Optimal trade execution in an order book model with stochastic liquidity parameters
OOptimal trade execution in an orderbook model with stochastic liquidityparameters
Julia Ackermann ∗ Thomas Kruse † Mikhail Urusov ‡ June 11, 2020
We analyze an optimal trade execution problem in a financial market withstochastic liquidity. To this end we set up a limit order book model in whichboth order book depth and resilience evolve randomly in time. Trading is al-lowed in both directions and at discrete points in time. We derive an explicitrecursion that, under certain structural assumptions, characterizes minimalexecution costs. We also discuss several qualitative aspects of optimal strate-gies, such as existence of profitable round trips or closing the position in onego, and compare our findings with the literature.
Keywords: optimal trade execution; limit order book; stochastic orderbook depth; stochastic resilience; discrete-time stochastic optimal control;long time horizon limit; profitable round trip; premature closure.
Primary: 91G10; 93E20. Secondary: 60G99.
Introduction
Market liquidity describes the extent to which buying (resp. selling) an asset movesthe price against the buyer (resp. seller). In an illiquid financial market large ordershave a substantial adverse effect on the realized prices. Typically, this effect is notconstant over time. Temporal variations of liquidity are partly driven by deterministictrends such as intra-day patterns. In addition, there exist random changes in liquidity ∗ Institute of Mathematics, University of Gießen, Arndtstr. 2, 35392 Gießen, Germany.
Email: [email protected],
Phone: +49 (0)641 9932113. † Institute of Mathematics, University of Gießen, Arndtstr. 2, 35392 Gießen, Germany.
Email: [email protected],
Phone: +49 (0)641 9932102. ‡ Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany.
Email: [email protected],
Phone: +49 (0)201 1837428. a r X i v : . [ q -f i n . T R ] J un uch as liquidity shocks that superimpose the deterministic evolution. To benefit fromtimes when trading is cheap, institutional investors continuously monitor the availableliquidity and schedule their order flow accordingly. The scientific literature on optimaltrade execution problems deals with the optimization of trading schedules, when aninvestor faces the task of closing a position in an illiquid market. Incorporating randomfluctuations of liquidity into models of optimal trade execution constitutes a highly activefield of research (see, e.g., [1, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 20, 22, 24, 26] andreferences therein).In this work we analyze a trade execution problem in a financial market model withlinear stochastic price impact and stochastic resilience. To be more specific, we considera block-shaped limit order book, where liquidity is uniformly distributed to the left andto the right of the mid-price. To account for stochastic liquidity, the depth of the orderbook is allowed to vary randomly in time. At initial time the investor observes thecurrent order book depth /γ > but has no precise knowledge about the order bookdepth at future times (only a probabilistic assessment). If the investor executes a trade of size ξ ∈ R at time , she incurs costs of size γ ξ . Moreover, the trade of size ξ shifts the mid-price of the order book by γ ξ . Observe that this deviation is positiveif and only if ξ > , i.e., if ξ is a buy order. In the period from time to the nexttrading time this deviation changes from γ ξ to D − = β γ ξ , where β > is apositive stochastic factor (unknown to the investor at time ). The factor β describesthe resilience of the order book: if β is close to the order book nearly fully recoversfrom the trade ξ , whereas if β is close to the impact of ξ persists. We highlighthere that we do not exclude the case, where the event { β > } has positive probability,which would reflect a possibility of self-exciting behavior of the market impact. At time the value of β is disclosed to the investor. Moreover, she observes the updated orderbook depth /γ > . Based on this information the investor executes a trade of size ξ which generates costs ( D − + γ ξ ) ξ and moves the deviation to D − + γ ξ . Bycontinuing this sequence of operations to arbitrary trading times k ∈ N we thus obtainour financial market model with stochastic price impact (described by a positive process γ = ( γ k ) k ∈ N ) and stochastic resilience (described by a positive process β = ( β k ) k ∈ N ).In this financial market we consider an investor who has to close a financial positionof size x ∈ R up to a given time N ∈ N . We assume that the investor is risk-neutral andaims at minimizing the overall trading costs. Apart from some technical integrabilityconditions we do not a priori impose any restrictions on trading strategies of the investor.In particular, even if the task is to sell a certain amount of assets (i.e., x > ), we allowfor trading strategies where the investor buys assets at some points in time.The above description of the model highlights that our setting is a certain discrete-timeformulation within the class of limit order book models, where the liquidity parametersare stochastic (i.e., both the price impact and the resilience are positive random pro-cesses). The approach to mathematically model liquidity via order book considerationswas initiated in [3], [4], [23] and [25]. Limit order book models with deterministicallytime-varying liquidity are studied in [2], [10] and [13], while stochastic liquidity is dis- We allow for both buy ( ξ ≥ ) and sell ( ξ ≤ ) orders. β and γ are random processes, while they aredeterministic functions of time in [2], [10] and [13].(b) In [10], [13] and [14], execution strategies are constrained in one direction, whiletrading in both directions is allowed in the present paper and in [2].