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The insider problem in the trinomial model: a discrete-time jump process approach

In an incomplete market underpinned by the trinomial model, we consider two investors : an ordinary agent whose decisions are driven by public information and an insider who possesses from the beginning a surplus of information encoded through a random variable for which he or she knows the outcome. Through the definition of an auxiliary model based on a marked binomial process, we handle the trinomial model as a volatility one, and use the stochastic analysis and Malliavin calculus toolboxes available in that context. In particular, we connect the information drift, the drift to eliminate in order to preserve the martingale property within an initial enlargement of filtration in terms of the Malliavin derivative. We solve explicitly the agent and the insider expected logarithmic utility maximisation problems and provide a hedging formula for replicable claims. We identify the insider expected additional utility with the Shannon entropy of the extra information, and examine then the existence of arbitrage opportunities for the insider.


Introduction
The aim of this paper is to investigate several aspects of insider trading in the trinomial model (see Boyle et al. [16], [17]). Adopting an initial enlargement approach, we will focus on the following issues and aims: (i) characterize the drift of information produced by moving to the enlarged filtration, (ii) provide an explicit expression of the leading strategies for replicable claims, (iii) compute the insider's benefit in the form of optimal portfolio and additional utility (iv) characterize the existence of arbitrage opportunities, (v) design, as an opening perspective, two simple models for an insider influenced market. Our findings extend to a discrete-time setting works by Amendinger et al. [6], [5], Imkeller et al. [7] [8], and Kohatsu-Higa et al. [30], [38].
The model we are considering consists of an ordinary agent and an insider who has confidential information (not available to the public) from which he or she benefits when carrying out financial transactions. Except for section 5, both investors are supposed to be small enough not to influence market prices. From the point of view of martingale theory adopted in this paper, the extra information is hidden in a random variable G of which the insider knows the outcome at the beginning of the trading interval, so that insider's filtration G is larger than F, the ordinary agent's. This framework is naturally connected to the theory of enlargement of filtration that one can roughly classify into two distinct approaches: the initial enlargement approach under Jacod's hypothesis, which assumes the equivalence between the conditional laws of G with respect to F and the law of G (see Jacod [34]), and the progressive enlargement approach with honest times (see Barlow [9], Jeulin and Yor [35]). All related results extend immediately in a discrete time setting as highlighted by Blanchet-Scalliet, Jeanblanc and Romero in [14], most of them simply stemming from Doob's decomposition. The theory has enjoyed a significant revival since the end of the 90's because of its applications in financial mathematics and notably to insider trading problems (see Kohatsu-Higa [37]). One of the questions that arises is how to optimize insider's expected utility and quantify his/her benefit. This is studied in Pikovsky and Karatzas [45], Amendinger et al. through [6], [5], as well as in Grorud and Pontier [28]. In [7] and [8], Imkeller et al. discover a crucial link between this quantity and information theory by identifying it with the Shannon entropy of the extra information. In [33], Imkeller connects these notions to Malliavin calculus by expressing the information drift as the logarithmic Malliavin trace of a conditional density characterizing the insider's advantage. The existence or not of arbitrage is another core question. Many papers deal with arbitrage opportunities under initial or progressive filtration enlargement; we can cite in particular Aksamit et al. [3], [4], Acciaio, Fontana and Kardaras [1], Choulli et al. [2], as well as Chau, Runggaldier and Tankov [19] in the frame of an incomplete market. Most of the aforementioned works use a continuous-time setting. Among the (much fewer) studies in discrete-time, we can mention the works of Choulli and Deng [20] in a progressive enlargement setting, of Blanchet-Scalliet, Hillairet and Jiao [13] for a successive enrichment by a family of enlargement of filtrations, and of Burzoni, Frittelli and Maggis in [18] in the frame of uncertainty models (without a unique probability reference measure).
In a related stream of research, insider trading appears as a by-product of portfolio management issues (see for instance Biagini and Øksendal [10], [11]). An extensive literature on related topics (see the books of Shreve [54], Pascucci and Runggaldier [44]) is available in the most famous complete discrete-time market model, namely the Cox-Ross-Rubinstein (CRR for short), also called binomial model. In such model, all claims are attainable; Privault exhibits (see chapter 1 in Privault [46] or the book [47]) an explicit formula of the replicating strategy in terms of the Malliavin derivative (on the Rademacher space) from the application of Clark-Ocone formula. Research about portfolio management in the incomplete trinomial model is scarcer. We can nevertheless mention the books of Delbaen and Schachermayer [24] or Björefeldt et al. [12], the survey of Runggaldier [50], the works of Dai and Lyuu [21], Glonti et al. [27]. From a slightly different perspective, an hedger can aim at maximizing his/her expected utility from the terminal wealth for a given utility function. A very popular method is based on tools from functional analysis and the formulation of a dual problem; the reader can refer to the survey of Schachermayer [51], and the reference book of Delbaen and Schachermayer [24]. The same question of utility optimization has been also addressed in incomplete markets in a "classical" sense (Hu, Imkeller and Muller [31]) and, more recently, within other types of incompleteness such as that arising from friction (see Bouchard and Nutz [15], Neufeld and Sikic [40]) or from uncertainty (see Nutz [41], Rasonyi and Meireles [48], Oblój and Wiesel [42]).
The approach developed in the present paper is original in two respects. First, and up to our knowledge, the insider trading problem has not been investigated so far in the trinomial model. What's more, through the ternary model we introduce, we give a new and innovative representation of the trinomial model, viewed here as a volatility one. Second, our approach differs radically from what has been done before in this context (some portfolio management related topics are studied in the trinomial model in [24], [50]). We use the alternative representation by the ternary model in order to unleash the full force on the Malliavin calculus framework (developed in Halconruy [29]) for the first time for exclusively financial purpose. The following results stand as the main contributions of this paper: -Theorem 4.1 points out the two filtration enlargement key tools i.e., the so-called drift of information to deal with the preservation of semimartingales, and the set of martingale preserving measures to cope with the problem of martingale conservation. Both contributions are developed up in the framework of stochastic analysis for binomial marked point processes built in Halconruy [29]. Furthermore the drift of information is connected to Malliavin's calculus in the same vein as Imkeller [32], [33]; -Theorem 4.6 provides an Ocone-Karatzas type formula for replicable claims, what sounds as the analogue of Proposition 1.14.4 in Privault [46] but in the trinomial/ternary model; -Theorem 4.10 gives an explicit computation of the expected additional logarithmic utility of the insider.
This can be interpreted as the Shannon entropy of the extra information like in the continuous case (see Amendinger, Imkeller and Schweizer [7]); -Proposition 4.15 answers the question of the existence of arbitrage opportunities for the insider within this frame.
-Proposition 5.1 respectively, Proposition 5.3 provides the expression of the maximal logarithmic expected utility for the agent respectively, the insider, in a model where insider's influence is characterized through an anticipating equation in differences satisfied by the discounted price sequence. Proposition 5.5 deals with the insider's optimization problem in a model, widely inspired by Kohatsu-Higa and Sulem [38], the asset price dynamics is generated by a linear difference equation explicitly dependent on the insider's own strategy.
The paper is organized as follows. Section 2 is devoted to preliminaries including the presentation of the ternary model as well as reminders of the stochastic analysis and Malliavin's calculus for binomial marked point processes. Section 3 addresses the problem of enlargement of filtration in this frame. Section 4 deals with the application of these tools to address insider's problem in the ternary model and gathers most of the main results. Section 5 draws the outlines of a possible extension to market influenced by a large insider. We formalize our conclusions and open up possible perspectives in Section 6. Most proofs are postponed to Section 7.
Notation. The following notation is used throughout the paper. Any interval of R is [a, b], for real numbers a, b such that a b.
. . , m} for any n, m ∈ N 0 such that n m.
1} and for all n ∈ N, any n-tuple of X n can be denoted by bold letters; for instance, (t n , k n ) = (t 1 , Given a process (X t ) t∈N defined on the underlying probability space (Ω , F, P), we let ∆ X t = X t − X t−1 be the increment of X at time t and we set ∆ X 0 = X 0 . In the following, E designates the expectation taken under the reference probability measure P. When dealing with other probability measure Q on (Ω , F), the expectation with respect to Q will be denoted E Q .

