Insider information and its relation with the arbitrage condition and the utility maximization problem
IInsider information and its relation with the arbitragecondition and the utility maximization problem
Bernardo D’Auria ∗† Jos´e Antonio Salmer´on ‡§ December 5, 2019
Abstract
Within the well-known framework of financial portfolio optimization, we analyzethe existing relationships between the condition of arbitrage and the utility maxi-mization in presence of insider information . We assume that, since the initial time,the information flow is altered by adding the knowledge of an additional randomvariable including future information. In this context we study the utility maxi-mization problem under the logarithmic and the Constant Relative Risk Aversion(CRRA) utilities, with and without the restriction of no temporary-bankruptcy.In particular, we show that the value of the insider information may be boundedwhile the arbitrage condition holds and we prove that the insider information doesnot always imply arbitrage for the insider by providing an explicit example.
Keywords— optimal portfolio; enlargement of filtration; value of the information; arbi-trage; No Free Lunch Vanishing Risk; risk neutral measure
In the field of financial mathematics, the problem of the optimal portfolio plays a crucial roleand in recent years it has been deeply analyzed in the literature. In its simplest form it consistsin finding the best strategy in order to maximize a given utility function at a fixed terminalfinite time.In the simplified settings of just two assets, one risk-less and one risky, the optimal portfolioproblem has been introduced and solved in [1] by considering both the logarithmic and riskadverse utilities. Later, a more general model was considered in [2].One fundamental ingredient in computing the optimal portfolio is the information flow thatthe agent employs in order to build her strategy. This flow is modeled mathematically by theconcept of filtration, and the restrictions on the agent choices are modeled by requiring that ∗ This research was partially supported by the Spanish Ministry of Economy and Competitive-ness Grants MTM2017-85618-P via FEDER funds and MTM2015-72907-EXP; both authors thank theNYUAD, Abu Dhabi (United Arab Emirates) for hosting them during the fall 2018. † UC3M, Department of Statistics, 28911, Legan´es, Spain & UC3M-BS, Institute of Financial BigData, 28903 Getafe, Spain ([email protected]). ‡ This author acknowledges financial support by an FPU Grant (FPU18/01101) of Spanish Ministeriode Ciencia, Innovaci´on y Universidades. § UC3M, Department of Statistics, 28911, Legan´es, Spain ([email protected]). a r X i v : . [ q -f i n . M F ] D ec INTRODUCTION her portfolio is adapted to this filtration. While in general the underlying filtration is the onenaturally generated by the set of risky assets, in the literature the interest has recently grownabout analyzing filtrations that contain additional information.The technique consisting in substituting the natural filtration by a larger one containingadditional information, is generally referred to as enlargement of filtration and has been intro-duced and studied in the seminal works [3, 4, 5, 6]. The concept of enlargement of filtrationwas first applied to the financial setting by [7], to describe situations in which the agent hasaccess to privileged information and to model the insider trading portfolios, and in [8], as a firstattempt to detect the use of insider information by applying statistical tests.In [7], various examples of initial enlargements were analyzed by computing the expectedadditional gain carried by the privileged information. It was shown that the knowledge of theprice of the stock at a given moment in the future implies an expected unbounded additionalprofit while the knowledge of an interval of values containing the future price only added abounded expected additional gain. For the case of an infinite interval, a direct proof of thisresult was given in [7] while, for the case of a finite interval, the result was only conjectured bythe support of numerical calculations. Later a series of remarkable works, [9, 10], employed theconcept of the Fischer Entropy Theory , shortly mentioned already in [7], to close the conjecture.In Theorem 4.10 below we prove again this result by the same techniques used in [7]. In [11],the Fischer Entropy Theory is generalized for a more broad class of enlargement filtration.In more recent years many results on insider trading models appeared, we just mention[12, 13, 14] and references therein.The above research indicates that there should exist a relation between the type of additionalinformation, such as if it is exact or it is of interval type, and the value that it carries in termsof its contribution to the maximal expected utility. A partial result to this question has beengiven in [9] where they look at the atomicity of the insider information. However there are manyquestions still open, such as understanding the existence of the arbitrage condition is related tothe boundedness of the value of the information.On this direction, [15] analyzes the relation between the arbitrage condition – in particu-lar the (NFLVR), see Proposition 2.10 below – and the integrability of the drift of the semi-martingale representation of the asset in the enlarged filtration, see Proposition 2.8 below. Theworks [16, 17, 18] studied the relations of a weak arbitrage condition, the No Unbounded Profitswith Bounded Risk property (NUPBR), with different types of enlargements of filtrations.It seems that an important ingredient in the analysis is the role played by the set of strategiesthat are allowed to be used. For example, the strategy constructed in [7] to play with theinformation given by an interval of future prices, does not take advantage of the arbitragecondition introduced by this information. This is due to the fact that the proposed strategyavoids the possibility for the insider agent to be for some moments in time in bankruptcy, whatwe later refer to as temporary-bankruptcy . However it is possible to construct, by removing thisconstraint, even simpler strategies that get advantage of the arbitrage condition and assure apositive gain, see for example Propositions 4.6 and 4.12 below, for the semi-infinite and thefinite interval respectively.In this work, we analyze the relations that may exist between the condition of arbitrageand the utility maximization problem under the special setting of enlargement of filtration. Wefirst study the condition of (NFLVR), introduced in [19], and its relationship with bankruptcy,by concluding that if the privileged information implies arbitrage, the investor can improve herprofit expectations by employing strategies that allow temporary-bankruptcy. Even if it mayseem counterintuitive, the privileged information guarantees that the trend will be correctedand the condition of bankruptcy will be only temporary.Finally we prove the result that the insider information does not always imply arbitrage,by constructing a counter example in Section 4.3. This is surprising if we consider that for
BASIC NOTIONS AND PRELIMINARIES the case of progressive enlargement, it has been proved that often the additional informationimplies arbitrage, for more information see [20, 21, 22].We verify that the introduction of the privileged information does not violate the Novikovcondition, Equation (2.4) below, therefore assuring the existence of an equivalent local martin-gale measure and therefore the absence of arbitrage.In Section 2, we provide the notation as well as the basic and preliminary notions that weadopt for the rest of the paper. It includes a more precise definition of the general frameworkin which the problem of the optimal portfolio is framed, such as the definition of arbitrage andthe concept of enlargements of filtration. In Section 3, we introduce the utility maximizationproblem, analyzing it under different conditions on the set of allowed strategies, such as theno-temporary-bankruptcy, and linking it with the arbitrage conditions. Section 4 provides threeexamples that show various cases of enlargement of filtration, where the arbitrage condition isanalyzed and the utility maximization problem is solved. Section 5 ends with some conclusions. As a general setup we assume to work in a probability space (Ω , F , F , P ) where F is theevent sigma-algebra, and F = {F t , t ≥ } is a right continuous filtration satisfying the usualconditions. We consider a financial market with a continuous R semi-martingale S = ( D, S )and, unless otherwise specified, the filtration F that is the natural one generated by S . Eachcomponent represents the prices of an asset in which the agent could invest. Usually, the firstone is assumed to be a risk-less asset driven by some interest rate r >
0, and the second one isgiven by a diffusion process whose coefficients µ and σ are F -adapted processes. In this paper,we generally assume that the dynamics of the semi-martingale S are given by the followingstochastic differential equations, dD t = D t r dt (2.1a) dS t = µ ( t, S t ) dt + σ ( t, S t ) dB t (2.1b)with D = 1 and where the process B = ( B t , ≤ t ≤ T ) is a standard F -adapted Brownianmotion. We fix T > R valued predictableand F -adapted process H = ( H , H ), we use the notation ( H · S ) t to denote the sum of thecomponents of the Itˆo integral, i.e.( H · S ) t = (cid:90) t H u dD u + (cid:90) t H u dS u t ∈ [0 , T ] . (2.2)We refer to the process D − S = (1 , D − S ) as the discounted price process. We define theprocess Θ = { ( M t , N t ) : 0 ≤ t ≤ T } as the number of shares that the agent owns of eachasset at any time t ∈ [0 , T ]. Assuming that, in the market, short-selling and borrowing moneyfrom the bank is allowed, there is no restriction on the values assumed by the process Θ. Toguarantee the existence of the Itˆo integral (Θ · S ) T , we are going to work with a restricted classof strategies, the ones satisfying the following definition. Definition 2.1.