(c) In [2], [10], [13] and [14], the resilience process (or function) β is assumed to be (0 , -valued, while we only require it to be positive in the present paper.In our setting we encounter several new qualitative effects, which are briefly mentionedbelow and discussed in more detail in the main body of the paper. Moreover, for eachof these effects, we identify its reason by constructing pertinent examples.We also mention [1], which is a continuous-time counterpart of our present paper.In this connection it is worth noting that our results do not follow from the results in[1], but both papers rather concentrate on studying different questions: e.g., [1] doesnot study the qualitative effects mentioned in the previous paragraph (and discussedbelow); instead we need to work with a challenging quadratic BSDE in [1] and extendthe continuous-time problem to incorporate execution strategies of infinite variation. Inparticular, some of the results of the present paper are required in [1] to derive, e.g., theappropriate problem formulation and the mentioned quadratic BSDE as continuous-timelimits of the corresponding discrete-time objects.In Theorem 2.1 we show that the optimal trading strategies and the minimal expectedtrading costs are characterized by a single stochastic process Y = ( Y n ) n ∈ Z ∩ ( −∞ ,N ] whichis defined via a backward recursion. We prove Theorem 2.1 by means of dynamic pro-gramming. To this end we put the trade execution problem into a dynamic frameworkand allow for arbitrary initial times n ∈ Z ∩ ( −∞ , N ] , arbitrary initial positions x ∈ R and arbitrary initial market deviations d ∈ R . In this setting we show that the minimalexpected overall execution costs amount to V n ( x, d ) = Y n γ n ( d − γ n x ) − d γ n . (1)In particular, for each n ∈ Z ∩ ( −∞ , N ] it follows that the random variable Y n takesvalues in (0 , and describes to which percentage the costs of closing one unit x = 1 attime n immediately can be reduced by executing this position optimally over { n, . . . , N } (given no initial market deviation d = 0 ). Accordingly, if Y n is close to / it is nearlyoptimal to close the position immediately in one go, whereas if Y n is close to it paysoff to split the position and to put only a small fraction in the market at time n .In the remainder of the article we discuss several qualitative and quantitative prop-erties of our market model and the trade execution problem. For instance, we analyzewhether our financial market admits price manipulation (in the sense of Huberman and3tanzl [21], see also [5] or [15]). A financial market is said to admit price manipula-tion if there exist round trip strategies (i.e., execution strategies that start in the initialposition x = 0 ) that generate profits in expectation. It follows immediately from (1)that if there is no initial market deviation (i.e., d = 0 ), then the market does not admitprice manipulation. However, for general d ∈ R we have that V n (0 , d ) = d γ n ( Y n − ) andthus in the case d (cid:54) = 0 there exist profitable round trips starting at time n if and only if Y n < . We show that if the investor has a directional view on the resilience process attime n (i.e., E [ β n +1 |F n ] (cid:54) = 1 , where F n represents the information available at time n ),then she can exploit the information d (cid:54) = 0 and construct profitable round trips (seeCorollary 4.3 and the subsequent discussion). This is in line with the results in [13] and[14], where β is assumed to take values in (0 , . Interestingly, in our model profitableround trips with d (cid:54) = 0 can in general exist even on a part of the event { E [ β n +1 |F n ] = 1 } and, moreover, even when there is no directional view on the resilience in all future timepoints (see Example 4.8).A further interesting effect that appears because we do not restrict the process β to take values in (0 , concerns the question under which conditions it is optimal toclose the position in one go. We notice that in the settings of [13] and [14], where, inparticular, β is (0 , -valued, it is never optimal to close the position prematurely (seeProposition A.3 in [14]). On the contrary, in our setting, closing the position prematurelycan be optimal even with deterministic β and γ (Example 5.5) but is never optimal withthe additional restriction for β to be (0 , -valued (Proposition 5.4). Moreover, in thesituation when closing the position prematurely is optimal, it can either be optimal tobuild up a new position at the next time point (Example 5.5) or not to trade any longer(the latter happens on the event { Y n = } , see Proposition 5.2 and Corollary 4.3). Onthe other hand, when we allow for stochastic β and γ , closing the position prematurelycan be optimal even with (0 , -valued β (Example 5.6). We, finally, notice that thedifference between the latter statement and the mentioned Proposition A.3 in [14] is dueto the fact that, in contrast to our current setting, in [14] the trading is constrained onlyin one direction. We refer to Table 1 for a more detailed discussion.Furthermore, we address the question of how much better in comparison to the im-mediate position closure the investor can perform if the time horizon is very large. Thatis, we analyze the behavior of the random sequence ( Y n ) n ∈{ ...,N − ,N } as n → −∞ . Ifliquidity increases on average (more precisely, if γ is a supermartingale) we show that ( Y n ) n ∈{ ...,N − ,N } converges a.s. and in any L p , p ∈ [1 , ∞ ) , to a [0 , / -valued randomvariable as n → −∞ (Proposition 2.2). If liquidity decreases on average, then, in general,the limit can fail to exist (Lemma 3.3). In a more specific setting, where the multiplica-tive increments of the price impact η k +1 = γ k +1 /γ k and the resilience factor β k +1 areindependent of the history up to time k and their expectations are homogeneous in time,the limit of ( Y n ) n ∈{ ...,N − ,N } as n → −∞ exists, is deterministic and can be identifiedexplicitly (Proposition 3.2). In particular, we see that the cost savings can range from (if E [ β k ] = 1 and E [ η k ] > ) to (if E [ β k ] < and E [ η k ] ≤ ).This article is organized as follows. In Section 1 we introduce the mathematical setting,state the stochastic control problem and provide its financial interpretation. In Section 24e solve the problem via dynamic programming, study the existence of the long-timelimit lim n →−∞ Y n of the characterizing process Y and discuss a few technical issues.A subsetting where Y becomes deterministic is examined in Section 3. In Section 4 westudy the existence of profitable round trips and in Section 5 we discuss when it is optimalto close the position prematurely; both sections describe several qualitative effects viageneral statements and examples. Appendix A contains the proof of Theorem 2.1. Twosimple lemmas on integrability, which we often use in our arguments, are included forconvenience in Appendix B.
1. A trade execution problem with stochastic marketdepth and stochastic resilience
In this section we introduce a financial market model where liquidity varies randomlyin time. We first give the comprehensive mathematical formulation of the model andsubsequently comment on its financial motivation.