Trinomial and ternary models
To begin with, we recall the main characteristics of the trinomial model underpinning our problem. This embodies a simple financial market modeled by two assets i.e. a couple of R + -valued processes (A t , S t ) t∈T , defined on the same filtered probability space (Ω , A , (F t ) t∈T , P) where (F t ) t∈T =: F is assumed to be generated by the canonical process, T := N 0 ∩ [0, T ] is the trading interval and T ∈ N is called the maturity. The market model thus defined will be labelled as (T, F, P, S). The riskless asset (A t ) t∈T is deterministic and is defined for some r ∈ R + (generally smaller than 1) and all t ∈ T by whereas the stock price which models the risky asset, is the F-adapted process (S T t ) t∈T with (deterministic) initial value S T 0 = 1 and such that for any t ∈ T * , We replace the trinomial model with a more computationally amenable one. At the very beginning, this formalism was in fact introduced to get around the impossibility (due to measurability issues) to state an Ocone-Karatzas formula from the Clark formula for independent random variables stated by Decreusefond and Halconruy ([23], Theorem 3.3). As suggested in the schema below, this surrogate model is based on a jump process and we name it ternary model.
Ternary model As a matter of fact, the family of random variables 1}-Bernoulli random variables of probability success p. They will stand for the direction (positive or negative) of the 1-high jumps of a process which construction follows. On the one hand, consider a binomial process (N t ) t∈T of intensity measure λ ∈ (0, 1) i.e., that can be written as N t = ∑ s∈T * 1 {T s t} whose t-th jump time is defined by T t = ∑ t s=1 ξ s , and where the inter-arrival variables {ξ t , t ∈ T * } are i.i.d. geometric random variables of parameter λ . On the other hand, let a family of i.i.d. {−1, 1}-Bernoulli variables {V t , t ∈ T * } that are independent of (N t ) t∈T , and for any t ∈ T * , set W t = V T t . Thus, the variables ∆ Y t stand for the increments of the process Y := (Y t ) t∈T * defined by Y 0 = 0, and for t ∈ T * , and which is a particular case of compound binomial process. The corresponding compensated compound process, denoted Y := (Y t ) t∈T , is defined by Y 0 = 0 and where p 1 := p, p −1 := 1 − p and λ := P({∆ N t = 1}) such that Y is a (P, F)-martingale. The key fact is that a good choice of the parameter couple (λ , p) makes the trinomial and ternary models equivalent in law. Indeed, by considering S 0 = S T 0 , some λ ∈ (0, 1), p T = λ p and The stock price of the ternary model is thus the F-adapted process (S t ) t∈T with (deterministic) initial value S 0 = 1 and such that for any t ∈ T * , where a and b are defined in (2.1). The sequence of discounted prices S := (S t ) t∈T is defined by S t = A −1 t S t (t ∈ T), where (A t ) t∈T is given by (2.1). Unless otherwise stated, the results for the insider problem will be established within the framework of the ternary model. All "expected results" will de facto hold in the trinomial model thanks to their equivalence in law, whereas the identities ensure a pathwise correspondence between the two models in stake.
Under this paradigm shift, the ternary model can be interpreted as a volatility model. Indeed, the parameter λ = P({∆ N t = 0}) ∈ (0, 1) can be viewed as the volatility of the model: The closer λ is to 0, the lower the probability that the stock price process changes and the lower the volatility. On the contrary, λ close to 1 means that the stock market process changes with a high probability, and in the extreme case λ = 1 the ternary model is no longer equivalent to the trinomial model but coincides with the Cox-Ross-Rubinstein (or binomial) model. For short, we may and shall write any probability measure M on (Ω , F) by Like the trinomial (see Runggaldier [50]), the ternary model stands for an incomplete market; the sequence of discounted prices (S t ) t∈T satisfies for t ∈ T * Consider a probability measure M equivalent to P and define for t ∈ T * , Then, (S t ) t∈T is a (M, F)-martingale provided the system is satisfied for all t ∈ T. In fact, there exist infinitely many solutions (m 0 t , m 1 t , m −1 t ) that form a simplex in R 3 . Noting that the coefficient in (P) are independent of t and F t , so do the solutions i.e., (m 0 The solutions form then a segment characterized by its extremal points i.e., the measures (independent of t) P 0 := (1, (r − a)/(b − a), (b − r)/(b − a)) and P 1 := (r/b, 1, 0), that are not equivalent to P but such that any convex combination P γ = γP 0 + (1 − γ)P 1 is (for γ ∈ (0, 1)). Furthermore, we can describe M F the set of (equivalent) F-martingale measures, that consists of probability measures (equivalent to P) with respect to which S is a F-martingale. In fact, there is a bijection between the set M F in our model, and its analogue in the trinomial one. Indeed, for T = 1, for a given measure P γ = γP 0 +(1−γ)P 1 = (λ γ , p γ , 1− p γ ) (γ ∈ (0, 1)), we can find a probability measure P γ,T which coincides with P γ : The set M F is the polyhedron which 2 T vertices are depicted by the extremal measure P j ( j ∈ {1, . . . , 2 T } =: J ) that be can be written as where (γ j s ) s∈T * ∈ {0, 1} T . An induction from the case T = 1 enables to show that the convex set of Fmartingale measures equivalent to P in our model, and the one existing in the trinomial are one-to-one.