Given the filtration T = {T t , t ≥ } , we define an allowed portfolio process as aprogressive T t -measurable process Θ T ( t, ω ) : [0 , T ] × Ω −→ R which satisfies (cid:82) T (cid:107) Θ T t σ ( t, S t ) (cid:107) dt < + ∞ almost surely. When T = F , we suppress the superscript notation. BASIC NOTIONS AND PRELIMINARIES The total wealth of an allowed portfolio process Θ is then defined as the following Itˆo integral X Θ t = M t D t + N t S t (2.3a)= X + (Θ · S ) t t ∈ [0 , T ] . (2.3b)The second equality indicates that Θ satisfies the self-financing condition . In this section we are going to define the concepts of arbitrage with the special care of denotingthe set of strategies that the agent is restricted to use. We do so by explicitly including inthe definition of the arbitrage conditions the set of allowed strategies. In the following we usethe notation A ( T ) to denote a general set of T -progressive admissible strategies, later we willreplace it with more concrete examples. Again when T = F , we omit to indicate the filtrationin the notation.For the following, we need some technical definitions. We refer to L ∞ (Ω , F T , P ) as the set of F T -measurable bounded random variables, equipped with the essential supremum norm (cid:107) X (cid:107) ∞ .We abbreviate it to L ∞ when the measure P is established. Finally we use L ∞ + (Ω , F T , P ) todenote the class of non-negative F T -measurable bounded random variables and L (Ω , F T , P )for the class of F T -measurable non-negative random variables. Following the classical definitionin [23, Definition 2.8], we define the set of contingent claims A -attainable at price 0 , K ( A ) = (cid:110) (Θ · S ) T | Θ ∈ A (cid:111) = { X Θ T − X | Θ ∈ A} . This set contains all possible random values of terminal wealth that an agent with initial 0wealth can reach at time T by only applying A -admissible strategies. Then, by allowing thepossibility of “throwing away money” at time T , we define the sets C ( A ) = K ( A ) − L = { (Θ · S ) T − f | Θ ∈ A , f ≥ } ,C ( A ) = C ( A ) ∩ L ∞ . together with the set C ( A ) as the closure of C ( A ) in L ∞ .Given the above, we are ready to state the most important definition in Arbitrage Theory.A complete review of this theory is given in [24]. Definition 2.2.
Given a semi-martingale S and a set of admissible strategies A we say that ( N A ) A : the condition of No Arbitrage holds on A when K ( A ) ∩ L ∞ + = { } . ( N F LV R ) A : the condition of No Free Lunch with Vanishing Risk holds on A when C ( A ) ∩ L ∞ + = { } .We also define the conditions of Arbitrage , ( A ) A , and Free Lunch with Vanishing Risk , ( F LV R ) A , as the complements of the conditions above, that is ( N A ) A and ( N F LV R ) A re-spectively. Since K ( A ) ⊂ C ( A ), it immediately follows that (NFLVR) implies (NA).With the following definitions, we describe the main sets of strategies that an agent is allowedto use. The reason to consider different sets of strategies is mainly due to the fact that some ofthem allow the agent to have negative total wealth for some moments in time – condition thatwe refer to as temporary-bankruptcy –, while others do not allow for this possibility. BASIC NOTIONS AND PRELIMINARIES Definition 2.3.
An allowed portfolio process Θ is called a-admissible , with a > , if (Θ · S ) t ≥− a, ∀ t ∈ [0 , T ] almost surely. If last inequality is strict we call Θ super a -admissible , and wesimply call it admissible if it is a-admissible for some a > . Definition 2.4.
The set of all admissible (resp. a -admissible) strategies is denoted by H (resp. H a ). We write H ∗ a for the set of super a -admissible strategies. In particular, we use H + for theset H ∗ X . We will mainly work with the set H and H + , the latter being the one that forbids thecondition of temporary-bankruptcy.In the following we recall the central result of the theory of pricing and hedging by no-arbitrage, that relates the condition of no arbitrage with the existence of an Equivalent LocalMartingale Measure (ELMM). A proof of this result, known as the
Fundamental Theorem ofAsset Pricing , is given in quite general settings in [23].
Definition 2.5.
The semi-martingale S satisfies (ELMM) if there is a probability measure Q on F equivalent to P such that the discounted price process is a local martingale with respect to (Ω , F , Q ) . Proposition 2.6 (Fundamental Theorem of Asset Pricing) . The semi-martingale S satisfies ( N F LV R ) H if and only if it satisfies (ELMM). The following result on change of probabilities provides a necessary condition for the exis-tence of an ELMM and it will be useful in the context of enlargements of filtration introducedin the next section.