Mathematical formulation
Let N ∈ N and let (Ω , F , ( F k ) k ∈ Z , P ) be a filtered prob-ability space. Denote L ∞− = (cid:84) p ∈ [1 , ∞ ) L p (Ω , F , P ) and L = (cid:83) ε> L ε (Ω , F , P ) . Let β = ( β k ) k ∈ Z and γ = ( γ k ) k ∈ Z be strictly positive adapted stochastic processes, calledthe resilience and the price impact process, respectively. Assume that β k , γ k ∈ L ∞− for all k ∈ Z . Furthermore, it turns out to be convenient to denote the multiplicativeincrements of γ by η n = γ n γ n − , n ∈ Z .For n ∈ Z ∩ ( −∞ , N ] and x ∈ R we call a real-valued adapted stochastic process ξ = ( ξ k ) k ∈{ n,...,N } satisfying x + (cid:80) Nj = n ξ j = 0 an execution strategy . We denote by A n ( x ) the set of all execution strategies ξ with ξ k ∈ L for all k ∈ { n, . . . , N } . For an executionstrategy ξ ∈ A n ( x ) we call the process X = ( X k ) k ∈{ n,...,N } satisfying X k = x + (cid:80) kj = n ξ j , k ∈ { n, . . . , N } the position path associated to ξ . For d ∈ R and ξ ∈ A n ( x ) we definethe deviation process D = ( D k − ) k ∈{ n,...,N } associated to ξ recursively by D n − = d and D k − = ( D ( k − − + γ k − ξ k − ) β k , k ∈ { n + 1 , . . . , N } . (2)Note that the process D = ( D k − ) k ∈{ n,...,N } is adapted. The value function V : Ω × ( Z ∩ ( −∞ , N ]) × R × R → R of the control problem is given by V n ( x, d ) = ess inf ξ ∈A n ( x ) E n (cid:34) N (cid:88) j = n (cid:16) D j − + γ j ξ j (cid:17) ξ j (cid:35) , n ∈ Z ∩ ( −∞ , N ] , x ∈ R , d ∈ R , (3)where the argument d is the starting point of the process D in (2), and E n [ · ] is ashorthand notation for E [ ·|F n ] . Financial interpretation
The numbers N ∈ N and n ∈ Z ∩ ( −∞ , N ] specify the endand the beginning of the trading period, respectively. The possible trading times are5iven by the set { n, . . . , N } . The number x ∈ R represents the initial position of theagent. A negative x < means that the agent has to buy | x | shares over the tradingperiod, while a positive x > means that the agent has to sell x shares over the tradingperiod. For an execution strategy ξ ∈ A n ( x ) the value of ξ k specifies the number ofshares bought by the agent at time k ∈ { n, . . . , N } . A negative value ξ k < means thatthe agent sells shares. For the associated position path X the value of X k representsthe agent’s position at time k ∈ { n, . . . , N } directly after the trade ξ k . Observe thatall position paths satisfy X N = 0 , i.e., the position is closed after the last trade at time N . The process D describes the deviation of the price of a share from the unaffectedprice caused by the past trades of the agent. Given a deviation of size D ( k − − directlyprior to the trade at time k − , the deviation directly after a trade of size ξ k − equals D ( k − − + γ k − ξ k − . In particular, the change of the deviation is proportional to the sizeof the trade and the proportionality factor is given by the price impact process γ . In thelanguage of the literature on optimal trade execution problems our model thus includesa linear price impact. This corresponds to a block-shaped limit order book, i.e., limitorders are uniformly distributed to the left and to the right of the mid-market price.The height of the order book at time k is given by /γ k . In particular, our model allowsthe height of the limit order book to evolve randomly in time and thereby capturesstochastic market liquidity. Note that since γ is positive, a purchase ξ k > increasesthe deviation whereas a sale ξ k < decreases it. In the period after the trade at time k − and before the trade at time k the deviation changes from D ( k − − + γ k − ξ k − to D k − = ( D ( k − − + γ k − ξ k − ) β k . In the literature on optimal execution the resilienceprocess β is often assumed to take values in (0 , and describes the speed with whichthe deviation tends back to zero between two trades. On the contrary, we assume β only to be positive. A value β k > describes the effect when the deviation continuesto move in the direction of the trade for some time after the trade. Note that the β factor evolves randomly in time. In particular, when making a decision about the sizeof the trade at time k − , the agent, in general, cannot predict the exact impact ofthis trade on the future price at time k . Note, however, that the agent observes therealization of β k before she makes the decision about the size of trade at time k . At eachtime k ∈ { n, . . . , N } the costs of a trade ξ k amount to ( D k − + γ k ξ k ) ξ k . This means thatthe price per share that the agent has to pay equals the mean of the deviation beforethe trade D k − and the deviation after the trade D k − + γ k ξ k . The control problem (3)thus corresponds to minimizing the expected costs of closing an initial position of size x within the trading period { n, . . . , N } given an initial deviation d .We conclude this section with some remarks on the well-posedness of the optimaltrade execution problem (3) and a possible extension of the model. Remark 1.1.
Let n ∈ Z ∩ ( −∞ , N ] , x, d ∈ R and ξ ∈ A n ( x ) . Then for the associateddeviation process ( D k − ) k ∈{ n,...,N } it holds that D k − ∈ L for all k ∈ { n, . . . , N } .We prove this claim by induction on k . Since D n − = d , the claim obviously holds truefor k = n . Consider the step { n, . . . , N − } (cid:51) k − → k ∈ { n + 1 , . . . , N } and notethat by the Minkowski inequality and (2), it is sufficient to show that D ( k − − β k ∈ L and γ k − ξ k − β k ∈ L . Since β k ∈ L ∞− and, by the induction hypothesis, D ( k − − ∈ , Lemma B.2 proves that D ( k − − β k ∈ L . For γ k − ξ k − β k observe first that γ k − β k ∈ L ∞− since both factors belong to L ∞− . Then, recall that ξ k − ∈ L andapply Lemma B.2 to obtain that γ k − ξ k − β k ∈ L . Remark 1.2.
Note that the value function is well-defined. To show this, we verifythat for all n ∈ Z ∩ ( −∞ , N ] , x, d ∈ R , ξ ∈ A n ( x ) each summand (cid:0) D j − + γ j ξ j (cid:1) ξ j , j ∈ { n, . . . , N } , is integrable.Since γ j ∈ L ∞− and ξ j ∈ L , it follows from Lemma B.2 that the product γ j ξ j is in L . By Remark 1.1, D j − ∈ L as well. Hence, D j − and γ j ξ j are square integrable andso is D j − + γ j ξ j . Furthermore, ξ j is square integrable as it is in L . The Cauchy-Schwarzinequality thus yields the integrability of (cid:0) D j − + γ j ξ j (cid:1) ξ j . Remark 1.3.
For n ∈ Z ∩ ( −∞ , N ] , x, d ∈ R and ξ ∈ A n ( x ) the deviation process D = ( D k − ) k ∈{ n,...,N } associated to ξ is given explicitly by D k − = d k (cid:89) l = n +1 β l + k (cid:88) i = n +1 γ i − ξ i − k (cid:89) l = i β l , k ∈ { n, . . . , N } . (4)This can be established by induction on k ∈ { n, . . . , N } . Remark 1.4.