Elements of stochastic analysis for marked binomial processes
Motivated by this paradigm shift, we provide here some results of stochastic analysis for marked binomial processes. They can be found in Halconruy [29] and are given in the particular case we are interested in here i.e., when the mark space E is reduced to the set {−1, 1}. In the underlying probability space (Ω , A , P) introduced earlier, we consider the simple measurable space (X, X ) where X := T * × E and T * is defined above. Let N X be the space of simple, integer-valued, finite measures on X and N X be the smallest σ -field of subsets of N X such that the mapping χ ∈ N X → χ(A) is measurable for all A ∈ X . We write η the underlying marked process associated to (Y t ) t∈T as the random element of N X such that where the families {V t , t ∈ T * } and {T t , t ∈ T} are defined in the previous subsection. By a slight abuse of notation, we shall write (t, k) ∈ η (respectively, (t, k) / ∈ η) in order to indicate that the point (t, k) ∈ X is (respectively, is not) charged by the random measure η. Note in particular that for all t ∈ T * , From now onwards, we may and will assume that A = σ (η) = F where F = (F t ) t∈T is the canonical filtration defined from η by (2.9) Let P η = P • η −1 be the image measure of P under η on the space (N X , N X ) i.e. the distribution of η; its compensator -the intensity of η -is the measure ν defined on X by Throughout, we denote by L 0 (Ω ) = L 0 (Ω , F) the class of real-valued measurable functions F on Ω .
Since F = σ (η), for any F ∈ L 0 (Ω ), there exists a real-valued measurable function f on N X such that X} is a family of measurable functions from N X × X to R and u is called the representative of u. By default, the representative of a random variable or a process will be noted by the corresponding gothic lowercase letter. For the purpose of notation, we shall write u = (u (t,k) ) (t,k)∈X .

Chaotic decomposition
Let {P j , j ∈ J } be the set of martingale measures equivalent to P and denote The following elaboration is rigorously the same as in Halconruy [29], and can be reproduced by taking E = {−1, 1} and P = P j for any given j ∈ J . The detailed proofs can be found in the cited paper, so that we will limit ourselves to providing the most significant elements of the construction for some fixed j ∈ J . Consider the families By its very definition, Y j is a (F, P j )-martingale. Besides, R j is the orthogonal (for the scalar product induced by P j ) family constructed from Z j via the Gram-Schmidt process. Contrary to the general framework investigated in Halconruy [29], the set X is here finite. So are all the random measures on X we consider, and there is no question of integrability: all functionals in stake are p-integrable (for any p ∈ N) on Ω with respect to P j . For any function f 0 defined on X, we set J j 0 ( f 0 ) = f 0 . For any n ∈ T * we denote respectively by F (X) n and F (X) •n , the space of functions defined on X n and the subspace of F (X) n composed of functions that are symmetric in their n variables i.e., such that for any permutation where " " denotes the first n − 1 variables of f n ((t 1 , k 1 ), . . . , (t n , k n )). Set H j 0 = R and for any n ∈ T * , let H j n be the subspace of L 0 (P) made of integrals of order n ∈ T * given by and called P j -chaos of order n. By replacing P by P j and each H n by H j n in Halconruy ([29], theorem 2.11) we can state the key representation L 0 (Ω ) = n∈T H j n . In other words, any marked binomial functional, that is any random variable of the form can be expanded in a unique way as (2.14) In the following we will refer to this expansion as the P j -chaotic decomposition of F.

Clark-Ocone formula
As a reminiscence of the Malliavin operator on the Poisson space, the add-one cost operator or Malliavin's derivative D is defined for any F ∈ L 0 (Ω ), (t, k) ∈ X by where the application π t : By rewriting Proposition 4.4 of Halconruy [29] with respect to the P j -decomposition, we get the analogue of the Clark-Ocone formula: for any F ∈ L 0 (Ω ), As a corollary, if (L t ) t∈T is a (F, P j )-martingale, for any (s,t) ∈ T 2 such that s < t,

Enlargement of filtration in a discrete setting
On the ternary model defined above, we introduce two agents with different levels of information; the first one, called insider, possesses from the beginning extra information whereas the second one, the ordinary agent, bases his/her investment decisions on the public flow. This difference translates mathematically into the introduction of two distinct filtrations: the ordinary agent information level corresponds to the initial filtration F (i.e. his/her knowledge at time t ∈ T is given by F t ) whereas the insider disposes at time t an information given by the σ -algebra G t defined via the initial enlargement where G is a F T -measurable random variable with values in a finite space (Γ , G ) that encodes the information overload enjoyed by the insider. His/her filtration is then denoted by G := (G t ) t∈T . The crucial point in order to make computations from insider's point of view, is the study of the preservation of (semi)martingales within this information enrichment. Throughout this section and with no loss of generality, we fix a measure P j ( j ∈ J ) in the non-void (see subsection 2.1) set of martingale measures M F . All forthcoming results hold for any j ∈ J . The enlarged market model where both the ordinary agent (with information flow given by F) and the insider (with information flow given by G) trade will be referred to as (T, F, G, P, S).

