above. As a result, we proved that [MATH] is also a super-martingale relative t... |
complete the proof of Lemma Theorem 3 On a measurable space [MATH] and a filtration [MATH] on it, let [MATH] be an arbitrary convex set of equivalent measures. For a random value [MATH] the random process [MATH] |
[MATH] is a local regular martingale relative to the convex set of equivalent measures [MATH] Proof. Let [MATH] be a certain subset of measures from [MATH] Denote |
[MATH] a convex set of equivalent measures [EQUATION] Due to Lemma [MATH] is a martingale relative to the set of measures [MATH] where [MATH] Let us consider an arbitrary measure [MATH] and let |
[EQUATION] Then [MATH] where [MATH] is a martingale relative to the set of measures [MATH] It is evident that [EQUATION] Since [MATH] the inequalities ( 46 ) give [MATH] Analogously, |
[MATH] From the equalities [MATH] [MATH] we obtain [MATH] Since the measure [MATH] is an arbitrary one it implies that [MATH] is a martingale relative to all measures from [MATH] |
Due to Theorem , it is a local regular super-martingale with the random process [MATH] Theorem is proved. Theorem 4 On a measurable space [MATH] and a filtration [MATH] on it, let [MATH] be an arbitrary convex set of equivalent measures. If [MATH] is an adapted random process satisfying conditions |
[EQUATION] then the random process [EQUATION] is a local regular super-martingale relative to the convex set of equivalent measures [MATH] |
Proof. Due to Theorem , the random process [MATH] is a martingale relative to the convex set of equivalent measures [MATH] Therefore, |
[EQUATION] [EQUATION] So, if to put [MATH] then [MATH] it is [MATH] -measurable and [MATH] It proves the needed statement. Corollary 1 |
If [MATH] [MATH] then [MATH] is a local regular martingale. Assume that [MATH] then [MATH] is a local regular super-martingale relative to a convex set of equivalent measures [MATH] |
Denote [MATH] the set of adapted processes [EQUATION] For every [MATH] let us introduce the set of adapted processes [EQUATION] [EQUATION] |
and [EQUATION] Corollary 2 Every random process from the set [MATH] where [EQUATION] is a local regular super-martingale relative to the convex set of equivalent measures [MATH] on a measurable space [MATH] with filtration [MATH] on it. |
Proof. The proof is evident. Theorem 5 On a measurable space [MATH] and a filtration [MATH] on it, let [MATH] be an arbitrary convex set of equivalent measures. Suppose that [MATH] is a nonnegative uniformly integrable super-martingale relative to a convex set of equivalent measures [MATH] then the necessary and suffic... |
Proof. Necessity. It is evident that if [MATH] belongs to [MATH] then it is a local regular super-martingale. Sufficiency. Suppose that [MATH] is a local regular super-martingale. Then there exists nonnegative adapted process [MATH] and a martingale [MATH] |
such that [EQUATION] Then [MATH] Since [MATH] we have [MATH] Let us put [MATH] Using the uniform integrability of [MATH] we can pass to the limit in the equality |
[EQUATION] as [MATH] Passing to the limit in the last equality, as [MATH] we obtain [EQUATION] Introduce into consideration a random value [MATH] |
Then [MATH] From here we obtain that [MATH] and [EQUATION] Let us put [MATH] It is easy to see that the adapted random process [MATH] belongs to [MATH] Therefore, for the super-martingale [MATH] the representation |
[EQUATION] is valid, where [MATH] belongs to [MATH] with [MATH] and [MATH] The same is valid for [MATH] with [MATH] This implies that [MATH] belongs to the set [MATH] Theorem |
is proved. Theorem 6 On a measurable space [MATH] and a filtration [MATH] on it, let [MATH] be an arbitrary convex set of equivalent measures. Suppose that the super-martingale [MATH] relative to the convex set of equivalent measures [MATH] satisfy conditions |
[EQUATION] then the necessary and sufficient conditions for it to be a local regular one is belonging it to the set [MATH] Proof. |
The necessity is evident. Sufficiency. Suppose that [MATH] is a local regular super-martingale. Then there exists a nonnegative adapted random process [MATH] and a martingale [MATH] |