Proposition 2.7 (Cameron-Martin-Girsanov) . Let θ = ( θ t , ≤ t ≤ T ) be an F predictableprocess, and such that E (cid:20) exp (cid:18) (cid:90) T θ t dt (cid:19)(cid:21) < + ∞ (Novikov’s Condition) , (2.4) then there exists a measure Q on (Ω , F T ) , equivalent to P , such that d Q d P = E (cid:18)(cid:90) T θ t dB t (cid:19) and W = ( B t − (cid:82) t θ u du, ≤ t ≤ T ) is a ( F , Q ) Brownian motion. The process B = ( B t , ≤ t ≤ T ) is the ( F , P ) Brownian motion appearing in (2.1b) . In this section we introduce the concepts required to model the portfolio of an insider agent.We assume that an insider has at her disposal more information than the one freely accessible,and we model this by enlarging the filtration with respect to which she can look for adaptedstrategies.To this end, we introduce the filtration G = {G t , t ≥ } that we assume larger than F , thatis F ⊂ G . In particular we focus on the case where the additional information is accessible sincethe initial time, that is G = F σ ( G ) where F σ ( G ) t = (cid:92) s>t ( F s ∨ σ ( G )) , (2.5)with G ∈ F T being a real random variable modeling the privileged information. BASIC NOTIONS AND PRELIMINARIES In order to assure that any F semi-martingale is also a G semi-martingale, what is known inthe literature as hypothesis (H’) , see [3], for the rest of this paper we state the following standingassumption, known as condition (A’) in [4], that is stronger as it implies the hypothesis (H’). Assumption:
The distribution of G is positive and σ -finite while the regular F t -conditionaldistributions almost surely verifies for t ∈ [0 , T ) the absolutely continuity condition P ( G ∈·|F t ) (cid:28) P ( G ∈ · ).The above assumption assures the existence of a jointly measurable process η g = ( η gt , ≤ t < T ), with ( g, t ) ∈ R × [0 , T ) such that P ( A |F t ) = (cid:82) A η gt P ( G ∈ dg ) for any A ∈ σ ( G ).The following result allows to compute the G -semi-martingale decomposition of an F -semi-martingale. Its proof is given in [4]. Proposition 2.8 (Jacod, 1985) . There exists a jointly measurable process α g = ( α gt , ≤ t < T ) ,with ( g, t ) ∈ R × [0 , T ) such that, • (cid:82) t ( α Gu ) du < + ∞ almost surely. • (cid:104) η G , B (cid:105) t = (cid:82) t η Gu α Gu du .and W t = B t − (cid:82) t α Gu du is a G -Brownian motion. Remark 2.9.
Using the second statement of the previous lemma, for t < T , we can write η Gt = E (cid:18) − (cid:90) t α Gu dB u (cid:19) , (2.6) where E ( X ) denotes the Dol´eans-Dade exponential of the semi-martingale X , see also [6, 25]. The following proposition gives a first relation between the (FLVR) arbitrage condition andthe α G process. For a proof see [15]. Proposition 2.10. If P (cid:16)(cid:82) T ( α Gt ) dt = ∞ (cid:17) > , then S satisfies ( F LV R ) H ( G ) . Using results from the previous section combined with the semi-martingale decompositiongiven in Proposition 2.8 we can construct a simple test to check the (NFLVR) H ( G ) condition inpresence of initial enlargement of filtration. This result can be also found in [26, Theorem 2.3]. Corollary 2.11.
Let G as in (2.5) , if the process α G = ( α Gt , ≤ t < T ) satisfies E (cid:20) exp (cid:18) (cid:90) T ( α Gt ) dt (cid:19)(cid:21) < + ∞ , (2.7) then ( N F LV R ) H ( G ) holds true.Proof. By the Fundamental Theorem of Asset Pricing, Proposition 2.6, the (NFLVR) condi-tion follows by the existence of an ELMM. Combining the Cameron-Martin-Girsanov theo-rem (Proposition 2.7) with the semi-martingale decomposition given in Proposition 2.8, the(
N F LV R ) H ( G ) follows by equation (2.7) that in this context is equivalent to the Novikov con-dition (2.4). UTILITY MAXIMIZATION PROBLEM In this section we begin to study the relationships between the set of strategies that the agentis allowed to employ with her maximum expected profit and in general with the conditions ofarbitrage. We start by introducing the general class of utility functions, and then by consideringthe associated maximal expected utility. The utility functions are chosen to satisfy the classicalassumptions, that is to be increasing, continuous, differentiable and strictly concave. Theseassumptions have perfect sense economically.
Definition 3.1.
Let T = {T t , t ≥ } be a given filtration and U γ ( x ) = ( x γ − /γ + γ , with γ ∈ [0 , , an utility function – for γ = 0 we actually mean the right limit as γ → , that is U ( x ) = ln( x ) –, we denote by v T γ ( A ) = sup Θ ∈A ( T ) E (cid:2) U γ ( X Θ T ) (cid:3) γ ∈ [0 , , the corresponding maximal expected utility of the agent constrained to work with the strategiesbelonging to the set A . For the extreme cases we use the notations u T ( A ) := v T ( A ) and v T ( A ) = v T ( A ) . Following the convention adopted above, we omit in the notation to include the filtration T whenever the underlying one is F . Remark 3.2.
The utility functions as introduced above are a slight modification of the classicalConstant Relative Risk Aversion (CRRA) functions, see [1, 27], by the additional constant term γ . We use this modified form as they preserve the same optimal portfolio while including theimproper linear utility function for γ = 1 . Proposition 3.3.
With the previous notation, the following inequalities hold, u T ( H + ) < v T γ ( H + ) < v T ( H + ) ≤ v T ( H ) Proof.
The first two inequalities follow by the fact that U γ ( x ) is strictly increasing in theparameter γ while the last one by the fact that H + ⊂ H .It can be shown that with a sufficiently large initial capital, arbitrage opportunities do notdepend on the bankruptcy, as the following result shows. Lemma 3.4.
Given a filtration T , the following equivalence hold ( N A ) H ( T ) ⇐⇒ { ( N A ) H + ( T ) , ∀ X ∈ R + } . Proof.
The necessary condition follows from the inclusion H + ⊂ H . For the sufficient one, westart by assuming that ( N A ) H + ( T ) holds true for all X ∈ R + and we prove it by contradiction.If ( A ) H ( T ) holds, then there exists a strategy Θ ∈ H ( T ) with the property that (Θ · S ) T ≥ P ((Θ · S ) T > >
0. By definition of the set H ( T ) we deduce that thereexists a constant a > ∀ t ∈ [0 , T ], (Θ · S ) t ≥ − a + (cid:15) almost surely, with some (cid:15) > X t = a + (Θ · S ) t , we see that we get X T > a almost surely thereforeimplying that ( A ) H + ( T ) holds for X = a and Θ ∈ H + T , that is a contradiction.We present a technical result that we will use later. Its proof is given in [15]. Lemma 3.5.
Let S = ( D, S ) be the pair of continuous semi-martingale satisfying ( N F LV R ) H ( T ) .If (Θ · S ) T ≥ − X almost surely with Θ allowed, then the process Θ ∈ H X . Finally, we get to the result that relates the condition of (NFLVR) H ( T ) with the expectedterminal utility under H + ( T ) and H ( T ). EXAMPLES Proposition 3.6.
Let U be an utility function such sup x ∈ R { U ( x ) = −∞} = 0 , then the follow-ing implications hold. ( N F LV R ) H ( T ) = ⇒ sup Θ ∈H ( T ) E (cid:2) U ( X Θ T ) (cid:3) = sup Θ ∈H + ( T ) E (cid:2) U ( X Θ T ) (cid:3) . Proof.