From F-martingales to G-semimartingales via Doob decomposition
In the continuous case, Jacod's condition indicates that the absolutely continuity of the conditional laws of G with respect to its law suffices for the preservation of semimartingales. In a discrete setting, no such assumption is required and any (P, F)-martingale is a (P, G)-semimartingale. Since the set Γ is finite, the conditional distributions of G for all t ∈ T • and the law of G are even equivalent. Indeed, any set C ∈ G is of the form C = c∈C {G = c} and, for any t ∈ T • , and has an expectation equal to 1. Note however that there is no equivalence on F T , since P({G ∈ ·}|F T ) = 1 {G∈·} is zero with positive probability (unless G is constant). Note that for any c ∈ Γ and j ∈ J , the process q c is a (P j , F)-martingale. Considering two (P, F)-martingales X and Z, we denote by ·, · P the angle bracket i.e., the F-adapted process such that The difficulties inherent in the preservation of semimartingales are directly lifted as a consequence of Doob's decomposition. Indeed it is clear (see Blanchet-Scalliet, Jeanblanc and Romero [14]) that any integrable process is a special semimartingale in any filtration with respect to which it is adapted. Furthermore, the hypotheses of section 2.2 in Blanchet-Scalliet et al [14] are fulfilled in our context; thus, for some j ∈ J and a given is a (P j , G)-martingale. Consequently any (P j , F)-martingale X j is a (P j , G)-special semimartingale, i.e., a G-adapted process which can be decomposed as X j = M j + V j where M j is a (P j , G)-martingale and V j is a G-predictable process. In particular, we can define the process Y For any t ∈ T • , j ∈ J , the σ -algebras F t and σ (G) are independent under the probability measure Q j t defined for any A t ∈ G t by 3. For any t ∈ T • , j ∈ J , the probability measure Q j t coincides with P j on (Ω , F t ) and with P on (Ω , σ (G)), so that for B t ∈ F t and C ∈ G , Remark 3.2 In some ways, our framework benefits from Jacod's condition at any time before expiry and the results are unsurprisingly inherited. In a slightly different context where insider's extra information flow is dynamically modeled through a successive initial enlargement of filtrations, Blanchet-Scalliet, Hillairet and Jiao propose in [13] a so-called successive density hypothesis to construct equivalent probability measures. They put their working density assumptions into perspective relative to Jacod's usual criterion; the very essence of this latter i.e., the study of the regularity of the conditional laws of G with respect to its law, seems an inescapable element to cope with an initial enlargement.
Identifying the set of measures with respect to which the sequence of discounted prices is a (·, G)-martingale is a crucial point in view of the evaluation of financial claims. In fact, for any t ∈ T • , j ∈ J , the measure Q j t thus defined is a martingale preserving measure, what can be justified by the following proposition.
. Moreover the set of (P j , F)-martingales and the set of (Q j t , F)-martingales, all on [[0,t]], are equal.
As evoked above, q G T is null (except if G is constant) with positive probability and the regular conditional laws of G are not equivalent to the law of G. In some way, the analogue of Assumption 2.1 in Amendinger [5] in our context is not satisfied at time T and the previous result does not hold on T.
4 Application to the insider's problem in the ternary model

Information drift and Malliavin derivative
We have seen in subsection 3.1 that martingales with respect to the initial filtration become semimartingales by moving to the enlarged one. This transfer is encoded by a particular process µ G,· , called the information drift and defined by (3.2). In the same vein as Imkeller [33], we can traduct its connection to the random variable G thanks to the Malliavin derivative D.
Theorem 4.1 Let j ∈ J . The information drift µ G, j defined in (3.2) for the P j -compensated compound process Y j (given in (2.11)) can be written as for any t ∈ T * , where the family {a j t,k, , (k, ) ∈ E 2 }, does not depend on t (since the variables ∆ Z (t,·) are identically distributed) and is defined for Remark 4.2 This result is the discrete analogue of the formula (17) in Imkeller [33]. Classical Malliavin's derivative (in the Wiener space) enjoys the chain rule, so that the formula exhibited by Imkeller elegantly reduces in the continuous case (with the corresponding notations) to µ G t = ∇ t log(p t (·, c))| c=G .

Portfolio optimization an additional expected utility of the insider
We consider an economic agent and an insider both disposing of x ∈ R * + euros at date t = 0 (initial budget constraint), for whom we want to determine the maximal expected logarithmic utility from terminal wealth. Let H be some filtration on (Ω , P), that may and shall be replaced by F, F or G later on. As a reminder, the value of a H-portfolio at time t ∈ T is given by the random variable where the so-called H-strategy ψ = (α t , ϕ t ) t∈T is a couple of H-predictable processes modelling respectively the amounts of riskless and risky assets held in the portfolio. A H-strategy ψ = (α, ϕ) is said to be self-financed if it verifies the condition: for any t ∈ T • . The discounted value of the H-portfolio at time t ∈ T is given by V t (ψ) = V t (ψ)/A t . Before going on, let us recall two straightforward and well-known facts. First, for any given H-predictable process ϕ there exists a unique H-predictable process α such that ψ = (α, ϕ) is a self-financing process (see for instance Lamberton and Lapeyre [39], proposition 1.1.3). On the other hand, the quantity of riskless asset indicated by the process (α t ) t∈T does not change the discounted value of the portfolio namely (V t (ψ)) t∈T since the discounted version of the asset (A t ) t∈T is deterministic constant equal to 1. Consequently, the knowledge of the initial investment α 0 = x (for x ∈ R * + ) and the process of risky asset amount (ϕ t ) t∈T is enough to compute the (discounted) value of the portfolio. Then, with a slight abuse of notation, we may and often shall identify the strategy ψ = (α, ϕ) with (α 0 , ϕ) = (x, ϕ). We designate by V t (ψ) (respectively, V t (ψ)) its value (respectively, discounted value) at time t ∈ T. A nonnegative H T -measurable random variable F (called claim) is replicable or reachable if there exists a H-predictable self-financed strategy ψ = (α 0 , ϕ) which corresponding portfolio value satisfies α 0 = V 0 (ψ) > 0 and V T (ψ) = F. Let S H (x) be the class of H-eligible strategies of initial value x i.e., Since T is bounded, E[log(V t (ψ))] < ∞ for any t ∈ T, that confirms that all elements of S H (x) are eligible.

Portfolio optimization in the ternary model
In this subsection, we are led to consider the optimization problem at any time t ∈ T * from the agent's point of view and from the insider's where u is a utility function, strictly increasing and strictly concave on R or R * + . Throughout, we consider u = log.

Link with portfolio optimization in the binomial model
A very substantiated expression and comprehensive look at arbitrage-related issues can be found in the book of Delbaen and Schachermayer [24]; our results obtained in the ternary model are linked to theirs found in the Cox-Ross-Rubinstein. To that end, let us first introduce, the "CRR-embedded" processes (X B t ) t∈T and (S B t ) t∈T by S B 0 = 1 and with λ B := P({(t, ±1) ∈ η}) = 1 and p B := P({(t, 1) ∈ η}) = p, which means for all t ∈ T * , P({X B t = 0}) = 0, and then X B t = ±1 P-almost surely. The law of S B t is given by P B = (1, p B , 1 − p B ) = (1, p, 1 − p) so that (S B t ) t∈T is identically distributed to the CRR-stock price process and stands for its "embedding" into our framework.
Consider an initial investment x ∈ R * + . Until the end of the subsection, let T = 1. As proved in Proposition 3.3.2 of Delbaen and Schachermayer [24], there exists a unique optimizing strategy ψ B such that is the terminal value of the discounted portfolio of initial value x and defined from the CRR-stock price i.e. (since T . Besides, any discounted portfolio based on the stock price of the ternary model, of initial value x and F-eligible strategy ψ has a terminal value V T (ψ) = x + ϕ T ∆ S T , and we have , so that, as heuristically explained in Delbaen and Schachermayer [24], the optimizing strategy obtained for the trinomial model (and then for the ternary model by the pathwise correspondence) coincides with the one computed in the binary model. We finally get (4.7) Unsurprisingly, Φ(x) = Φ B (x) for λ = 1, which reflects that the CRR and the ternary model coincide in that case.