such that [EQUATION] The inequalities [MATH] give the inequalities [EQUATION] From the inequalities ( 58 ) it follows that the super-martingale [MATH] is a uniformly integrable one relative to the convex set of equivalent measures [MATH] . The martingale [MATH] relative to the convex set of equivalent measures [MATH] i... |
Then [MATH] Since [MATH] we have [MATH] Let us put [MATH] Using the uniform integrability of [MATH] and [MATH] we can pass to the limit in the equality |
Then [MATH] From here we obtain that [MATH] and for the super-martingale [MATH] the representation [EQUATION] is valid, where [MATH] |
From the last representation it follows that the super-martingale [MATH] belongs to the set [MATH] Theorem is proved. Corollary 3 |
Let [MATH] be a [MATH] -measurable integrable random value, [MATH] and let there exist [MATH] such that [EQUATION] where [MATH] Then a super-martingale [MATH] is a local regular one relative to the convex set of equivalent measures [MATH] where |
[EQUATION] [EQUATION] Proof. It is evident that [MATH] Therefore, the super-martingale [EQUATION] is a local regular one relative to the convex set of equivalent measures [MATH] |
Corollary is proved. Optional decomposition for super-martingales relative to the complete convex set of equivalent measures. In this section we introduce the notion of complete set of equivalent measures and prove that non negative super-martingales are local regular ones with respect to this set of measures. For this... |
Theorem 7 The necessary and sufficient conditions of the local regularity of the nonnegative super-martingale [MATH] relative to a convex set of equivalent measures [MATH] are the existence of [MATH] -measurable random values [MATH] such that |
[EQUATION] Proof. The necessity. Without loss of generality, we assume that [MATH] for a certain real number [MATH] Really, if it is not so, then we can come to the consideration of the super-martingale [MATH] Thus, let [MATH] be a nonnegative local regular super-martingale. Then there exists a nonnegative adapted rand... |
[MATH] such that [MATH] [EQUATION] Let us put [MATH] Then [MATH] and from the equalities ( 73 ) we obtain [MATH] It is evident that the inequalities ( 72 ) are valid. |
The sufficiency. Suppose that the conditions of Theorem are valid. Then [MATH] Introduce the denotation [MATH] Then [MATH] [MATH] The last equality and inequalities give |
[EQUATION] Let us consider the random process [MATH] where [MATH] Then [MATH] [MATH] Theorem is proved. 5.1 Space of finite set of elementary events. |
In this subsection we assume that a space of elementary events [MATH] is finite one, that is, [MATH] and we give a new proof of the optional decomposition for super-martingales relative to the complete convex set of equivalent measures. This proof does not use topological arguments as in |
Let [MATH] be a certain algebra of subsets of the set [MATH] and let [MATH] be an increasing set of algebras, where [MATH] [MATH] Denote [MATH] a convex set of equivalent measures on a measurable space [MATH] Further, we assume that the set [MATH] contains an element [MATH] It is evident that every algebra [MATH] is [M... |
Let [MATH] Then for [MATH] the representation [EQUATION] is valid. Consider the difference [MATH] Then [EQUATION] [EQUATION] where [MATH] as [MATH] and [MATH] for [MATH] |
From the equalities ( 76 ), ( 77 ) we obtain [EQUATION] [EQUATION] Denote [MATH] the contraction of the set of measures [MATH] on the algebra [MATH] Introduce into the set [MATH] the metrics |
[EQUATION] where [MATH] is a partition of [MATH] on [MATH] subsets, that is, [MATH] The maximum in the formula ( 80 is all over the partitions of the set [MATH] belonging to the [MATH] -algebra [MATH] |
Definition 3 On a measurable space [MATH] a convex set of equivalent measure [MATH] we call complete if for every [MATH] the closure of the set of measures [MATH] in the metrics ( 80 ) contains the measures |
[EQUATION] for every [MATH] and [MATH] Lemma 10 Let a convex family of equivalent measures [MATH] be a complete one and the set [MATH] contains an element [MATH] Then for every non negative [MATH] -measurable random value [MATH] there exists a real number [MATH] such that |
[EQUATION] Proof. On the set [MATH] the functional [MATH] is a continuous one, where [MATH] is the closure of the set [MATH] in the metrics [MATH] |