One can also include an unaffected price process in the model. In-deed, if the unaffected price process is given by the square integrable martingale S =( S k ) k ∈ Z ∩ ( −∞ ,N ] , then, for all n ∈ Z ∩ ( −∞ , N ] , x ∈ R and ξ ∈ A n ( x ) , with the notation X n − = x , we get E n (cid:34) N (cid:88) j = n S j ξ j (cid:35) = E n (cid:34) N (cid:88) j = n S j ( X j − X j − ) (cid:35) = E n (cid:34) − xS n − N − (cid:88) j = n X j ( S j +1 − S j ) (cid:35) = − xS n . It follows that for all n ∈ Z ∩ ( −∞ , N ] and x, d ∈ R the expected costs generated by anexecution strategy ξ ∈ A n ( x ) with the deviation process ( D k − ) k ∈{ n,...,N } of (2) satisfy E n (cid:34) N (cid:88) j = n (cid:16) S j + D j − + γ j ξ j (cid:17) ξ j (cid:35) = − xS n + E n (cid:34) N (cid:88) j = n (cid:16) D j − + γ j ξ j (cid:17) ξ j (cid:35) . (5)Hence, minimizing E n (cid:104)(cid:80) Nj = n (cid:0) S j + D j − + γ j ξ j (cid:1) ξ j (cid:105) is equivalent to (3).
2. Characterization of minimal costs and optimalstrategies
The following result provides a solution to the stochastic control problem (3). It showsthat the value function and the optimal strategy in (3) are characterized by a singleprocess Y that is defined via a backward recursion.7 heorem 2.1. Assume that for all n ∈ Z ∩ ( −∞ , N ] we have β n , γ n , γ n ∈ L ∞− and thatfor all n ∈ Z ∩ ( −∞ , N − it holds that E n (cid:104) β n +1 η n +1 (cid:105) < a.s. and, with α n = 1 − E n (cid:104) β n +1 η n +1 (cid:105) ,we have α n ∈ L ∞− . Let ( Y n ) n ∈ Z ∩ ( −∞ ,N ] be the process that is recursively defined by Y N = and Y n = E n [ η n +1 Y n +1 ] − ( E n [ Y n +1 ( β n +1 − η n +1 )]) E n (cid:104) Y n +1 η n +1 ( β n +1 − η n +1 ) + (cid:16) − β n +1 η n +1 (cid:17)(cid:105) , n ∈ Z ∩ ( −∞ , N − . (6) Then it holds for all n ∈ Z ∩ ( −∞ , N ] , x, d ∈ R that V n ( x, d ) = Y n γ n ( d − γ n x ) − d γ n and < Y n ≤ . (7) Moreover, for all x, d ∈ R the (up to a P -null set) unique optimal trade size is given by ξ ∗ n ( x, d ) = E n [ Y n +1 ( β n +1 − η n +1 )] E n (cid:104) Y n +1 η n +1 ( β n +1 − η n +1 ) + (cid:16) − β n +1 η n +1 (cid:17)(cid:105) (cid:18) x − dγ n (cid:19) − dγ n , n ∈ Z ∩ ( −∞ , N − , (8) and ξ ∗ N ( x, d ) = − x , and we have ξ ∗ n ( x, d ) ∈ L ∞− for all n ∈ Z ∩ ( −∞ , N ] and x, d ∈ R .In particular, for all n ∈ Z ∩ ( −∞ , N ] , x, d ∈ R the process ξ ∗ = ( ξ ∗ k ) k ∈{ n,...,N } recursively defined by X ∗ n − = x, D ∗ n − = d , ξ ∗ k = ξ ∗ k (cid:0) X ∗ k − , D ∗ k − (cid:1) , X ∗ k = X ∗ k − + ξ ∗ k , D ∗ ( k +1) − = (cid:0) D ∗ k − + γ k ξ ∗ k (cid:1) β k +1 , k ∈ { n, . . . , N } (9) is a unique optimal strategy in A n ( x ) for (3) . The proof of Theorem 2.1 is deferred to Appendix A.We can give the following interpretation to the process Y from Theorem 2.1: Supposethat at time n ∈ Z ∩ ( −∞ , N ] the task is to sell x = 1 share given an initial deviationof d = 0 . Then immediate execution of the share generates the costs γ n . The optimalexecution strategy incurs the expected costs V n (1 ,
0) = γ n Y n (recall (7)). So, the randomvariable Y n : Ω → [0 , describes to which percentage the costs of selling the unitimmediately can be reduced by executing the position optimally.In the next proposition we study the existence of the long-time limit lim n →−∞ Y n . Proposition 2.2.
Let the assumptions of Theorem 2.1 be in force. Fix any p ∈ [1 , ∞ ) .(i) The sequence ( γ n Y n ) n ∈ Z ∩ ( −∞ ,N ] converges a.s. and in L p as n → −∞ to a finitenonnegative random variable.(ii) If ( γ n ) n ∈ Z ∩ ( −∞ ,N ] is a supermartingale, then the sequence ( Y n ) n ∈ Z ∩ ( −∞ ,N ] convergesa.s. and in L p as n → −∞ to a finite nonnegative random variable. ( γ n ) n ∈ Z ∩ ( −∞ ,N ] is a supermartingale in (ii) means that the liq-uidity in the model increases in time (in average). In Lemma 3.3 below ( γ n ) n ∈ Z ∩ ( −∞ ,N ] isa submartingale and ( Y n ) n ∈ Z ∩ ( −∞ ,N ] does not converge. This shows that the claim in (ii)does not in general hold in the situation when the liquidity in the model decreases intime. Proof. (i) It follows from (6) that for all n ∈ Z ∩ ( −∞ , N − it holds Y n ≤ E n [ η n +1 Y n +1 ] = γ n E n [ γ n +1 Y n +1 ] . Thus, ( γ n Y n ) n ∈ Z ∩ ( −∞ ,N ] is a submartingale. Hence it converges a.s. as n → −∞ due to the backward convergence theorem. Moreover, ( γ n Y n ) n ∈ Z ∩ ( −∞ ,N ] isa positive sequence in L ∞− , and, by the Jensen inequality, ( γ n Y n ) p ≤ E n [( γ N Y N ) p ] , n ∈ Z ∩ ( −∞ , N ] , hence the sequence (( γ n Y n ) p ) n ∈ Z ∩ ( −∞ ,N ] is uniformly integrable. Thisimplies the convergence in L p .(ii) If ( γ n ) n ∈ Z ∩ ( −∞ ,N ] is a supermartingale, then it converges a.s. as n → −∞ to a R ∪ { + ∞} -valued random variable, denoted by γ −∞ , due to the backward convergencetheorem. As the process ( γ n ) is positive, γ −∞ is, in fact, [0 , + ∞ ] -valued. Furthermore,it holds E (cid:2) γ −∞ { γ −∞ =0 } (cid:3) ≥ E (cid:2) γ N { γ −∞ =0 } (cid:3) ≥ . Together with the fact that γ N > a.s., this implies γ −∞ > a.s. It now followsfrom (i) that ( Y n ) n ∈ Z ∩ ( −∞ ,N ] converges a.s. as n → ∞ . As the sequence ( Y n ) n ∈ Z ∩ ( −∞ ,N ] is bounded (being (0 , ] -valued), it also converges in L p .The next remark provides an improved upper bound for Y . Remark 2.3 (Upper bound for Y ) . Under the assumptions of Theorem 2.1, for all n ∈ Z ∩ ( −∞ , N ] it holds that γ n Y n = V n (1 , . For an initial position of size at time n ∈ Z ∩ ( −∞ , N ] a possible execution strategy is to sell the whole unit at a point