Agent's portfolio optimization
Let us start with the case T = 1. As suggested in the previous subsection, we can deduce the solution of the optimization problem (4.4) from that of (4.6) where (S B t ) t∈T is the CRR-price sequence embedded in the ternary model via (S B ). Our result directly lies on martingale and duality methods usually used to deal with utility optimization in incomplete markets (see Karatzas et al. [36]). A simple translation of the results of Delbaen and Schachermayer (see [24], example 3.3.2) into our frame leads to Φ B (x) = log((1 + r)x) − E[log(P 0 /P B )] =: log((1 + r)x) + c B since P 0 = (1, (r − a)/(b − a), (b − r)/(b − a)) =: (1, p 0 , 1 − p 0 ) stands for the unique CRR risk-neutral measure written in the pattern of the ternary model. Then the solution of (4.4) is given by Φ F (x) =: log((1 + r)x) + c F where, by using (4.7), where P F 1 = (1 − λ , λ p 0 , λ (1 − p 0 )). As proved earlier, the optimal strategy obtained for the ternary model is that found in the binomial frame. The case of logarithmic utility is completely solved through in Delbaen and Schachermayer ([24], example 3.3.2) and we will only give a brief proof. The results related to agent's portfolio optimization are summed up in the following proposition. Proposition 4.3 (Agent's portfolio optimization, T = 1) For T = 1, the maximal expected logarithmic utility up to expiry for the agent with initial wealth x is given by where the density of P F 1 with respect to P is defined by This is reached for a unique F-eligible strategy Proof The identity (4.9) is obtained by using (4.8) and inserting Φ B (x) = log((1 + r)x) + E[log(dP/d P F 1 )] into (4.7). That directly leads to V F x,T = x · (dP/d P F 1 ) for the terminal value of the discounted optimal portfolio. The optimal trading strategy is obtained by solving the equation As suggested in Delbaen and Schachermayer ([24], example 3.3.5), the optimization problem can be solved at expiry by extending the previous results via the principle of dynamic programming. This procedure sounds as a multiplicative concatenation of the one-step model which corresponds to the case T = 1. The choice of this procedure can be justified by the independence of the increments of the underlying jump process (N t ) t∈T . Let us consider some T ∈ N such that T 2. Then, we define on F T the measure P G such that Proposition 4.4 (Agent's portfolio optimization at time t ∈ T * , T 2) For some T ∈ N, T 2, the maximal expected logarithmic utility up to time t ∈ T * for the agent with initial wealth x is given by where the density of P F with respect to P is defined by (4.10). This is reached for a unique F-eligible Remark 4.5 Stated for the log function, an analogue result (by making the necessary changes) can be stated for any utility function smooth enough to accommodate a duality approach. It is worth noting that Rasonyi and Stettner propose in [49] a discrete-time alternative and directly probabilistic method to state the existence of optimal strategies for maximizing non smooth utility functions (that satisfy a mild "asymptotic elasticity" condition) and for possibly unbounded price processes.
The following theorem provides an expression of the hedging strategy for replicable claims and in terms of Malliavin derivative.
The application of the latter theorem to F = (1 + r) T V F x,T gives an expression of the optimizing strategy ψ = (x, ϕ) for the agent: ϕ 0 = 0, and for any t ∈ T * , Note that coefficients e · and ρ do not depend on t since the ∆ S t are i.i.d. under P. We can check that x and the initial budget constraint is respected. The expression (4.11) of the amount of risky asset that is needed to replicate F is the transposition into our frame of the Ocone-Karatzas formula stated in the continuous case by Ocone and Karatzas [43], and in the CRR model by Privault ([46], proposition 1.14.4). The two expressions are closely resembling and differ only in the different expression of the gradient in each context.

Remark 4.7
The previous result holds for replicable claims. In fact, in incomplete markets, there exist no-redundant claims that carry an intrinsic risk; thus, optimizing a portfolio means minimizing this risk as introduced by Föllmer and Sondermann in [26]. In that case the minimizing-risk strategy can be constructed using the Kunita-Watanabe projection technique with respect to the initial reference measure (in the martingale case) or to the so-called minimal martingale measure (see Föllmer and Schweizer [25], Schweizer [52]). In a discrete setting, in order to approximate any claim by the total gain from trade (given in terms of a stochastic integral with respect to the stock process) in L 2 , Schweizer introduces in [53] the variance optimal signed martingale measure. This measure coincides with the (discrete version of) minimal martingale measure both in the martingale and deterministic mean-variance tradeoff cases. In the same spirit, portfolio optimization in the trinomial model (still superseded by the ternary model) is handled in Halconruy ( [29], theorem 5. 16), what provides the loss quadratic minimizing strategy for non-replicable claims. By the way, we retrieve here the expression of the minimizing strategy of theorem 5.16 by letting in (4.11)