We assume there exists Θ ∈ H that E [ U ( X Θ T )] > sup Θ ∈H + ( T ) E (cid:2) U ( X Θ T ) (cid:3) . Then, Θ (cid:54)∈ H + and X Θ t (cid:54)≥ t ∈ [0 , T ]. Like (NFLVR) H ( T ) holds, lemma 3.5 says that X Θ T < E [ U ( X Θ T )] = −∞ follows. In this section, we show several examples to highlight the differences that may or may not existby playing with one set of strategies or another. We focus on the following specific model,deeply studied in the literature, see in particular [1, 7], in where the risky asset is given by aGeometric Brownian motion. In equation (2.1b), we set the drift and volatility processes to µ ( t, S t ) = η t S t and σ ( t, S t ) = ξ t S t , so that the resulting model is dD t = D t r dt, (4.1a) dS t = S t ( η t dt + ξ t dB t ) (4.1b)where r > η = ( η t , ≤ t ≤ T ) and ξ = ( ξ t , ≤ t ≤ T ), that we refer to as the proportional drift and volatility, are assumed to be adapted tothe natural filtration of the process B with η , ξ and 1 /ξ bounded. Whenever the agent playswith Θ ∈ H + , the wealth process X is almost surely strictly positive, and therefore we can usean alternative form to express the SDE (2.3b), that is dX t X t = (1 − π t ) dD t D t + π t dS t S t . (4.2)where π t = S t N t /X t and N t is defined in (2.3a). The following result, proved in [1] and adaptedhere to our notation, gives the optimal strategies and the corresponding maximal expectedutilities for the cases when the agent has no additional information (i.e. under the filtration F ) and works with the strategies that do not allow bankruptcy (i.e. in the set H + ). For moredetails we refer the reader to [7]. Theorem 4.1 (Merton, 1969) . Under the filtration F the optimal strategy is arg sup Θ ∈H + E (cid:2) U γ ( X Θ T ) (cid:3) = 11 − γ η t − rξ t , γ ∈ [0 , and the maximal expected utility is given, with γ ∈ [0 , , by u ( H + ) = ln X + rT + 12 (cid:90) T E (cid:34)(cid:18) η t − rξ t (cid:19) (cid:35) dt logarithmic utility , (4.3a) v γ ( H + ) = X γ γ E (cid:34) exp (cid:32) γrT + 12 γ − γ (cid:90) T (cid:18) η t − rξ t (cid:19) dt (cid:33)(cid:35) + γ − γ CRRA utility . (4.3b) EXAMPLES In presence of insider information, as expected, the optimal strategies change since the agentmay take advantage of the additional information she has privileged access to. The followingresult computes the same quantities as in Theorem 4.1 but under the initial enlarged filtration G .The process α G = ( α Gt , ≤ t < T ) appearing in the statement comes from the semi-martingaledecomposition given in Proposition 2.8. The result for the logarithmic utility has been provedin [7] while the one for general CRRA utilities can be obtained by solving the correspondingHJB equation. Details can be found, for example, in [27]. Theorem 4.2.
Under the filtration G the optimal strategy is arg sup Θ ∈H + ( G ) E [ U γ ( X T )] = 11 − γ (cid:18) η t − rξ t + α Gt ξ t (cid:19) , γ ∈ [0 , and the maximal expected utility is given, with γ ∈ [0 , , by u G ( H + ) = ln X + rT + 12 (cid:90) T E (cid:34)(cid:18) η t − rξ t (cid:19) (cid:35) dt + 12 (cid:90) T E (cid:2) ( α Gt ) (cid:3) dt logarithmic utility , (4.4a) v G γ ( H + ) = X γ γ E (cid:34) exp (cid:32) γrT + 12 γ − γ (cid:90) T (cid:18) η t + α Gt ξ t − rξ t (cid:19) dt (cid:33)(cid:35) + γ − γ CRRA utility . (4.4b)From the result above it follows that u G ( H + ) < + ∞ if and only if (cid:82) T E (cid:2) ( α Gt ) (cid:3) dt < + ∞ .As it is known, for the natural filtration F , an ELMM can be found and we conclude thatNFLVR H ( F ) holds true. In the following result, we specify if the maximal expected utility of theagent that plays with the filtration F for the different utilities that we have defined is boundedor not. Proposition 4.3.
Under the modeling assumptions (4.1) , v F γ ( H + ) < + ∞ for ≤ γ < and v F ( H ) = v F ( H + ) = + ∞ .Proof. For 0 ≤ γ < η and 1 /ξ and thefollowing expression v γ ( H + ) = X γ γ exp( γrT ) E (cid:34) exp (cid:32) γ − γ (cid:90) T (cid:18) η t − rξ t (cid:19) dt (cid:33)(cid:35) + γ − γ < + ∞ . (4.5)For γ = 1, using Jensen’s inequality , we get v ( H + ) = lim γ → v γ ( H + ) ≥ lim γ → X γ γ exp (cid:32) γrT + 12 γ − γ E (cid:34)(cid:90) T (cid:18) η t − rξ t (cid:19) dt (cid:35)(cid:33) + γ − γ = + ∞ . (4.6) G with v G γ ( H + ) = ∞ for γ ∈ [0 , Fixing G = B T , we consider the enlargement of filtration G = F σ ( B T ) . This implies that theinsider agent knows since the time t = 0 the final value B T ( ω ) for any ω ∈ Ω. The semi-martingale decomposition in the filtration G is dB t = α B T t dt + dW t , where α B T t = B T − B t T − t , ≤ t < T (4.7) EXAMPLES and the process W = ( W t , ≤ t ≤ T ) is a G -Brownian motion. Applying Itˆo calculus, if we uselogarithmic utility and the strategy set H + ( G ), equation (4.2) has the following exact solutionln X T = ln X + (cid:90) T (cid:18) r (1 − π t ) + η t π t + π t ξ t B T − B t T − t − π t ξ t (cid:19) dt + (cid:90) T π t ξ t dW t . By pointwise maximizing the expectation of the equation above, we get the optimal strategy π ∗ t ( B T ) = arg sup π t ∈H + ( G ) E [ln X T ] = η t − rξ + 1 ξ t B T − B t T − t (4.8)that implies the following pathwise capital gain X T = X exp (cid:18) rT + 12 (cid:90) T ( π ∗ t ( B T ) ξ t ) dt + (cid:90) T π ∗ t ( B T ) ξ t dW t (cid:19) (4.9) Proposition 4.4.