in time k ∈ { n, . . . , N } . If there is no initial deviation, i.e., d = 0 , it follows that the expectedcosts of such a strategy amount to E n (cid:2) γ k (cid:3) . This implies that Y n ≤ min k ∈{ n,...,N } E n [ γ k ]2 γ n ,which improves the bound Y n ≤ provided by Theorem 2.1.Besides some integrability assumptions, Theorem 2.1 requires that E n (cid:104) β n +1 η n +1 (cid:105) < a.s.for all n ∈ Z ∩ ( −∞ , N − . The next remark discusses this assumption. Remark 2.4 (Discussion of the structural assumption) . The assumption E n (cid:104) β n +1 η n +1 (cid:105) < a.s. for all n ∈ Z ∩ ( −∞ , N − in Theorem 2.1 is a certain structural assumption whichensures that minimization problem (3) is strictly convex. More precisely, under thisassumption the coefficients a n in front of ξ in (42) (see Appendix A) and the randomvariables Y n in (6) stay positive at all times. In this remark we show that, on the onehand, this assumption is in general not necessary for that, but, on the other hand, itguarantees that the problem preserves the structure with increasing number of timesteps. To this end we consider a two-period version of the problem and distinguish Here we use the convention ∞ · . Y N = and observe that with (42) it holds for all x, d ∈ R V N − ( x, d ) = ess inf ξ ∈S N − (cid:26) E N − [ η N + 1 − β N ] γ N − ξ E N − (cid:20) (1 − β N ) d − (cid:18) β N η N − (cid:19) γ N x (cid:21) ξ + E N − (cid:20) γ N x − β N dx (cid:21) (cid:27) . (10)Next, observe that the process Y defined by (6) is given at time N − by Y N − = E N − (cid:104) η N (cid:105) − ( E N − [ β N − η N ]) E N − [ η N − β N + 1] = E N − [ η N ] − ( E N − [ β N ]) E N − [ η N − β N + 1] . (11)Moreover, the Cauchy-Schwarz inequality ensures that ( E N − [ β N ]) ≤ E N − (cid:104) β N η N (cid:105) E N − [ η N ] and hence it holds that E N − [ β N ] − E N − [ η N ] ≤ ( E N − [ β N ]) E N − [ η N ] ≤ E N − (cid:20) β N η N (cid:21) . (12)In particular, we get the following statements.(i) On the event (cid:110) E N − [ β N ] − E N − [ η N ] > (cid:111) the minimization problem in (10) is ill-posed inthe sense that it is strictly concave and one can generate infinite gains (in the limit) bychoosing strategies with | ξ | → ∞ .(ii) On the event (cid:110) E N − [ β N ] − E N − [ η N ] < < ( E N − [ β N ]) E N − [ η N ] (cid:111) there exists a minimizer in (10).The random variable Y N − is, however, negative. As a consequence, in view of (38), oneneeds to impose further conditions on β N − and η N − to ensure that the coefficient a N − is positive and that the minimization problem at time N − is well-posed.(iii) On the event (cid:110) ( E N − [ β N ]) E N − [ η N ] < (cid:111) , which is bigger than (cid:110) E N − (cid:104) β N η N (cid:105) < (cid:111) (see (12)),there exists a minimizer in (10) and, moreover, Y N − ∈ (0 , ] (see (11)).Observe, however, that replacing the assumption E n (cid:104) β n +1 η n +1 (cid:105) < a.s. with the weakerone ( E n [ β n +1 ]) E n [ η n +1 ] < a.s. for all n ∈ Z ∩ ( −∞ , N − does not in general allow to performthe backward induction, as the structure of the problem can be lost already on the step N − → N − . Namely, Y N − can be strictly less than (in contrast to Y N = ), while E N − (cid:104) β N − η N − (cid:105) can be strictly bigger than (even assuming ( E N − [ β N − ]) E N − [ η N − ] < a.s.), and wedo not necessarily get positivity of a N − (see (38)).The next remark reveals the following property of optimal strategies: Irrespectively ofthe position x and the deviaton d prior to the trade at time n , the ratio between positionand deviation after the trade ξ ∗ n ( x, d ) is given by an F n -measurable random variable z n (that does not depend on ( x, d ) ). 10 emark 2.5 (Optimal deviation-position ratio) . In the setting of Theorem 2.1 theoptimal position path can be characterized in terms of its ratio to the associated deviationprocess. More precisely, let z = ( z n ) n ∈ Z ∩ ( −∞ ,N ] be the R ∪ {∞} -valued adapted processgiven by z n = γ n E n [ Y n +1 ( β n +1 − η n +1 )] E n (cid:104)(cid:0) Y n +1 − (cid:1) β n +1 η n +1 − Y n +1 β n +1 + (cid:105) , n ∈ Z ∩ ( −∞ , N − , z N = ∞ , (13)where we set a = ∞ whenever a ∈ R \ { } . Notice that the fraction defining z n , n ∈ Z ∩ ( −∞ , N − , a.s. does not produce because E n (cid:20)(cid:18) Y n +1 − (cid:19) β n +1 η n +1 − Y n +1 β n +1 + 12 (cid:21) − E n [ Y n +1 ( β n +1 − η n +1 )]= E n (cid:20) (cid:18) − β n +1 η n +1 (cid:19) + Y n +1 η n +1 ( β n +1 − η n +1 ) (cid:21) > a.s.under the assumptions of Theorem 2.1. Then for all n ∈ Z ∩ ( −∞ , N − , x, d ∈ R , d (cid:54) = γ n x , the ratio between the deviation d + γ n ξ ∗ n ( x, d ) and the position x + ξ ∗ n ( x, d ) directly after the optimal trade equals d + γ n ξ ∗ n ( x, d ) x + ξ ∗ n ( x, d ) = γ n E n [ Y n +1 ( β n +1 − η n +1 )] E n [ Y n +1 ( β n +1 − η n +1 )] + E n (cid:104) (cid:16) − β n +1 η n +1 (cid:17) + Y n +1 η n +1 ( β n +1 − η n +1 ) (cid:105) = z n , (14)which does not depend on the pair ( x, d ) except the requirement d (cid:54) = γ n x (the latter isto exclude the deviation-position ratio , see (8)). Likewise, for all x, d ∈ R , d (cid:54) = γ N x ,the deviation-position ratio after the terminal trade equals d + γ N ξ ∗ N ( x, d ) x + ξ ∗ N ( x, d ) = ∞ = z N . It is worth noting that the process z can take value ∞ also before the terminal time N and it is even possible that z takes finite values after being infinite (see Section 5 formore detail).
