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[EQUATION] is a local regular one, where [MATH] are constants. Proof. From the inequalities ( 150 ) it follows that [MATH] Consider the random value [MATH] Due to Lemma 12 |
[EQUATION] It is evident that [MATH] Since [MATH] then [EQUATION] Theorem and the inequalities ( 152 ) prove Theorem 12 Consequence 1 |
If a super-martingale [MATH] relative to a complete convex set of equivalent measures [MATH] satisfy conditions [MATH] where [MATH] are constant, then it is local regular. |
Proof. The super-martingale [MATH] [MATH] is a nonnegative one and satisfies the conditions [EQUATION] From Theorem 11 it follows the validity of the local regularity for the super-martingale [MATH] |
therefore, for the super-martingale [MATH] the local regularity is also true. Local regularity of majorized super-martingales. In this section, we give the elementary proof that a majorized super-martingale relative to the complete set of equivalent measures is local regular one. |
Theorem 13 On a measurable space [MATH] with a filtration [MATH] on it, let the set [MATH] be a complete convex set of equivalent measures on [MATH] and the set [MATH] contains an element [MATH] Then every bounded super-martingale [MATH] relative to the complete convex set of equivalent measures [MATH] is a local regul... |
Proof. From Theorem 13 conditions, there exists a constant [MATH] such that [MATH] Consider the super-martingale [MATH] Then [MATH] Due to Consequence for the super-martingale [MATH] the local regularity is true. So, the same statement is valid for the super-martingale [MATH] Theorem 13 is proved. |
The next Theorem is analogously proved as Theorem 13 Theorem 14 On a measurable space [MATH] with filtration [MATH] on it, let the set [MATH] be a complete convex set of equivalent measures on [MATH] and the set [MATH] contains an element [MATH] Then a super-martingale [MATH] relative to the complete convex set of equi... |
[EQUATION] for certain constants [MATH] is a local regular one. Application to Mathematical Finance. Due to Corollary , we can give the following definition of the fair price of contingent claim [MATH] relative to a convex set of equivalent measures [MATH] |
Definition 6 Let [MATH] be a [MATH] -measurable integrable random value relative to a convex set of equivalent measures [MATH] such that for some [MATH] and [MATH] |
[EQUATION] Denote [MATH] We call [EQUATION] the fair price of the contingent claim [MATH] relative to a convex set of equivalent measures [MATH] if there exists [MATH] and a sequences [MATH] |
[MATH] satisfying the conditions: [MATH] [MATH] by probability, as [MATH] and such that [EQUATION] Theorem 15 Let the set [MATH] be uniformly integrable one relative to every measure [MATH] Suppose that for a nonnegative [MATH] -measurable integrable contingent claim [MATH] relative to every measure [MATH] there exist ... |
[EQUATION] then the fair price [MATH] of contingent claim [MATH] exists. For [MATH] the inequality [EQUATION] is valid. If [MATH] and a super-martingale [MATH] is a local regular one, then [MATH] |
Proof. If [MATH] then Theorem 15 is proved. Suppose that [MATH] Then there exists a sequence [MATH] and [MATH] such that [EQUATION] |
Due to the uniform integrability [MATH] we obtain [EQUATION] Using again the uniform integrability of [MATH] and going to the limit in ( 160 ) we obtain |
[EQUATION] From the inequality [MATH] it follows the inequality ( 159 ). If [MATH] and [MATH] is a local regular super-martingale, then |
[EQUATION] where a martingale [MATH] is a nonnegative one and [MATH] Introduce into consideration a random value [MATH] where [MATH] Then [MATH] belongs to the set [MATH] and |
[EQUATION] From this it follows that [MATH] Let us prove that [MATH] is a fair price for certain evolutions of risk and non risk assets. Suppose that the evolution of risk asset is given by the law [MATH] and the evolution of non risk asset is given by the formula [MATH] |
As proved above, for [MATH] there exists [MATH] such that the inequality [EQUATION] is valid. Let us put [EQUATION] [EQUATION] It is evident that [MATH] |
Therefore, the super-martingale [EQUATION] is a local regular one. It is evident that [EQUATION] where [EQUATION] [EQUATION] [EQUATION] |
For the martingale [MATH] the representation [EQUATION] is valid, where [MATH] Let us consider the trading strategy [MATH] where |
[EQUATION] [EQUATION] It is evident that [MATH] are [MATH] measurable and the trading strategy [MATH] satisfy self-financed condition |