Let G = B T , then v G ( H ) = v G γ ( H + ) = + ∞ , ∀ γ ∈ [0 , . In addition thestronger result, lim t → T ln( X π ∗ ( B T ) t ) = + ∞ , holds almost surely.Proof. In [7], it was proved that E (cid:104) ln X π ∗ ( B T ) T (cid:105) is not bounded. Using Proposition 3.3 we getthe result for the first statement.For the second statement, we denote the function α B T t in (4.7) by simply α t and we use thedefinition β t = ( η t − r ) / ( ξ t ).From (4.9) we have that, for 0 < s < t < T ,log X T X e rT = 12 (cid:90) t ( β s − α s ) ds + (cid:90) t ( β s + α s ) dB s (4.10)Using the semi-martingale decomposition (4.9), see also [28] and [29, Chapter VI], we have (cid:90) t α s dB s = (cid:90) t B T − B s T − s dB s = B T (cid:90) t T − s dB s − (cid:90) t B s T − s dB s . (4.11)By applying the Itˆo’s lemma to the functions x/ ( T − t ) and x / ( T − t ) respectively, we get (cid:90) t T − s dB s = B t T − t − (cid:90) t B s ( T − s ) ds (cid:90) t B s T − s dB s = 12 B t T − t − (cid:90) t B s ( T − s ) + 1 T − s ds that substituted in (4.11) give (cid:90) t α s dB s = 12 (cid:90) t B s − B T B s ( T − s ) ds − B t − B T B t T − t + 12 (cid:90) t T − s ds . (4.12)In addition 12 (cid:90) t α s ds = 12 (cid:90) t B s − B T B s + B T ( T − s ) ds By substituting (4.12) in (4.10) and by manipulating the expression we finally getlog X T X e rT = 12 (cid:90) t β s ds + (cid:90) t β s dB s + 12 B T T −
12 ( B T − B t ) T − t + 12 (cid:90) t T − s ds . We have ( B T − B t ) / ( T − t ) ∼ χ , so all terms but the integral on the right end side havewell-defined limits as t → T . Since the integral diverges almost surely we get the result. EXAMPLES The fact that the optimal expected logarithmic utility is almost surely unbounded impliesthat the filtration G admits arbitrage opportunities. However the strategy π ∗ ( B T ) is quitecomplicated to implement as, according to (4.8), it is of finite quadratic variation, while for thisexample, it is easy to see that the arbitrage can be achieved more easily by employing strategieswith a more clear financial meaning, such as buy, hold and sell. We give in the following aprecise definition of this kind of strategies as given in [23, Definition 7.1]. Definition 4.5.
A simple predictable strategy is a linear combination of processes of the form Θ t = M ( T ,T ] , where M is F T measurable and T and T are finite stopping times with respectto the filtration F T . When stopping time T happens, the agent buys a quantity M ∈ F T of shares of the riskyasset at price S T , borrowing the corresponding money from the riskless asset. Until T , she holds her position and then she sells her shares at price S T .For the following proposition, we assume that the proportional volatility process is constantand the proportional drift process is deterministic. Under this simplifying assumption, theinsider information on B T is equivalent to the one on S T according to the following equation S T = S exp (cid:18)(cid:90) T (cid:18) η t − ξ (cid:19) dt + ξB T (cid:19) . Proposition 4.6.
Let G = B T and X > , then ( A ) H + ( G ) holds.Proof. Let 0 < (cid:15) < S T and we define the following stopping time. τ (cid:15) = inf { S t < e − r ( T − t ) ( S T − (cid:15) ) } . (4.13)On the set { τ (cid:15) < T } , the agent invests her money in the risky asset at time τ (cid:15) . This strategy ismodeled by, N t = X e r τ (cid:15) S τ (cid:15) { t ≥ τ (cid:15) } . On { τ (cid:15) < T } , X T = e rT X S T S T − (cid:15) and on its complement X T = e rT X . As P ( τ (cid:15) < T ) >
0, weconclude that X T ≥ e rT X almost surely and P ( X T > e rT X ) > . The Figure 1 shows an example of the situation that is described in the proof. When thestopping time happens, the insider trader invests in the stock as she knows almost surely thatshe will realize a positive profit. τ (cid:15) tS t e t ( e t − (cid:15) ) Figure 1: An example of realization of the stopping time τ (cid:15) defined in (4.13). Corollary 4.7.
Let G = B T , then we have ( A ) H ( G ) , ( F LV R ) H + ( G ) and ( F LV R ) H ( G ) . EXAMPLES G with v ( H + ) < ∞ and v G γ ( H + ) = ∞ for γ ∈ (0 , Here we analyze another example of enlargement of filtrations. It was introduced by [7] to studythe case when the insider trader knows a lower or an upper bound of the stock price in a certainfuture horizon time. We use the initial enlargement G = F σ ( G ) with G = { c ≤ B T ≤ c } . (4.14)In the following, we show that the additional information carried by the filtration G impliesa finite terminal logarithmic utility, and therefore different for the case analyzed in Section 4.1.We start by computing the explicit expression of the drift α G = ( α Gt , ≤ t < T ) appearingin the G -semi-martingale decomposition given in Proposition 2.8. The proof is deferred to theappendix. Proposition 4.8.
Let G as in (4.14) , then α gt = ( − g √ T − t Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) − Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) P ( G = g | B t ) , ≤ t < T and more explicitly α gt = √ T − t Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) − Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) Φ (cid:16) − c − B t √ T − t (cid:17) + Φ (cid:16) c − B t √ T − t (cid:17) when g = 0 , √ T − t Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) − Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) Φ (cid:16) c − B t √ T − t (cid:17) − Φ (cid:16) c − B t √ T − t (cid:17) when g = 1 . (4.15) where Φ denotes the distribution of a standard Gaussian random variable. Before the main result of this section we need to introduce the following technical lemmawhose proof is deferred to the appendix.
Lemma 4.9.
The integral of the function I ( x, t ) defined as I ( x, t ) = 1 √ T − t [Φ (cid:48) ( z ) − Φ (cid:48) ( z )] [Φ ( z ) − Φ ( z )] [Φ ( − z ) + Φ ( z )] , (4.16) in the variable x ∈ R is uniformly bounded for t ∈ [0 , T ] . In (4.16) , we have used the followingdefinitions z = ( c − x ) / √ T − t and z = ( c − x ) / √ T − t . The following result shows that the logarithmic utility optimization allows for a finite opti-mum. This result was first conjectured in [7], where they supported the conjecture via numericalresults. Then the conjecture was solved in the general entropy setting by [9]. Here we give amore direct proof on the line of the arguments given in [7].
Theorem 4.10.
Let G as in (4.14) , then u G ( H + ) < ∞ .Proof. By using the expression of α G , given in [7, Equation (4.25)], we have E [( α Gt ) ] = 12 π √ T − t √ πt (cid:90) R I ( x, t ) e − x / dx EXAMPLES where I ( x, t ) is defined in (4.16) and the volatility process is bounded. By (4.4a), it is enoughto prove that, for some constant K > E [( α Gt ) ] ≤ K (cid:112) t ( T − t ) . This follows by Lemma 4.9.
Proposition 4.11.
Let G as in (4.14) , then v G γ ( H + ) = + ∞ for γ ∈ [1 / , .Proof. By Corollary 4.13 the (
F LV R ) H ( G ) condition holds and therefore an ELMM can notexist. This implies that the Novikov condition, given in (2.7), is not satisfied. We conclude that E (cid:104) exp (cid:16) (cid:82) T ( α G ) dt (cid:17)(cid:105) = + ∞ . Moreover, if γ ≥ /
2, it follows that γ/ (1 − γ ) ≥ v γ ( H + ) = X γ γ exp( γrT ) E (cid:34) exp (cid:32) γ − γ (cid:90) T (cid:18) η t − rξ t + α Gt (cid:19) dt (cid:33)(cid:35) + γ − γ ≥ X γ γ exp( γrT ) E (cid:34) exp (cid:32) (cid:90) T (cid:18) η t − rξ t + α Gt (cid:19) dt (cid:33)(cid:35) + γ − γ = + ∞ . Now, we are going to enunciate the results on arbitrage for the strategy sets H ( G ) and H + ( G ). Using the solution of the process S at time T , under constant proportional volatilityand proportional deterministic drift processes, this problem is equivalent to, G = (cid:110) b ≤ S T ≤ b (cid:111) , b i = S exp (cid:18)(cid:90) T (cid:18) η t − ξ (cid:19) dt + ξ c i (cid:19) . (4.17)So, under these hypothesis of the coefficient processes, we can link the privileged informationbetween the process B = ( B t , ≤ t ≤ T ) and S = ( S t , ≤ t ≤ T ). In the following statements,we use the assumption and notation given in (4.17). Proposition 4.12.