Insider's portfolio optimization
Let us first tackle the problem (4.5) for some t ∈ T * ,• and x ∈ R * + . Insider's portfolio optimization can be performed by repeating mutatis mutandis the solution of agent's problem. This means replacing the underlying filtration F by G, and identifying the set of G-eligible strategies. As seen in subsection 3.2, for all j ∈ J , any (P j , F)-martingale is a (Q j t , G)-martingale on [[0,t]] where Q j t is defined by (3.3). In particular, the discounted price process where ψ G := (x, ϕ G ) is a G-eligible strategy. Moreover, for T = 1, the set of G-martingale measures consists of the convex combinations of Q G,0 and Q G,1 . In particular we have We also define Q G 1 , the "analogue" of P F 1 for the insider i.e., the probability measure such that (4.14) The transposition of Proposition 4.4 into insider's paradigm gives the following result. The maximal expected logarithmic utility up to time t ∈ T * ,• for the insider with initial wealth x is given by where (d Q G t /dP)| G t is given by (4.14). This is reached for a unique G-eligible strategy of discounted value Proof The case T = 1 can be treated in the same fashion as for agent's problem, by substituting Q G 1 to P F 1 . For T 2 and any t ∈ T * ,• , (∆ S s ) s∈[[0,t]] is a (Q j t , G)-martingale. Using the same arguments as in agent's problem, we can construct Q G t from the probability measure Q G 1 as suggested by (4.14). Then the result follows simply by replacing F and P F |F t respectively by G and Q G t in Proposition 4.4. The result follows.
Since there is absolutely no reason why a (P, F)-martingale should be a (Q j T , G)-martingale on T, we need to address the problem at the deadline T with another kind of argument. For some x ∈ R * + , let Φ G 1 (x) be the solution of (4.5) when T = 1. For all s,t ∈ T such that s < t, we define F s,t = (F r ) s+1 r t .  For any x ∈ R * + define Φ G T (x) by considering (4.5) at the deadline T . Then,

Additional expected utility of the insider in the ternary model
Insider's additional expected logarithmic utility up to time t ∈ T is defined by Given two probability measures defined on the same mesurable space (Ω , H) where H = (H t ) t∈T , D H (P||Q) designates the relative entropy of P with respect to Q on H and is defined by Define also for any t ∈ T • , Ent(G) and Ent(G | H t ) by that respectively stand for the entropy and the conditional entropy with respect to the filtration H of the random variable G. Here stands our third main result.
Theorem 4.10 The insider's additional expected logarithmic utility up to time t ∈ T • is given by Remark 4.11 Thus, we get a result akin to Theorem 4.1 in Amendinger, Imkeller and Schweizer [7] in the Black-Scholes model (and for a discrete random variable G) i.e., the additional expected logarithmic utility of the insider can be expressed in terms of relative entropy. The measure P appearing in [7] is replaced in our frame by the equivalent measure P F under which the price process is F-martingale. Our result can also be compared to Theorem 5.12 in Ankirchner et al. [8]; under an initial enlargement (continuous) setting, they provide an expression of the additional utility of the insider in terms of the relative difference of the enlarged filtration with respect to the initial one; in fact, that also coincides with the Shannon entropy between (with the corresponding notations) G and Id F T where Id A is the application ω ∈ (Ω , A ) → ω ∈ (Ω , F T ) defined for any sub-algebra A ⊂ F T . Moreover, in the continuous case, the result still holds at the deadline T by taking the limit when t goes to T . Here, we need to appeal to Proposition 4.9 to state the following result.

Corollary 4.12
The insider's additional expected logarithmic utility at expiry is given by Proof By definition, from Proposition 4.9 and where the identity Φ F 1 (x F ) = Φ F T (x) can be easily stated by adapting this latter to agent's optimization problem. The result follows.

Arbitrages
An important question that arises is whether an arbitrage is produced by the enlargement of insider's filtration. Roughly speaking, an investor has an arbitrage opportunity also called free lunch if he or she can hope to make profit without taking some risk. An overview of the question of arbitrage and its complementary notion of No Free Lunch (with Vanishing Risk) -NFL(VR) for short -in a discrete setting can be found in Dalang, Morton and Willinger [22]. Several equivalent definitions coexist and we recall here theirs. Definition 4.13 (Arbitrage, see Dalang et al. [22]) In a market model (T, F, P, S), an arbitrage opportunity is a F-predictable self-financed trading strategy ψ such that V 0 (ψ) = 0, V T (ψ) 0 P-a.s. and V T (ψ) > 0 with positive probability. The market model (T, F, P, S) has no-arbitrage or is said to be arbitrage-free if it contains no arbitrage opportunities i.e., if for all F-predictable self-financed trading strategies ψ with V 0 (ψ) = 0 and V T (ψ) 0 P-a.s., we have V T (ψ) = 0 almost surely. Furthermore, it can be shown (see again Dalang et al. [22]) that (T, F, P, S) is arbitrage-free if there exists a positive (P, F)-martingale M = (M t ) t∈T , with M 0 = 1 such that SM is a (P, F)-martingale. From this starting point, Choulli and Deng define and give in [20] some conditions for the stability of the non-arbitrage condition in a discrete-time model with a progressive enlargement. Blanchet-Scalliet et al. design the notion of model free in the setting of a discrete-time model (T, F, G, P, ·) with an enlargement of filtrations, but without specifying the price process. We adapt it slightly to support our coming result. Definition 4.14 Given two filtrations F and G such that F ⊂ G and I a subset of T, the enlarged model (T, F, G, P, ·) is arbitrage-free on the time horizon I if there exists a positive (P, G)-martingale M = (M t ) t∈I , called (I, F, G, P, ·)-deflator, such that for any (P, F)-martingale X, (X t M t ) t∈I is a (P, G)-martingale.
In particular, the model (T, F, G, P, S) is arbitrage-free on the time horizon I, if there exists a positive (P, G)-martingale M = (M t ) t∈I , called (I, F, G, P, S)-deflator, such that (S t M t ) t∈I is a (P, G)-martingale.

Proposition 4.15
Let j ∈ J . The model (T, F, G, P j , ·) is not arbitrage-free on the time horizon T. Assume that S T is σ (G)-measurable. The model (T, F, G, P j , S) is not arbitrage-free on time horizon T but it is on time horizon T • and u G, j is a (T • , F, G, P j , S)-deflator.
The assumption S T ∈ σ (G) is natural; indeed, the case G = 1 {S T ∈[[a,b]]} (a, b ∈ R + , a b) is a classic example and a similar one will be discussed in the next subsection.

Remark 4.16
This result can be put in perspective with (older) analogues in other contexts. In the continuous case, Amendinger proves in [5] that under Jacod's hypothesis, insider's model is arbitrage free. Our result is therefore quite comparable to this, since Jacod's criterion is de facto fulfilled on T • . In a discrete setting, Choulli and Deng set up in [20] the necessary and equivalent conditions on a not public information G -incorporated in the market through a progressive enlargement of filtration -to preserve the non-arbitrage condition in a market. No surprisingly, the stability of non-arbitrage is ensured within mild conditions on any interval [[0, τ]] (which is comparable to the case I = T • here) where τ is a stopping time. The preservation of non-arbitrage condition on [[τ, +∞[[ is provided in the case where τ is an honest time.