[EQUATION] Moreover, the capital corresponding to the self-financed trading strategy [MATH] is given by the formula [EQUATION] Herefrom, [MATH] |
Further, [EQUATION] The last proves Theorem 15 From ( 162 ) and Corollary the Theorem 16 follows. Theorem 16 Suppose that the set [MATH] contains only [MATH] linear independent elements [MATH] If there exist [MATH] and [MATH] such that |
[EQUATION] where [EQUATION] then the fair price [MATH] of the contingent claim [MATH] exists, where [MATH] is [MATH] measurable and integrable relative to every measure [MATH] |
[MATH] Proof. The proof is evident, as the set [MATH] is a uniformly integrable one relative to every measure from [MATH] Corollary 4 |
On a measurable space [MATH] with filtration [MATH] on it, let [MATH] be a non negative local regular super-martingale relative to a convex set of equivalent measures [MATH] If the set [MATH] is uniformly integrable relative to every measure [MATH] then the fair price of contingent claim [MATH] exists. |
Proof. From the local regularity of super-martingale [MATH] we have [MATH] Therefore, [MATH] where [MATH] From the last it follows that the conditions of Theorem 15 are satisfied. Corollary is proved. |
On a probability space [MATH] let us consider an evolution of one risk asset given by the law [MATH] where [MATH] is a random value taking values in [MATH] Suppose that |
[MATH] is a filtration on [MATH] and [MATH] is [MATH] -measurable random value. We assume that the non risk asset evolve by the law [MATH] |
Denote [MATH] the set of all martingale measures being equivalent to the measure [MATH] We assume that the set [MATH] of such martingale measures is not empty and the effective market is non complete, see, for example, |
So, we have that [EQUATION] The next Theorem justify the Definition Theorem 17 Let a contingent claim [MATH] be a [MATH] -measurable integrable random value with respect to every measure from [MATH] and the conditions of the Theorem 16 are satisfied with [MATH] Then there exists self-financed trading strategy [MATH] th... |
where [MATH] is a fair price of contingent claim [MATH] Proof. Due to Theorems 15 16 , for [MATH] there exists [MATH] such that the inequality |
[EQUATION] is valid. Let us put [EQUATION] [EQUATION] It is evident that [MATH] Therefore, the super-martingale [EQUATION] is a local regular one. It is evident that |
[EQUATION] where [EQUATION] [EQUATION] [EQUATION] Due to Theorem 20 , for the martingale [MATH] the representation [EQUATION] is valid. Let us consider the trading strategy [MATH] where |
[EQUATION] [EQUATION] It is evident that [MATH] are [MATH] -measurable ones and the trading strategy [MATH] satisfy the self-financed condition |
[EQUATION] Moreover, a capital corresponding to the self-financed trading strategy [MATH] is given by the formula [EQUATION] Herefrom, [MATH] |
Further, [EQUATION] Therefore [MATH] Theorem 17 is proved. In the next Theorem we assume that the evolutions of risk and non risk assets generate incomplete market |
, that is, the set of martingale measures contains more that one element. Theorem 18 Let an evolution [MATH] of the risk asset satisfy the conditions [MATH] where the constants [MATH] satisfy the inequalities [MATH] and let the non risk asset evolution be deterministic one given by the law [MATH] The fair price of Stan... |
is given by the formula [EQUATION] The fair price of Standard European Put Option with the payment function [MATH] is given by the formula |
[EQUATION] Proof. In Theorem 18 conditions, the set of equations [MATH] has the solutions [MATH] It is evident that [MATH] and [MATH] since |
[EQUATION] Let us prove the needed formula. Consider the inequality [EQUATION] where [MATH] Or, [EQUATION] Suppose that [MATH] satisfies the inequality |
[EQUATION] If [MATH] satisfies additionally the equality [EQUATION] then for all [MATH] 212 ) is valid. From ( 214 ) we obtain for [MATH] |
[EQUATION] If [MATH] then [EQUATION] since [MATH] From here we obtain [EQUATION] It is evident that [MATH] satisfies the inequality ( 213 ). |
If [MATH] then [MATH] and from ( 211 ) we can put [MATH] Then, the formula ( 212 ) is valid for all [MATH] Let us prove the formula ( 209 ) for Standard European Put Option. If [MATH] it is evident that [MATH] and [MATH] since |
[EQUATION] Let us prove the needed formula. Consider the inequality [EQUATION] Or, for [MATH] [EQUATION] If [MATH] is a solution of the equality |
[EQUATION] then for all [MATH] 220 ) is valid. From ( 221 ) we obtain for [MATH] [EQUATION] Therefore, [EQUATION] since [MATH] From here we obtain |
[EQUATION] If [MATH] then [MATH] and from ( 219 ) we can put [MATH] Then, ( 220 ) is valid for all [MATH] The Theorem 18 is proved. |
Some auxiliary results. On a measurable space [MATH] with filtration [MATH] on it, let us consider a convex set of equivalent measures [MATH] |
Suppose that [MATH] is a set of random values belonging to the set [MATH] Introduce [MATH] martingales relative to a set of measures [MATH] |
[MATH] where [MATH] Denote by [MATH] a set of all martingale measures equivalent to a measure [MATH] that is, [MATH] if [EQUATION] |
It is evident that [MATH] and [MATH] is a convex set. Denote [MATH] a certain fixed measure from [MATH] and let [MATH] be a set of finite valued random values on a probability space [MATH] taking values in [MATH] |
Let [MATH] be a set of finite valued predictable processes [MATH] where [MATH] takes values in [MATH] and [MATH] is [MATH] -measurable random vector. Introduce into consideration a set of random values |
[EQUATION] [EQUATION] Lemma 13 The set of random values [MATH] is a closed subset in the set of finite valued random values [MATH] relative to the convergence by measure [MATH] |
The proof of the Lemma 13 see, for example, Introduce into consideration a subset [EQUATION] of the set [MATH] where [MATH] Let [MATH] be a subset of the set [MATH] |
[EQUATION] Denote also a set [EQUATION] where [MATH] is a set of bounded nonnegative random values. Let [MATH] be the closure of [MATH] in [MATH] metrics. |
Lemma 14 If [MATH] and such that [MATH] then for [MATH] the representation [EQUATION] is valid for a certain finite valued predictable process |
[MATH] Proof. If [MATH] then Lemma 14 is proved. Suppose that [MATH] then there exists a sequence [MATH] such that [MATH] where [MATH] Since |
[MATH] we have [MATH] From here we obtain [MATH] Therefore, [MATH] by measure [MATH] On the basis of Lemma 13 , a set [EQUATION] |
[EQUATION] is a closed subset of [MATH] relative to the convergence by measure [MATH] From this fact, we obtain the proof of Lemma 14 , since there exists the finite valued predictable process [MATH] such that for [MATH] the representation |
[EQUATION] is valid. Theorem 19 Let [MATH] If for every [MATH] then there exists finite valued predictable process [MATH] such that for [MATH] the representation |
[EQUATION] is valid. Proof. If [MATH] then ( 235 ) follows from Lemma 14 . So, let [MATH] does not belong to [MATH] As in Lemma 14 [MATH] is a closure of [MATH] in [MATH] metrics for the fixed measure [MATH] The set [MATH] is a closed convex set in [MATH] Consider the other convex closed set that consists from one elem... |
[EQUATION] and the inequalities [MATH] [MATH] are valid. Since [MATH] is a convex cone we can put [MATH] From the condition [MATH] we have [MATH] From ( 236 ) and the inclusions [MATH] we have [MATH] Introduce a measure |
[EQUATION] Then, we have [EQUATION] Let us choose [MATH] where [MATH] is an indicator of a set [MATH] We obtain [EQUATION] So, [MATH] is a martingale measure that belongs to the set [MATH] which is a set of absolutely continuous martingale measures. Let us choose [MATH] and consider a measure [MATH] A measure [MATH] an... |
Theorem 20 For every martingale [MATH] relative to the set of measures [MATH] there exists a predictable random process [MATH] such that for [MATH] the representation |
[EQUATION] is valid. Proof. For fixed natural [MATH] let us consider the random value [MATH] Since [EQUATION] then [MATH] satisfies the conditions of Theorem 19 and, therefore, belongs to [MATH] so, there exists a sequence |
[MATH] such that [EQUATION] From here, we obtain [EQUATION] But [MATH] Hence, we obtain that both [MATH] and [MATH] converges by measure [MATH] to [MATH] and [MATH] correspondingly. There exists a subsequence [MATH] such that [MATH] converges everywhere to predictable process [MATH] From here, we have [MATH] and [MATH]... |
[EQUATION] Theorem 20 is proved. Conclusions. In the paper, we generalize Doob decomposition for super-martingales relative to one measure onto the case of super-martingales relative to a convex set of equivalent measures. For super-martingales relative to one measure for continuous time Doob’s result was generalized i... |
Section 2 contains the definition of local regular super-martingales. Theorem gives the necessary and sufficient conditions of the local regularity of super-martingale. In spite of its simplicity, the Theorem appeared very useful for the description of the local regular super-martingales. |