3. Processes with independent multiplicativeincrements
In this section we restrict attention to resilience and price impact processes that satisfy (PIMI) for all k ∈ Z the random variables η k +1 and β k +1 are independent of F k . Recall that η n = γ n γ n − , n ∈ Z .
11n this case it turns out that the process Y from Theorem 2.1 is deterministic. Corollary 3.1.
Assume (PIMI), that for all n ∈ Z ∩ ( −∞ , N ] we have β n , γ n , γ n ∈ L ∞− and that for all n ∈ Z ∩ ( −∞ , N − it holds that E (cid:104) β n +1 η n +1 (cid:105) < . Let Y = ( Y n ) n ∈ Z ∩ ( −∞ ,N ] be the process from Theorem 2.1 that is recursively defined by Y N = and (6) . Then Y is deterministic, (0 , ] -valued and satisfies the recursion Y n = E [ η n +1 ] Y n +1 − Y n +1 ( E [ β n +1 ] − E [ η n +1 ]) Y n +1 E (cid:104) ( β n +1 − η n +1 ) η n +1 (cid:105) + (cid:16) − E (cid:104) β n +1 η n +1 (cid:105)(cid:17) , n ∈ Z ∩ ( −∞ , N − . (15) Furthermore, formula (8) for optimal trade sizes in the state ( x, d ) ∈ R takes the form ξ ∗ n ( x, d ) = Y n +1 ( E [ β n +1 ] − E [ η n +1 ]) Y n +1 E (cid:104) ( β n +1 − η n +1 ) η n +1 (cid:105) + (cid:16) − E (cid:104) β n +1 η n +1 (cid:105)(cid:17) (cid:18) x − dγ n (cid:19) − dγ n , n ∈ Z ∩ ( −∞ , N − , (16) and ξ ∗ N ( x, d ) = − x .Proof. Recursion (15) follows by a straightforward induction argument. Formula (16) isan immediate consequence of the fact that Y is deterministic.In the next proposition we discuss the long-time limit lim n →−∞ Y n assuming (PIMI)and a sort of time-homogeneity (only for expectations). Proposition 3.2.
Suppose that the assumptions of Corollary 3.1 hold true and that ¯ β = E [ β n +1 ] , ¯ η = E [ η n +1 ] and ¯ α = E (cid:104) β n +1 η n +1 (cid:105) do not depend on n ∈ Z ∩ ( −∞ , N − .1. If ¯ β = 1 , we have ¯ η > , and it holds for all n ∈ Z ∩ ( −∞ , N ] that Y n = .2. If ¯ η ≤ , we have ¯ β < , and the sequence Y = ( Y n ) n ∈ Z ∩ ( −∞ ,N ] converges mono-tonically to as n → −∞ .3. If ¯ β (cid:54) = 1 and ¯ η > , the sequence Y = ( Y n ) n ∈ Z ∩ ( −∞ ,N ] converges monotonically to (1 − ¯ α ) (¯ η − − ¯ α ) (¯ η −
1) + (cid:0) ¯ β − (cid:1) ∈ (cid:18) , (cid:19) (17) as n → −∞ . Discussion of Proposition 3.2
Suppose that at time n we have x = 1 share to selland the initial deviation is d = 0 . The immediate selling of the share incurs the costs γ n .The optimal execution strategy produces the expected costs V n (1 ,
0) = γ n Y n (recall (7)).So, in other words, the question about the long-time limit lim n →−∞ Y n is the questionof how much better in comparison to the immediate selling we can perform if our timehorizon is very big.In general, dividing a large order into many small orders and executing them in con-secutive time points can be profitable compared to the immediate execution because ofthe following reasons: 121) the price impact process γ penalizes trades at different times in a different waywhenever γ is nonconstant,(2) the resilience process β changes the deviation process D between the trades whenever β is not identically .From this viewpoint the claims of Proposition 3.2, which deals with the “time-homogeneousin expectation (PIMI) case”, are naturally interpreted as follows. If the resilience is inexpectation ( ¯ β = 1 ), then neither of the above reasons suggests dividing a large orderinto many small orders (notice that, in this case, the price impact process γ is increasingin average, as ¯ η > ). We can asymptotically get rid of the execution costs in the caseof nonincreasing price impact (in the sense ¯ η ≤ ). Notice that, in this case, the priceimpact is allowed to be constant, but we anyway profit from the resilience, which, inexpectation, drives the deviation back to zero between two trades ( ¯ β < ). Finally, inthe remaining case of a nontrivial resilience and a geometrically increasing price impact(in the sense ¯ β (cid:54) = 1 and ¯ η > ) we cannot fully get rid of the execution costs regardlessof how big our time horizon is. Proof of Proposition 3.2.