Let G as in (4.14) , then ( A ) H + ( G ) holds.Proof. The result follows by a similar argument of Proposition 4.6.
Corollary 4.13.
Let G as in (4.14) , then we have ( A ) H ( G ) , ( F LV R ) H + ( G ) and ( F LV R ) H ( G ) . Proposition 4.14.
Let G as in (4.14) , then v G ( H ) = + ∞ .Proof. Let 0 < (cid:15) < b / τ (cid:15) , τ (cid:15) = inf { S t < e − r ( T − t ) ( b − (cid:15) ) } . (4.18)and the strategy Θ = { ( M t , N t ) : 0 ≤ t ≤ T } with N t = M X e r τ(cid:15) S τ(cid:15) { t ≥ τ (cid:15) } , for some constant M >
0. On { τ (cid:15) < T } , X T = e rT X M S T b − (cid:15) + e rT X (1 − M ) and on its complement X T = e rT X ,hence, E [ X Θ T ] = e rT X + e rT X M (cid:20) S T b − (cid:15) − (cid:21) P ( τ (cid:15) < T ) , which is not bounded as M → ∞ for a fixed (cid:15) > EXAMPLES G with v G γ ( H + ) < ∞ for γ ∈ [0 , and v G ( H + ) = + ∞ In this section we show that the acquisition of additional information by an insider agent doesnot directly implies that she can take advantage of an arbitrage. Indeed we show that evenknowing information about the terminal price of the risky asset (we model this by a functionof B T ) may not lead to an arbitrage condition.We use the initial enlargement G = F σ ( G ) with G = (cid:110) B T ∈ ∪ + ∞ k = −∞ [2 k − , k ] (cid:111) (4.19)implying that the insider trader only knows if the Brownian motion will end up in a particularinfinite union of intervals of size one. Following the same arguments of Proposition 4.8, nextresult gives a closed form expression of the new drift of the G -semi-martingale decomposition.It’s proof is deferred to the appendix. Proposition 4.15.
Let G be as in (4.19) , then α gt = ( − g √ T − t P ( G = g | B t ) (cid:32) + ∞ (cid:88) k = −∞ Φ (cid:48) (cid:18) k − B t √ T − t (cid:19) − Φ (cid:48) (cid:18) k − − B t √ T − t (cid:19)(cid:33) , ≤ t < T and more explicitly α gt = √ T − t (cid:80) + ∞ k = −∞ Φ (cid:48) (cid:16) k − B t √ T − t (cid:17) − Φ (cid:48) (cid:16) k − − B t √ T − t (cid:17)(cid:80) + ∞ k = −∞ Φ (cid:16) − k − B t √ T − t (cid:17) + Φ (cid:16) k − − B t √ T − t (cid:17) when g = 0 , √ T − t (cid:80) + ∞ k = −∞ Φ (cid:48) (cid:16) k − − B t √ T − t (cid:17) − Φ (cid:48) (cid:16) k − B t √ T − t (cid:17)(cid:80) + ∞ k = −∞ Φ (cid:16) k − B t √ T − t (cid:17) − Φ (cid:16) k − − B t √ T − t (cid:17) when g = 1 . (4.20) where Φ denotes the distribution of a standard Gaussian random variable. Before stating the main result of this section we state the following version of the MeanValue Theorem for definite integrals, whose proof may be found in [30, Lemma 2.1].
Lemma 4.16.
Let c < c , and f ( · ) a differentiable function. There exists ξ ∈ ( c , c ) suchthat c − c (cid:90) c c f ( y ) dy = f ( c ) + f ( c )2 − c − c (cid:90) c c (cid:18) y − c + c (cid:19) f (cid:48) ( y ) dy = f ( c ) + f ( c )2 − (cid:18) ξ − c + c (cid:19) f (cid:48) ( ξ ) . (4.21)The following result shows that under the enlargement by the random variable in (4.19) theinsider does not get any possibility of arbitrage. Theorem 4.17.
Let G be as in (4.19) , then (NFLVR) G holds.Proof. By Corollary 2.11, it is enough to prove that the process α G satisfies the Novikov con-dition (2.7). Using the Jensen inequality we get E (cid:20) exp (cid:18) (cid:90) T ( α Gt ) dt (cid:19)(cid:21) ≤ T (cid:90) T E (cid:20) exp (cid:18) T α Gt ) (cid:19)(cid:21) dt , EXAMPLES so we are left to find a bound for the right hand side of the expression above. The expectationcan be computed as E (cid:20) exp (cid:18) T α Gt ) (cid:19)(cid:21) = (cid:88) g ∈{ , } (cid:90) R exp (cid:18) T α gt ) (cid:19) P ( G = g | B t = x ) d P ( B t ≤ x ) . (4.22)Using the similarity of the integrals for g ∈ { , } , we focus on getting a uniform bound,integrable with respect to t for the case g = 1.Let A = ∪ + ∞ k = −∞ [2 k − , k ] be the set appearing in (4.19). If x ∈ A , then P ( G = 1 | B t = x )is far from zero and the numerator of α t is uniformly convergent as t → T and bounded by 2 inthe interval [0 , T ]. Bounding the term P ( G = 1 | B t = x ) by 1 we get the following inequality E (cid:20) exp (cid:18) T α t ) (cid:19)(cid:21) ≤ K + (cid:90) A c exp (cid:18) T α t ) (cid:19) d P ( B t ≤ x ) , where K > c , c ], to latergeneralize the argument to the whole set A c .In the following we assume that x (cid:54)∈ [ c , c ] and z i = ( c i − x ) / √ T − t with i ∈ { , } . ByLemma 4.16, there exists ζ ∈ [ c , c ] such that P ( B T ∈ [ c , c ] | B t = x ) = c − c √ T − t Φ (cid:48) ( z ) + Φ (cid:48) ( z )2+ (cid:18) ζ − x √ T − t − c + c − x √ T − t (cid:19) c − c √ T − t exp (cid:18) − ( ζ − x ) T − t ) (cid:19) . (4.23)Using the expression above to find an alternative form of the following term,1 √ T − t Φ (cid:48) ( z ) − Φ (cid:48) ( z ) P ( B T ∈ [ c , c ] | B t = x ) = (Φ (cid:48) ( z ) − Φ (cid:48) ( z )) / (Φ (cid:48) ( z ) + Φ (cid:48) ( z ))( c − c ) / (cid:0) ζ − c + c (cid:1) c − c √ T − t exp (cid:16) − ( ζ − x )22( T − t ) (cid:17) Φ (cid:48) ( z )+Φ (cid:48) ( z ) (4.24)that allows to show that it is bounded for all t ∈ [0 , T ]. It is enough to look at the limit as t → T . The numerator is easily shown to be bounded as0 ≤ lim t → T Φ (cid:48) ( z ) − Φ (cid:48) ( z )Φ (cid:48) ( z ) + Φ (cid:48) ( z ) ≤ lim t → T Φ (cid:48) ( z )Φ (cid:48) ( z ) + Φ (cid:48) ( z ) ≤ . Therefore the boundedness follows from the fact that the denominator is sum of a constantterm and another one that goes to zero. Indeed, assuming w.l.o.g. that x < c < c – the othercase being equivalent by the symmetry of the function Φ (cid:48) –, we havelim t → T c − c √ T − t exp (cid:16) − ( ξ − x ) T − t ) (cid:17) Φ (cid:48) ( z ) + Φ (cid:48) ( z ) = lim t → T √ π c − c √ T − t exp (cid:16) − ( ξ − x ) T − t ) (cid:17) exp (cid:16) − ( c − x ) T − t ) (cid:17) + exp (cid:16) − ( c − x ) T − t ) (cid:17) = lim t → T √ π c − c √ T − t exp (cid:16) − ( ξ − x ) − ( c − x ) T − t ) (cid:17) (cid:16) − ( c − x ) − ( c − x ) T − t ) (cid:17) = 0 . Fixing t sufficiently near to T , we have shown that1 √ T − t Φ (cid:48) ( z ) − Φ (cid:48) ( z ) P ( B T ∈ [ c , c ] | B t = x ) ≤ c − c , x (cid:54)∈ [ c , c ] . CONCLUSIONS Replacing the interval [ c , c ] by the set A = ∪ + ∞ k = −∞ [2 k − , k ], and by repeating the sameargument above, we get that α t = (cid:80) + ∞ k = −∞ Φ (cid:48) (cid:16) k − − x √ T − t (cid:17) − Φ (cid:48) (cid:16) k − x √ T − t (cid:17) P ( B T ∈ A | B t = x ) ≤ , x ∈ A c . Finally, E (cid:20) exp (cid:18) T α t ) (cid:19)(cid:21) ≤ K + (cid:90) A c exp (cid:18) T (cid:19) d P ( B t ≤ x ) < + ∞ and the result follows. Proposition 4.18.
Let G be as in (4.19) , then v G γ ( H + ) < + ∞ for ≤ γ < and v G ( H ) = v G ( H + ) = + ∞ .Proof. Similarly to the arguments used in Proposition 4.3, we can show the result starting fromequations (4.4).
Table 1: Summary of the utility maximization problems analyzed in Section 4.Sect. T u T ( H + ) u T ( H ) v T γ ( H + ) v T γ ( H ) v T ( H + ) = v T ( H ) Arbitrage cond4 F < + ∞ = + ∞ (NA) F G < + ∞ = + ∞ (NA) G G < + ∞ (in H + for γ ≥ / = + ∞ (A) G G = + ∞ (A) G In this paper we analyzed the relations between the arbitrage conditions, the utility max-imization problems and the enlargements of filtration. In particular we considered all theseconcepts with the respect to the class of strategies the agent may employ in maintaining herportfolio, by focusing on the general admissible class, H , and the one that does not allow fortemporary-bankruptcy, H + .In terms of arbitrage, by Lemma 3.4, there is practically no difference between working withthe class H and the class H + . However in terms of utility maximization, the difference becomesclear as we found cases in which the (logarithmic) utility maximization is finite even in presenceof arbitrage. We have included examples of this type and Table 1 shows a brief summary of theresults.In particular it shows with the example of Section 4.3 that an enlargement of initial typedoes not always imply arbitrage for the insider. Acknowledgments
This research was partially supported by the Spanish Ministry of Economy and Competitivenessgrants MTM2017-85618-P (via FEDER funds), MTM2015-72907-EXP and FPU18/01101. Wethank the referees for the insightful comments that helped to improve the presentation of thiswork.
EFERENCES Conflict of interest
The authors declare no conflicts of interest in this paper.
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Appendix
Proof of Proposition 4.8.
From (2.6) it follows that the process α G = ( α Gt , ≤ t < T ) satisfiesthe following relation dη t = η t α Gt dB t , (5.1) EFERENCES where η is the Radon-Nikodym derivative between the law of G = { c ≤ B T ≤ c } conditionedto the the σ -algebra F t and its unconditional law, i.e, η t ( g ) = P ( G = g |F t ) P ( G = g ) , g ∈ { , } . (5.2)Substituting (5.2) in (5.1) we try to get the an expression for the process α G .The conditional probability mass function of G computed for { G = 1 } may be expressed interms of Gaussian distributions as P ( G = 1 |F t ) = Φ (cid:18) c − B t √ T − t (cid:19) − Φ (cid:18) c − B t √ T − t (cid:19) . (5.3)By Itˆo’s lemma applied to the function Φ (cid:0) ( a − x ) / ( √ T − t ) (cid:1) , with a ∈ R , we have that d Φ (cid:18) a − B t √ T − t (cid:19) = − √ T − t Φ (cid:48) (cid:18) a − B t √ T − t (cid:19) dB t . (5.4)Substituting (5.4) in (5.3) and applying linearity it follows that d P ( G = 1 |F t ) = 1 √ T − t (cid:18) Φ (cid:48) (cid:18) c − B t √ T − t (cid:19) − Φ (cid:48) (cid:18) c − B t √ T − t (cid:19)(cid:19) dB t and finally, dη t η t = 1 √ T − t Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) − Φ (cid:48) (cid:16) c − B t √ T − t (cid:17) P ( G = 1 |F t ) dB t , (5.5)and we get the result. The case { G = 0 } follows on the same lines. Proof of Lemma 4.9.