For this purpose we investigate the structure of super-martingales of special types relative to the convex set of equivalent measures, The main result of the section 3 is Lemma , which allowed proving Lemma , giving the sufficient conditions of the existence of a martingale with respect to a convex set of equivalent me... |
Theorem describes all local regular non negative super-martingales of the special type ( 30 ) relative to the convex set of equivalent measures, |
In the Theorem , we give the sufficient conditions of the existence of the local regular martingale relative to an arbitrary set of equivalent measures and arbitrary filtration. After that, we present in Theorem the important construction of the local regular super-martingales which we sum up in Corollary . Theorem |
proves that every majorized super-martingale belongs to the described class ( 53 ) of the local regular super-martingales. Theorem gives a variant of the necessary and sufficient conditions of local regularity of non negative super-martingale relative to a convex set of equivalent measures. Definition determines a clas... |
Corollary contains the important construction of the local regular super-martingales playing the important role in the definition of the fair price of contingent claim relative to a convex set of equivalent measures. The Definition is a fundamental one for the evaluation of risks in incomplete markets. Theorem 15 gives... |
# Source: arxiv 1806.05799 # Title: CIA-Towards a Unified Marketing Optimization Framework for e-Commerce Sponsored Search # Sections: all # Downloaded: 2026-03-02T09:23:01.577772+00:00 |
CIA: Towards a Unified Marketing Optimization Framework for e-Commerce Sponsored Search Abstract As the largest e-commerce platform, Taobao helps advertisers reach billions of search queries each day via sponsored search, which has also contributed considerable revenue to the platform. An efficient bidding strategy to ... |
Introduction Sponsored search has provided considerable revenue for universal search engines such as Google, Bing and vertical e-commerce websites like Taobao and Amazon. The 2017 Tmall Double Eleven shopping festival sees total sales of [MATH] billion ( ), showing the prosperity of online shopping and associated adver... |
Taobao Search Advertising (TSA) is a suite of commercial advertising service . Each advertiser maintains a tree-like TSA account as shown in Fig. . The account has a balance and consists of several budget-limited marketing campaigns, while a campaign accommodates a set of ADs to be promoted. At the most fine-grained le... |
Search queries initiated by consumers retrieve relevant ADs through exact or broad keyword matching. Generally, all the ADs are ranked by the product of bid and predicted Click Through Rate (CTR) . Once clicked, they will be charged the minimal price to maintain their position (Generalized Second Price, GSP), to maximi... |
Apart from keyword selection, advertisers can optimize their marketing demands in TSA by way of cautiously determining strategic bids. Nevertheless, such optimization is intractable due to the shortage of auction competition information, computation, and storage capacity. On the contrary, the advertising platform is om... |
In this paper, we cater to various advertiser demands with a unified impression-level bidding based demand optimization framework called Customer Intelligent Agent (CIA) . CIA keeps advertisers’ knobs on keyword-level bids to convey varied-level Return-On-Investment (ROI) expectations. Meanwhile, the impression-level b... |
The rest of this paper is organized as follows. In Section we give a more detailed analysis of related work on bidding optimization. The architecture of CIA is given in Section where we formulate the optimization problems with their constraints. Section discusses the influence of CIA bidding in the GSP mechanism. And w... |
Related work Most existing bidding strategies designed by DSPs aim to optimize only one single advertiser’s profits by assuming that the competitive environment is stationary ( ). In the case of unconstrained budget, truthful bidding in GSP has proved to achieve the Nash-Equilibrium for a single item auction ( ). In ( ... |
Other novel work also exists in display advertising. Considering the sequential decision processes in impression bidding, Cai et al. utilized reinforcement learning to bid ( ). With the assumption of i.i.d impression features, they derived an optimal real-time bidder under budget and auction volume constraints. A simil... |