Computations in the case G = 1 {S T S 0 }
In that case, the insider knows from the start whether it is worth investing in the risky asset; this one appears in fact riskless for him or her since he or she knows the outcome of G and then whether S T is greater or less than S 0 . One of two things must be true: either G = 1 namely the discounted stock price does not increase and it is better to invest the entire capital in the asset A, or G = 0 and investing in S is more profitable. As a reminder the budget constraint can be written as and be transposed at time T = 1 to α G 1 + ϕ G 1 S 0 = x, by readjusting the portfolio under the self-financing condition. We get clearly that the G-eligible optimal strategy ψ = ( α, ϕ) is given by Besides, the maximal expected logarithmic utility of the agent is provided at any time t ∈ T by (4.9). That of the insider is given in Proposition 4.8 where Q G |G t is defined by (4.14) and (q G t ) t∈T • is defined by where we have used that the variable S T /S t has the same law as S T −t . Moreover we can write S t = S 0 ) follows a trinomial law of parameters t (number of trials), P(ζ 1 1 ) = λ p and P(ζ 1 −1 ) = λ (1 − p). Let n be the maximal integer such that (1 + b) n (1 + r) T . Last, we get an explicit expression of the insider's additional expected logarithmic utility at expiry.

Proposition 4.17
In the case G = 1 {S T S 0 } , the additional expected logarithmic utility of a insider having initially x ∈ R * + is given at expiry T by and where we have defined for t ∈ {T − 1, T }, χ t = n − }, and the couple (n + , n − ) ∈ N 2 0 satisfies n + + n − T − 1 and n + − n − n + 1. This explicit example highlights that insider's additional expected logarithmic utility depends on the parameter λ . As could be anticipated, the bigger λ ∈ (0, 1), the more volatile the model and the greater the benefit from the surplus information.

Possible extension to a simple insider influenced market model
In this last part we address insider trading from another point of view, widely inspired by the frameworks of Kohatsu-Higa et al. [30], [38]. This section is not intended to provide a general and conclusive result, but merely to introduce two simple insider influenced models and to open perspectives for a possible future work on them. In fact, we no longer postulate that the asset price process is adapted to the ordinary (and small) agent's filtration but that it depends on the insider who is assumed to be large enough to get a leverage on the market. Each forthcoming subsection is devoted to the study of an insider influenced market model: in the first one this asset price dynamics is described by an anticipating equation in differences whereas it involves explicitly the insider's strategy in the second one.

Anticipating equation in differences
First, we study the insider's and small ordinary agent's behaviours in a market where the influence of the insider is incorporated in the definition of the underlying "canonical" random variables ∆ Z ·,· which are no longer F-adapted but G-adapted. The dynamics of the discounted price satisfies the equation in differences: for any t ∈ T * , where the family {a j t,k, , t ∈ T * , (k, ) ∈ E 2 } is given by (4.1) so that Y For all G-predictable processes ϕ, we define in the influenced market (IM) model, V IM t (ψ) (respectively, V IM t (ψ)) the discounted value (respectively, the value) at time t ∈ T of the portfolio of strategy ψ = (x, ϕ) i.e.
We consider for any t ∈ T * the optimization problem from the agent's point of view and from the insider's Proposition 5.1 Within the influenced insider model described by (5.1), the maximal expected logarithmic utility up to time t ∈ T * for the ordinary agent with initial wealth x is given by This is reached for a unique strategy ψ IM,F = (x, ϕ IM,F ) where ϕ IM,F ≡ 0.

Remark 5.2
In this anticipating paradigm where the insider becomes the reference, the ordinary agent has a restrictive access to the information. As a result, it becomes -in average -unsafer for him/her to invest in the risky asset on which he or she has only partial information, and the optimal strategy is to invest all the initial wealth in the risk-free asset.
In the influenced market model, the benchmark filtration is that of insider. His/her behaviour is then comparable to the ordinary agent's, and we can deduce the following result.

Proposition 5.3
Within the influenced insider model described by (5.1), the maximal expected logarithmic utility up to time t ∈ T * for the insider with initial wealth x is given by where P G is the measure defined on G T such that This is reached for a unique strategy which discounted value at time t is V G Proof Within this framework, the sequence (S t ) t∈T is a (P j , G)-martingale for all j ∈ J . It suffices to apply Proposition 4.8 by replacing F by G.

Influential insider's strategy
Let θ be some G-predictable process that will depict the amount of risky asset in insider's portfolio. In this part, the influence of the insider is incorporated in the price process dynamics via θ . For any t ∈ T * , where c ∈ R * + and for all t ∈ T * h(t, ·) is a positive function. We can interpret these parameters as follows. As expected, the bigger c is, the bigger the demand and the higher the price of the risky asset. The term ∑ t s=1 h(t, s)θ s compensates the influence of the insider by considering the variations of his/her strategy with respect to past time averages (weighted through h) of the same strategy. The random variables ∆ Z (t,·) are defined as in (2.10) and then are F t -measurable. Let S IM G (x) be the class of influenced admissible strategies for the influential insider with initial wealth x, defined by where, for (x, θ ) ∈ S IM G (x), and some κ ∈ (0, 1), Up to now, we assume that the insider's strategy (x, θ ) is in S IM G (x). In the same vein as Hata and Kohatsu-Higa we consider the maximization problem that measures a change in the model structure with respect to the perturbation ζ . The case c > 0, h ≡ 0 means the insider has a long term effect on the stock price. Indeed, the bigger T , the bigger Θ θ ,ζ . The case where c > 0 and h is a positive function corresponds to an insider with medium term effects on price dynamics. As a matter of fact, a relevant determination of h enables to compensate the effects of the insider's strategy and to weaken the impact of change of his/her portfolio at the beginning of the trading interval.

Conclusions and perspectives
In this paper we have provided the discrete counterpart of three continuous-time results of insider trading and portfolio management, as to say we have obtained: (i) an expression of the information drift that encapsulates the insider's extra knowledge via the Malliavin derivative of the conditional laws of the random variable G with respect to the initial filtration, (ii) the statement of an Ocone-Karatzas formula for replicable claims, (iii) the resolution of the insider's expected utility maximization problem and the connection of his/her advantage to information theory via Shannon entropy. We have also stated (iv) the lack of arbitrage opportunities for the insider and the existence of a deflator on the time horizon T • and (v) the design of two simple models of an insider influenced market and the interpretation of a small ordinary agent's behaviour in one of these new paradigms. All these outcomes are derivative of a change of model, where a marked binomial process is substituted for the underlying sequence of i.i.d. random variables (in the trinomial model), so that the results are established at a lower cost by simply using the stochastic analysis tools provided in this framework. Besides, all the instruments are still available when the mark space E is countable, so that our statements should naturally extend to any multinomial market model without any additional assumption.
In the present paper we have restricted ourselves to the study of the stability of non-arbitrage condition by moving to insider's paradigm in the "classical" sense (using the definition of Dalang, Morton and Willinger [22]). It would be interesting to confront our formalism with other types of incompleteness such as friction (investigated by Bouchard and Nutz [15], Neufeld and Sikic [40]) or with uncertainty (see Nutz [41], Rasonyi and Meireles [48], Obloj and Wiesel [42]). Except for section 5, our results hold under the assumption that the insider is "small" enough so that his/her trading decisions do not affect the long/medium term evolution of prices in the model. In section 5, we have investigated two simple influenced market models and have given in particular the agent's point of view in a market impacted by the insider's decisions. In particular, another question arises as to whether there is a partial equilibrium in the sense defined by Hata and Kohatsu [30]; it could be investigated in a further work.