From (15), we have Y n = ¯ ηY n +1 − Y n +1 (cid:0) ¯ β − ¯ η (cid:1) Y n +1 (cid:0) ¯ α − β + ¯ η (cid:1) + (1 − ¯ α ) , n ∈ Z ∩ ( −∞ , N − . (18)Define g : [0 , ∞ ) → R ,g ( y ) = ¯ ηy − y (cid:0) ¯ β − ¯ η (cid:1) y (cid:0) ¯ α − β + ¯ η (cid:1) + (1 − ¯ α ) , y ∈ [0 , ∞ ) . (19)Note that ¯ α < by assumption and that ¯ α − β + ¯ η ≥ ( ¯ β − ¯ η ) ¯ η ≥ because ¯ β ¯ η ≤ ¯ α bythe Cauchy-Schwarz inequality. Let y ≥ . Then g (cid:48) ( y ) = ¯ η − (cid:0) ¯ β − ¯ η (cid:1) y (cid:0) y (cid:0) ¯ α − β + ¯ η (cid:1) + (1 − ¯ α ) (cid:1) − y (cid:0) ¯ α − β + ¯ η (cid:1)(cid:0) y (cid:0) ¯ α − β + ¯ η (cid:1) + (1 − ¯ α ) (cid:1) = ¯ η − (cid:0) ¯ β − ¯ η (cid:1) y (cid:0) ¯ α − β + ¯ η (cid:1) + y (1 − ¯ α ) (cid:0) y (cid:0) ¯ α − β + ¯ η (cid:1) + (1 − ¯ α ) (cid:1) . Hence, g (cid:48) ( y ) > is equivalent to ¯ η (cid:18) y (cid:0) ¯ α − β + ¯ η (cid:1) + 12 (1 − ¯ α ) (cid:19) > (cid:0) ¯ β − ¯ η (cid:1) (cid:0) y (cid:0) ¯ α − β + ¯ η (cid:1) + y (1 − ¯ α ) (cid:1) . ¯ η > and note that ( ¯ β − ¯ η ) ¯ η = ¯ β ¯ η − β + ¯ η . This yields the equivalent statement < y (cid:0) ¯ α − β + ¯ η (cid:1) + y (cid:0) ¯ α − β + ¯ η (cid:1) (1 − ¯ α ) + (1 − ¯ α ) − (cid:0) ¯ β − ¯ η (cid:1) ¯ η y (cid:0) ¯ α − β + ¯ η (cid:1) − (cid:0) ¯ β − ¯ η (cid:1) ¯ η y (1 − ¯ α )= y (cid:0) ¯ α − β + ¯ η (cid:1) (cid:18) ¯ α − ¯ β ¯ η (cid:19) + y (cid:18) ¯ α − ¯ β ¯ η (cid:19) (1 − ¯ α ) + (1 − ¯ α ) (cid:18) y (cid:18) ¯ α − ¯ β ¯ η (cid:19) + 1 − ¯ α (cid:19) + y (cid:18) ¯ α − ¯ β ¯ η (cid:19) (cid:0) ¯ β − ¯ η (cid:1) ¯ η . Since ¯ α < and ¯ β ¯ η ≤ ¯ α , this always holds true for y ≥ . It follows that g is strictlyincreasing on [0 , ∞ ) .Recall that < Y n ≤ for all n ∈ Z ∩ ( −∞ , N − and Y N = . In particular, Y N − ≤ Y N . The recursion Y n = g ( Y n +1 ) , n ∈ Z ∩ ( −∞ , N − (cf. (18) and (19)),implies that the sequence Y is nondecreasing. Hence, the limit lim n →−∞ Y n exists andbelongs to [0 , ] . Moreover, it is the largest fixed point of g in [0 , ] . Indeed, since g is increasing, for the largest fixed point ¯ y of g in [0 , ] , we have that y ≥ ¯ y implies g ( y ) ≥ g (¯ y ) = ¯ y . Hence, ¯ y is a lower bound of Y . We obtain that lim n →−∞ Y n ≥ ¯ y andis a fixed point of g , which means that lim n →−∞ Y n = ¯ y .1. Suppose that ¯ β = 1 . The claim that ¯ η > follows from ¯ β ¯ η ≤ α < . A directcalculation shows that g (cid:0) (cid:1) = . Since Y N = , it follows that Y n = for all n ∈ Z ∩ ( −∞ , N ] .2. Suppose that ¯ η ≤ . First notice that ¯ β ≤ ¯ η ¯ α < ¯ η ≤ and hence ¯ β < . Nowit follows from (19) that for all y > we have g ( y ) < y . This yields that is theonly fixed point of g on [0 , ∞ ) and hence lim n →−∞ Y n = 0 .3. Suppose that ¯ β (cid:54) = 1 and ¯ η > . In this case ¯ y = (1 − ¯ α ) (¯ η − − ¯ α ) (¯ η −
1) + (cid:0) ¯ β − (cid:1) ∈ (cid:18) , (cid:19) (20)is a further fixed point of g and the only one in (0 , ∞ ) . Indeed, for y ∈ (0 , ∞ ) thecondition g ( y ) = y is equivalent to y (cid:16)(cid:0) ¯ β − ¯ η (cid:1) − (¯ η − (cid:0) ¯ α − β + ¯ η (cid:1)(cid:17) = 12 (1 − ¯ α ) (¯ η − . (21)From the fact that (cid:0) ¯ β − ¯ η (cid:1) − (¯ η − (cid:0) ¯ α − β + ¯ η (cid:1) = (1 − ¯ α ) (¯ η −
1) + (cid:0) ¯ β − (cid:1) > (1 − ¯ α ) (¯ η − > we deduce (20), which completes the proof.14he following lemma provides an example where the process Y = ( Y n ) n ∈ Z ∩ ( −∞ ,N ] defined by Y N = and (6) does not converge. In this example the price impact process γ is a submartingale (cf. the discussion following Proposition 2.2). Lemma 3.3.