From this it follows that the equality [EQUATION] is valid. Denote [MATH] Then [EQUATION] For those [MATH] for which [MATH] and those [MATH] for which [MATH] the inequality ( 87 ) is as follows |
[EQUATION] From ( 88 ) we obtain the inequalities [EQUATION] Since the inequalities ( 89 ) are valid for every [MATH] as [MATH] and since the set of such elements is finite, then if to denote |
[EQUATION] then we have [EQUATION] From the definition of [MATH] we obtain the inequalities [EQUATION] Now if [MATH] for some [MATH] then in this case [MATH] All these inequalities give |
[EQUATION] Multiplying on [MATH] the inequalities ( 93 ) and summing over all [MATH] we obtain the needed inequality. Lemma 10 is proved. |
Theorem 8 Suppose that the conditions of Lemma 10 are valid. Then every non negative super-martingale [MATH] relative to a convex set of equivalent measures [MATH] satisfying conditions |
[EQUATION] is a local regular one, where [MATH] are constants. Proof. Consider the random value [MATH] Due to Lemma 10 [EQUATION] |
It is evident that [MATH] Since [MATH] then [EQUATION] Theorem and the inequalities ( 96 ) prove Theorem Theorem 9 On a finite space of elementary events [MATH] with a filtration [MATH] on it, every super-martingale [MATH] relative to the complete convex set of equivalent measures [MATH] is a local regular one if the s... |
Proof. It is evident that every super-martingale [MATH] is bounded. Therefore, there exists a constant [MATH] such that [MATH] From this it follows that the super-martingale [MATH] is a nonnegative one and satisfies the conditions |
[EQUATION] It implies that the conditions of Theorem are satisfied. Theorem is proved. Theorem 10 Let [MATH] be a complete convex set of equivalent measure on a measurable space [MATH] with a filtration [MATH] on it. Suppose that [MATH] and [MATH] is a martingale relative to the set of measures [MATH] Let [MATH] be a s... |
Proof. Let the sequence [MATH] be a convergent one to the measure [MATH] then for [MATH] [EQUATION] The functionals [MATH] on the set [MATH] for all |
[MATH] are continuous ones relative to the metrics [MATH] defined by the formula ( 80 ). Going to the limit in the equality ( 98 ), as [MATH] we obtain |
[EQUATION] The last implies that [MATH] Theorem 10 is proved. 5.2 Countable set of elementary events. In this subsection, we generalize the results of the previous subsection onto the countable space of elementary events. Let [MATH] be a certain [MATH] -algebra of subsets of the countable set of elementary events [MATH... |
Denote [MATH] a set of equivalent measures on the measurable space [MATH] Further, we assume that the set [MATH] contains an element [MATH] Suppose that the [MATH] -algebra [MATH] is [MATH] |
Introduce into consideration the martingale [MATH] Then for [MATH] the representation [EQUATION] is valid. Consider the difference [MATH] |
Then [EQUATION] [EQUATION] where [MATH] as [MATH] and [MATH] [MATH] From the equalities ( 101 ), ( 102 ) we obtain [EQUATION] [EQUATION] |
Denote [MATH] the contraction of the set of measures [MATH] on the [MATH] -algebra [MATH] Introduce into the set [MATH] the metrics |
[EQUATION] where [MATH] is a partition of [MATH] on [MATH] subsets, that is, [MATH] The supremum in the formula ( 105 ) is all over the partitions of the set [MATH] belonging to the [MATH] -algebra [MATH] |
Definition 4 On a measurable space [MATH] with a filtration [MATH] on it, a convex set of equivalent measure [MATH] we call complete one if for every [MATH] |
the closure of the set of measures [MATH] in the metrics ( 105 ) contains the measures [EQUATION] for every [MATH] and [MATH] Lemma 11 |
Let a family of measures [MATH] be complete and the set [MATH] contains an element [MATH] Then for every non negative bounded [MATH] -measurable random value [MATH] there exists a real number [MATH] such that |
[EQUATION] Proof. On the set [MATH] the functional [MATH] is a continuous one relative to the metrics [MATH] where [MATH] is the closure of the set [MATH] in this metrics. From this it follows that the equality |
[EQUATION] is valid. Denote [MATH] Then [EQUATION] The last inequalities can be written in the form [EQUATION] For those [MATH] for which [MATH] and those [MATH] for which [MATH] the inequality ( 113 ) is as follows |
[EQUATION] From ( 114 ) we obtain the inequalities [EQUATION] Two cases are possible: a) for all [MATH] [MATH] b) there exists [MATH] such that [MATH] |