In sponsored search, the work on joint optimization of keyword-level bids and budget allocation over a campaign (several campaigns shared a budget) ( ) resembles our scenario. More keyword-level sponsored search bidding work can be found in ( ). However, keyword-level bidding does not distinguish impressions from the s... |
CIA Framework in Taobao Sponsored Search Marketing demands at TSA can be classified into two categories, i.e., basic and compound demands. Basic demands include acquisitions of impressions, clicks and conversions, which are ubiquitous in online advertising. Particularly, for e-commerce platforms like Taobao, compound d... |
Fig. illustrates how CIA framework works within the TSA system. Advertisers express their marketing demands on different ADs via keyword-level bids or direct AD-level take-rate (abbreviated as [MATH] ) and CIA optimizes impression-level bids respecting the demands. Optimizer balances the take-rate and advertising costs... |
Advertisers’ demands are optimized under CIA in equation ( ). Regardless of various demands, a reasonable budget constraint is necessary. Additionally, advertisers also bid with the expectation of conversions (adding to collections/shopping carts/purchase) in e-commerce. Each dollar spent should account for some return... |
[EQUATION] While optimization objectives are exactly demonstrated in Tab. , the constraints, despite roughly depicted above, are still challenging to be specified. Firstly, budget constraints are set only at the campaign level in TSA, while AD-level [MATH] to regulate impression-level bidding requires a reasonable AD-l... |
3.1 Bridging keyword-level bid settings and constraints Generally, advertisers choose the keywords of an AD and set fixed bids. Given [MATH] auctions ( [MATH] ) which involves the AD, the accumulated ROI can be calculated as: |
[EQUATION] where [MATH] is the click cost and can be approximated with the corresponding keyword bid [MATH] to obtain a lower bound of ROI while [MATH] stands for the item price. Following the above formula, we actually obtain an expected ROI from the original bid settings and auction log. Implicitly, this ROI reflects... |
Meanwhile, it is sufficient to guarantee the cumulative ROI if each advertiser bids with an acceptable ROI in each auction. An impression-level [MATH] satisfies that, |
[EQUATION] where [MATH] is the cost upon clicking. Let [MATH] , then [MATH] . In GSP it usually satisfies that [MATH] , therefore it is sufficient to bid as, |
[EQUATION] Furthermore, the advertiser’s changing keyword-level bids can easily propagate to [MATH] given the historical auction log. |
[EQUATION] where [MATH] corresponds to the perturbations on its input argument. In the above analysis, we take the auction log of an AD, arriving an AD-level [MATH] . Campaign/keyword-level [MATH] s can be derived likewise with corresponding logs. [MATH] can be updated daily with historical auction logs or in an online... |
Relying on the historical auction log, a reasonable AD-level virtual budget can also be obtained as: [EQUATION] If the marketing environment is stationary across days, we believe that such a budget shall be fair and sufficient for the advertiser. In this way, we do not need to explicitly split the campaign budget into ... |
To optimize various campaign-level demands, CIA introduces an additional cost regulation factor [MATH] to adjust AD-level [MATH] in a campaign, leading to a generalized bidding formula: |
[EQUATION] Generally, an [MATH] in the range of [MATH] can roughly guarantee that the advertiser will gain a higher ROI than previous keyword-level bidding. However, for the approximations used, the de facto effective [MATH] might be larger than [MATH] . Additionally, different demands might lead to a quite different f... |
3.2 Demands optimization An AD-level [MATH] simplifies advertisers’ bidding control. Meanwhile, campaign-level demand is the most natural way for advertisers to manage. To this end, in this section we tackle the two most representative demands of advertisers, i.e., campaign-level GMV (conversion) optimization and style... |
In the following formulation, for a campaign with [MATH] ADs (indexed by [MATH] ), we denote the expected daily GMV and cost of ADs if bidding with CIA as [MATH] [MATH] . Under the same auction environment, corresponding keyword-level bids will lead to a cost of [MATH] . And the cost regulation vector is denoted as [MA... |
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