We start by splitting R in three intervals ( −∞ , c ], ( c , c ) and [ c , ∞ ),then we prove that on each interval the integral is finite. Interval ( −∞ , c ] : We apply a change of variable in z and express z = z + ( c − c ) / √ T − t . We let s t := ( c − c ) / √ T − t and call its minimum in t as s >
0. We get (cid:90) c −∞ I ( x, t ) dx = (cid:90) + ∞ [Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t )] [Φ ( z + s t ) − Φ ( z )] [Φ ( − z − s t ) + Φ ( z )] dz = (cid:90) + ∞ (cid:32) [Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t )] Φ ( z + s t ) − Φ ( z ) + [Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t )] Φ ( − z − s t ) + Φ ( z ) (cid:33) dz . (5.6)We continue by showing that both terms are finite. We first consider the first term. (cid:90) + ∞ [Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t )] Φ ( z + s t ) − Φ( z ) dz ≤ (cid:90) + ∞ [Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t )] Φ ( z + s ) − Φ( z ) dz ≤ (cid:90) + ∞ [Φ (cid:48) ( z )] Φ ( z + s ) − Φ( z ) dz = (cid:90) [Φ (cid:48) ( z )] Φ ( z + s ) − Φ( z ) dz + (cid:90) + ∞ [Φ (cid:48) ( z )] Φ ( z + s ) − Φ( z ) dz . The integral in [0 ,
1] is clearly bounded. While for the interval [1 , + ∞ ], we apply a comparisoncriteria with the function f ( z ) = 1 /z , as follows,lim z →∞ z [Φ (cid:48) ( z )] Φ ( z + s ) − Φ( z ) = lim z →∞ √ π z (cid:2) exp (cid:0) − z / (cid:1)(cid:3) (cid:82) z + s z exp( − u / du = lim z →∞ √ π z (cid:2) exp (cid:0) − z / (cid:1)(cid:3) − z (cid:2) exp (cid:0) − z / (cid:1)(cid:3) exp( − ( z + s ) / − exp( − z / z →∞ √ π z exp (cid:0) − z / (cid:1) − z exp (cid:0) − z / (cid:1) exp( − z s ) exp( − s / − → . EFERENCES In the second equality above, we used L’Hopital Rule and we conclude that the integral is finiteon (1 , + ∞ ).As for the second term in (5.6), we have the following bound, (cid:90) + ∞ [Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t )] Φ( − z − s t ) + Φ( z ) dz ≤ (cid:90) + ∞ [Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t )] Φ( z ) dz ≤ (cid:90) + ∞ (cid:2) Φ (cid:48) ( z ) − Φ (cid:48) ( z + s t ) (cid:3) dz ≤ (cid:90) + ∞ (cid:2) Φ (cid:48) ( z ) (cid:3) dz = 1 √ . Interval [ c , + ∞ ) : We proceed in the same way as above, now applying a change of variablein z . (cid:90) + ∞ c I ( x, t ) dx = (cid:90) −∞ [Φ (cid:48) ( z − s t ) − Φ (cid:48) ( z )] [Φ ( z ) − Φ ( z − s t )] [Φ ( − z ) + Φ ( z − s t )] dz = (cid:90) −∞ (cid:32) [Φ (cid:48) ( z − s t ) − Φ (cid:48) ( z )] Φ ( z ) − Φ ( z − s t ) + [Φ (cid:48) ( z − s t ) − Φ (cid:48) ( z )] Φ ( − z ) + Φ ( z − s t ) (cid:33) dz (5.7)We show that both terms in (5.7) are finite. For the first one we have (cid:90) −∞ [Φ (cid:48) ( z − s t ) − Φ (cid:48) ( z )] Φ ( z ) − Φ ( z − s t ) dz ≤ (cid:90) −∞ [Φ (cid:48) ( z − s t ) − Φ (cid:48) ( z )] Φ ( z ) − Φ ( z − s ) dz ≤ (cid:90) −∞ [Φ (cid:48) ( z )] Φ ( z ) − Φ ( z − s ) dz , and applying the same reasoning as before, we conclude that the integral is finite. Then for thesecond term we have (cid:90) −∞ [Φ (cid:48) ( z − s t ) − Φ (cid:48) ( z )] Φ ( − z ) + Φ ( z − s t ) dz ≤ (cid:90) −∞ [Φ (cid:48) ( z − s t ) − Φ (cid:48) ( z )] Φ ( − z ) dz ≤ (cid:90) −∞ (cid:2) Φ (cid:48) ( z ) (cid:3) dz = 1 √ . Interval ( c , c ) : We proceed by applying a change of variable, and we arbitrarily chooseto do it in the variable z . We get (cid:90) c c I ( x, t ) dx = (cid:90) s t (cid:20) [Φ (cid:48) ( z ) − Φ (cid:48) ( z − s t )] Φ( z ) − Φ( z − s t ) + [Φ (cid:48) ( z ) − Φ (cid:48) ( z − s t )] Φ( − z ) + Φ( z − s t ) (cid:21) dz (5.8)and again we show that both integrals in (5.8) are bounded. For the first integral we have (cid:90) s t [Φ (cid:48) ( z ) − Φ (cid:48) ( z − s t )] Φ( z ) − Φ( z − s t ) dz = 2 (cid:90) s t / [Φ (cid:48) ( z ) − Φ (cid:48) ( z − s t )] Φ( z ) − Φ( z − s t ) dz ≤ (cid:90) s t / [Φ (cid:48) ( z )] Φ( z ) − Φ( z − s ) dz ≤ (cid:90) ∞ [Φ (cid:48) ( z )] Φ( z ) − Φ( z − s ) dz where the first equality holds because the function we are integrating is symmetric with respect s t /
2. The last integral is finite as it is trivially so on [0 ,
1] and using a comparison criteria withthe function f ( z ) = 1 /z it is also integrable on [1 , + ∞ ]. In a similar way we analyze the secondintegral in (5.8) and by symmetry of the function with respect to the s t / (cid:90) s t [Φ (cid:48) ( z ) − Φ (cid:48) ( z − s t )] Φ( − z ) + Φ( z − s t ) dz = 2 (cid:90) s t / [Φ (cid:48) ( z ) − Φ (cid:48) ( z − s t )] Φ( − z ) + Φ( z − s t ) dz . EFERENCES Then we compute the following boundΦ (cid:48) ( z ) − Φ (cid:48) ( z − s t ) = 1 √ π (cid:20) exp (cid:18) − z (cid:19) − exp (cid:18) − ( z − s t ) (cid:19)(cid:21) = 1 √ π exp (cid:18) − z (cid:19) (cid:20) − exp (cid:18) − s t − z s t (cid:19)(cid:21) ≤ √ π exp (cid:18) − z (cid:19) , where the last inequality holds because s t − z s t ≥ ≤ z ≤ s t / (cid:90) s t [Φ (cid:48) ( z ) − Φ (cid:48) ( z − s t )] Φ( − z ) + Φ( z − s t ) dz ≤ (cid:114) π (cid:90) s t / exp (cid:0) − z (cid:1) Φ( − z ) + Φ( z − s t ) dz ≤ (cid:90) s t / exp (cid:0) − z (cid:1) Φ( − z ) dz ≤ (cid:90) exp (cid:0) − z (cid:1) Φ( − z ) dz + (cid:90) + ∞ exp (cid:0) − z (cid:1) Φ( − z ) dz . The first integral is trivially bounded. For the second to be bounded, we apply a comparisoncriteria with the function f ( z ) = 1 /z . Putting together the given bounds we may bound theintegral in (5.8) and the proof is finished. Proof of Proposition 4.15.
We write P ( G = 1 |F t ) = + ∞ (cid:88) k = −∞ Φ (cid:18) k − B t √ T − t (cid:19) − Φ (cid:18) (2 k − − B t √ T − t (cid:19) . (5.9)Applying (5.4) to each term appearing in (5.9) and using the Dominated Convergence The-orem for semi-martingales [29, Theorem IV.32] we get d P ( G = 1 |F t ) = + ∞ (cid:88) k = −∞ √ T − t (cid:18) Φ (cid:48) (cid:18) k − B t √ T − t (cid:19) − Φ (cid:48) (cid:18) (2 k − − B t √ T − t (cid:19)(cid:19) dB t ..