First, let us consider the case a). Since the inequalities ( 115 ) are valid for every [MATH] as [MATH] and [MATH] then if to denote |
[EQUATION] we have [MATH] and [EQUATION] From the definition of [MATH] we obtain the inequalities [EQUATION] Now, if [MATH] for some [MATH] then in this case [MATH] All these inequalities give |
[EQUATION] Consider the case b). From the inequality ( 115 ), we obtain [EQUATION] The last inequalities give [EQUATION] Let us define [MATH] |
Then from ( 120 ) we obtain [EQUATION] From the definition of [MATH] we have [EQUATION] The inequalities ( 122 ), ( 123 ) give [EQUATION] |
Multiplying on [MATH] the inequalities ( 119 ) and the inequalities ( 124 ) on [MATH] and summing over all [MATH] we obtain the needed inequality. The Lemma 11 is proved. |
Theorem 11 Suppose that the conditions of Lemma 11 are valid. Then every non negative super-martingale [MATH] relative to a convex set of equivalent measures [MATH] satisfying the conditions |
[EQUATION] is a local regular one, where [MATH] are constants. Proof. From the conditions ( 125 ) it follows that [MATH] Consider the random value [MATH] Due to Lemma 11 |
[EQUATION] It is evident that [MATH] Since [MATH] then [EQUATION] Theorem and the inequalities ( 127 ) prove Theorem 11 5.3 An arbitrary space of elementary events. |
In this subsection, we consider an arbitrary space of elementary events and prove the optional decomposition for non negative super-martingales. |
Let [MATH] be a certain [MATH] -algebra of subsets of the set of elementary events [MATH] and let [MATH] be an increasing set of the [MATH] -algebras, where [MATH] |
Denote [MATH] a set of equivalent measures on a measurable space [MATH] We assume that the [MATH] -algebras [MATH] and [MATH] are complete relative to any measure [MATH] |
Further, we suppose that the set [MATH] contains an element [MATH] Let [MATH] Consider the difference [MATH] We assume that every [MATH] belongs to the [MATH] -algebra [MATH] and [MATH] |
For the random value [MATH] there exists not more then a countable set of the real number [MATH] such that [MATH] where [MATH] It is evident that [MATH] Suppose that |
[MATH] Introduce for every [MATH] two subsets [MATH] [MATH] of the set [MATH] Denote [MATH] the contraction of the set of measures [MATH] on the [MATH] -algebra [MATH] Introduce into the set [MATH] the metrics |
[EQUATION] where [MATH] is a partition of [MATH] on [MATH] subsets, that is, [MATH] The supremum in the formula ( 128 ) is all over the partitions of the set [MATH] belonging to the [MATH] -algebra [MATH] |
Definition 5 On a measurable space [MATH] with filtration [MATH] on it, a convex set of equivalent measure [MATH] we call complete if for every [MATH] the closure in metrics ( 128 ) of the set of measures [MATH] |
contains the measures [EQUATION] for [MATH] and [MATH] Lemma 12 Let a convex family of equivalent measures [MATH] be a complete one and the set [MATH] contains an element [MATH] Then for every non negative bounded [MATH] -measurable random value [MATH] there exists a real number [MATH] such that |
[EQUATION] is valid. Denote [MATH] Then [EQUATION] The last inequalities can be written in the form [EQUATION] The inequality ( 136 ) for the measures ( 132 ) is as follows |
[EQUATION] [EQUATION] From ( 137 ) we obtain the inequalities [EQUATION] [EQUATION] Two cases are possible: a) for all [MATH] [MATH] b) there exists [MATH] such that [MATH] |
First, let us consider the case a). Since the inequalities ( 138 ) are valid for every [MATH] as [MATH] and [MATH] then if to denote |
[EQUATION] Consider the case b). From the inequality ( 138 ), we obtain [EQUATION] [EQUATION] The last inequalities give [EQUATION] |
Let us define [MATH] Then from ( 144 ) we obtain [EQUATION] From the definition of [MATH] we have [EQUATION] The inequalities ( 147 ), ( 148 ) give |
[EQUATION] Since the set [MATH] has probability one, Lemma 12 is proved. Theorem 12 Suppose a convex set of equivalent measures [MATH] is a complete one and the conditions of Lemma 12 |
are valid. Then every non negative super-martingale [MATH] relative to a convex set of equivalent measures [MATH] satisfying conditions |
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