Proofs of Section 3
Proof of Proposition 3.1 The proof follows closely that of Proposition 2.3 in Amendinger, Imkeller and Schweizer [7]. For any t ∈ T • , B t ∈ F t , C ∈ G , and some j ∈ J , where we have used, since for any (t, c) ∈ T • × Γ the random variable q c t is not null P-a.s., that: This yields by the definition of Q j t given in the theorem, Taking B t = Ω , and then C = Γ provides and enables to establish 2. and 3. Let s ∈ [[0,t − 1]], B s ∈ F s and A s = B s ∩ {G ∈ C} an element of G s . Then, identical computations as in (7.1) but by conditioning with respect to F s (in the first equality) lead to so that the process u G, j = L j /q G is a (P, G)-martingale on T • . It results that 1/q G is a (P j , G)-martingale on T • by noting that where we have used that the σ -algebras F s ⊂ F t and σ (G) are independent under Q j t in the first and last lines, that P j coincides with Q j t on F t in the second one, and that (M s ) 1 s t is a (P j , F)-martingale in the third one. Then, (M s ) 1 s t is a (P j , G)-martingale on [[0,t]]. Since P j = Q j t on (Ω , F t ), the sets of (P j , F)and (Q j t , F)-martingales on [[0,t]] are equal. The proof is complete.

Proofs of Section 4
Proof of Theorem 4.1 Fix j ∈ J and consider the process µ G, j defined in (3.2) by taking X = Y. The proof directly derives from the Clark-Ocone formula (2.18) applied to all (P j , F)-martingales q c with c ∈ Γ . Taking s = t − 1 provides As stated in Lemma 1.4 of Blanchet et al. [14], for two F-adapted processes U and K, U, K P 0 = 0 and ∆ U, for all t ∈ T * . Then we get, for any c ∈ Γ , where we have got the second line by conditioning with respect to F t−1 and by defining the family a j t,1,1 = λ j t p j t (1 − λ j t p j t ), a j t,1,−1 = 0, a j t,−1,1 = (λ j t ) 2 p j t (1 − p j t ) and a j t,−1,−1 = Hence the result.
Proof of Proposition 4. 4 We define for all s,t ∈ T such that s < t, F s,t = (F r ) s+1 r t . The expression of Φ F T (x) can be deduced from the identity Θ F t (x) = Φ F T −t (x), together with the solution of the following induction system where the supremum is taken over the strategies ψ t = (x, ϕ t ) ∈ S F t−1,t (x) and ϕ t ∈ R. For t = T − 1, since the ∆ S t are independent and identically distributed, E [log (x + ϕ ∆ S 1 )] , and so on by downward induction. As Θ F T −1 (x) = log(x(1 + r)) − E[log[(d P F /dP)|F T −1,T ]], the iteration of (4.9) provides so that by letting t = T − s with s ∈ T * , Moreover, holds for any s ∈ T * so that the maximal expected utility and portfolio value at expiry are obtained by taking s = T .
since by definition (5.4), we have (dP/d P F )|F t = (dP/d P F )|G t . On the other hand, using the definition (4.14) of Q G t , we also get Moreover, since Γ is finite, where we get the second equality by conditioning on F t . Hence the result.
Proof of Proposition 4.15 Let j ∈ J . To prove that (T, F, G, P j ) is not arbitrage-free on T, we proceed as in Blanchet-Scalliet et al. (see [14], lemma 2.3). For any t ∈ T, let U t = E [G|F t ]. If a (T, F, G, P j )-deflator M exists, the process UM must be a (P, G)-martingale and then U t M t = E [U T M T |G t ]. Since G ∈ G t for all t ∈ T and M is a (P, G)-martingale, we have U T ∈ G t and E [U T M T |G t ] = U T M t that leads by taking t = 0 to U T M 0 = U 0 M 0 and contradicts U T = G / ∈ F 0 . If S T ∈ σ (G), we can prove in a similar way that (T, F, G, P j , S) is not arbitrage-free on T by noting that S T ∈ G t for all t ∈ T. Let us show that for a given j ∈ J , u G, j defines a (T • , F, G, P j , S)-deflator on I = T • . For any t ∈ T • , E[u G, j ∆ S t |G t−1 ] = E Q j We have also used that S 0 = 1, the definition (2.5) of S and got the third equality thanks to the i.i.d. where the last line is a consequence of (7.2). Moreover, the application ψ ∈ S F (x) → E[log(V IM T (ψ))] is strictly concave as the composition of function log and linear applications, that ensures the uniqueness of maximizing strategy. It is easy to see that (x, ϕ * ) with ϕ * ≡ 0 reaches the maximum. Since all variables are i.i.d. we can extend by dynamic programming the result to any deadline T ∈ N, T 2. so that L((1 − κ)θ 0 + κθ 1 ) (1 − κ)L(θ 0 ) + κL(θ 1 ) and the application (x, θ ) ∈ S IM G (x) → L(θ ) is strictly concave. Using the same arguments as Kohatsu-Higa and Sulem [38], we can show that for all (x, θ ), (x, ξ ) ∈ S IM G (x) and ε ∈ (0, 1), Letting then 1 + ε = 1/(1 − ε), we have that, combining with (7.5) gives Then for all (x, θ ), (x, ξ ) ∈ S IM G (x), we have L(ξ ) − L(θ ) D ξ L(θ ) − D θ L(θ ). Applying this to θ = θ * together with (7.4) gives L(ξ ) − L(θ * ) 0, for all (x, ξ ) ∈ S IM G (x) and proves that θ * is optimal. The expression of Ξ G (T ) follows by incorporating (5.7) in (5.6). The proof is complete.