Suppose that the assumptions of Corollary 3.1 hold true. Let ¯ β , ¯ β , ¯ η , ¯ η ∈ (0 , ∞ ) and ¯ α , ¯ α ∈ (0 , such that for all k ∈ N it holds ¯ β = E [ β N − k − ] = 1 , ¯ β = E [ β N − k ] (cid:54) = 1 , ¯ η = E [ η N − k − ] , ¯ η = E [ η N − k ] > , ¯ α = E (cid:104) β N − k − η N − k − (cid:105) and ¯ α = E (cid:104) β N − k η N − k (cid:105) .Then, γ is a submartingale and Y = ( Y n ) n ∈ Z ∩ ( −∞ ,N ] does not converge as n → −∞ .In particular, the sequence Y is not monotone.Proof. Note first that ¯ β = 1 and ¯ α < imply that ¯ η > by the Cauchy-Schwarz in-equality. It follows from < ¯ η = E [ η N − k − ] = E N − k − [ η N − k − ] = E N − k − (cid:104) γ N − k − γ N − k − (cid:105) = γ N − k − E N − k − [ γ N − k − ] and < ¯ η = γ N − k − E N − k − [ γ N − k ] for all k ∈ N that γ isa submartingale.For i ∈ { , } , denote by g i the function defined by (19) with ¯ β = ¯ β i , ¯ η = ¯ η i and ¯ α = ¯ α i . Recall that g , g are strictly increasing and note that for k ∈ N , we have Y N − k − = g ( Y N − k − ) and Y N − k − = g ( Y N − k ) . Furthermore, the equations g i ( y ) = y , i ∈ { , } , are (non-degenerate) quadratic ones, hence the functions g i have at mosttwo fixed points. We conclude that the only fixed points of g are and , and the onlyfixed points of g are given by and ¯ y ∈ (cid:0) , (cid:1) from (20). We also notice that g ( y ) > y for y ∈ (cid:0) , (cid:1) .We prove by induction that Y N − m > ¯ y for all m ∈ N . The case m = 0 is clear. Forthe induction step N (cid:51) m → m + 1 ∈ N , if m is even, we have Y N − m − = g ( Y N − m ) >g (¯ y ) = ¯ y . If m is odd, it holds Y N − m − = g ( Y N − m ) > g (¯ y ) > ¯ y .It can further be proven inductively that Y N − m ≥ Y N − m − for all m ∈ N since g , g are increasing and Y N − ≤ = Y N .Therefore, the subsequences ( Y N − k ) k ∈ N and ( Y N − k − ) k ∈ N of Y are decreasing in k ∈ N and bounded from below by ¯ y , which implies that the limits ¯ Y ( e ) = lim k →∞ Y N − k ≥ ¯ y and ¯ Y ( o ) = lim k →∞ Y N − k − ≥ ¯ y exist. Taking limits on both sides of Y N − k − = g ( Y N − k ) , we obtain ¯ Y ( o ) = g (cid:0) ¯ Y ( e ) (cid:1) by continuity of g . Similarly, it holds that ¯ Y ( e ) = g (cid:0) ¯ Y ( o ) (cid:1) . Now, if ¯ Y ( e ) and ¯ Y ( o ) were equal, then ¯ Y ( e ) = ¯ Y ( o ) would be a commonfixed point of g and g and hence , which is a contradiction to ¯ Y ( e ) ≥ ¯ y > . We thusconclude that Y does not converge.
4. Round trips
Let n ∈ Z ∩ ( −∞ , N − . Execution strategies in A n (0) are called round trips. It followsfrom Theorem 2.1 that if initially the agent has no position in the asset, i.e., x = 0 at15ime n ∈ Z ∩ ( −∞ , N ] , then the minimal costs amount to V n (0 , d ) = d γ n (cid:18) Y n − (cid:19) (22)for all d ∈ R . In particular, it holds that V n (0 ,
0) = 0 , i.e., without initial deviation ofthe price process the agent cannot make profits in expectation. In other words, thereare no profitable round trips whenever d = 0 . The existence of profitable round tripsis sometimes also referred to as price manipulation (see, e.g., [5], [15] or [21]). In thisregard, if there is no initial deviation of the price process (i.e., d = 0 ), then our modeldoes not admit price manipulation.Below we study existence of profitable round trips when the price of a share deviatesfrom the unaffected price, i.e., it holds d (cid:54) = 0 . We thus assume d (cid:54) = 0 in this section.Recall from (7) that the random variable Y n is (0 , ] -valued. Together with (22), thisimplies the following classification: • on { Y n < } there exist profitable round trips, • on { Y n = } there are no profitable round trips.Thus, the question reduces to finding a tractable description of the event { Y n = } . Wefirst characterize this event in Proposition 4.1 and discuss several consequences of thischaracterization. The proof of Proposition 4.1 is postponed to Subsection 4.1. Proposition 4.1.
Let the assumptions of Theorem 2.1 be satisfied. Then we have (cid:26) Y n = 12 (cid:27) = (cid:26) E n [ Y n +1 ] = 12 , E n [ β n +1 ] = 1 (cid:27) , n ∈ Z ∩ ( −∞ , N − , where here and below we understand the equalities for events up to P -null sets. Corollary 4.2.
Under the assumptions of Theorem 2.1 it holds (cid:26) Y N − = 12 (cid:27) = { E N − [ β N ] = 1 } . Proof.
The result is immediate because Y N = . Corollary 4.3.
Under the assumptions of Theorem 2.1 we have the following inclusionsfor n ∈ Z ∩ ( −∞ , N − :1. { Y n = } ⊆ { Y n +1 = } (equivalently, { Y n +1 < } ⊆ { Y n < } ) and2. { Y n = } ⊆ { E n [ β n +1 ] = 1 } ⊆ { E n [ β n +1 ] ≥ } ⊆ { E n [ η n +1 ] > } (equivalently, { E n [ η n +1 ] ≤ } ⊆ { E n [ β n +1 ] < } ⊆ { E n [ β n +1 ] (cid:54) = 1 } ⊆ { Y n < } ). The proof of Corollary 4.3 is given in Subsection 4.1.16 iscussion
In the literature on optimal execution it is often assumed that the resilienceprocess β takes values in (0 , . In this case we always have profitable round tripswhenever d (cid:54) = 0 , as we know that the deviation will go towards zero due to the resilienceand we can make use of it in constructing a profitable round trip (cf. Remark 8.2 in [13]and the discussion after Model 8.3 in [14]). Formally, this fact follows from Corollary 4.3.A natural generalization of this fact to the case of (only) positive β is the inclusion { E n [ β n +1 ] (cid:54) = 1 } ⊆ { Y n < } (again Corollary 4.3). The intuition is that on the event { E n [ β n +1 ] (cid:54) = 1 } we “expect” in which direction the deviation will go in the absence oftrading. A new qualitative effect in our setting is that the situation of nonexistence ofprofitable round trips is possible. The previous discussion explains that we necessarilyneed to be on the event { E n [ β n +1 ] = 1 } for the non-existence of profitable round trips. Asomewhat unexpected effect is, however, that the inclusion { Y n = } ⊆ { E n [ β n +1 ] = 1 } can be strict and hence there might exist profitable round trips on the event { E n [ β n +1 ] =1 } (see Examples 4.6 and 4.8 below for a more precise discussion). In particular, wecannot distinguish Y n = from Y n < on the basis of E n [ β n +1 ] alone, and, indeed, theexact characterization of the event { Y n = } also includes E n [ Y n +1 ] (see Proposition 4.1).In more detail, we have the following picture. At time N − we distinguish between Y N − = from Y N − < on the basis of E N − [ β N ] alone (Corollary 4.2). To discuss thestep n + 1 → n we consider the partition of Ω into two disjoint events (in F n ) Ω = (cid:26) E n [ Y n +1 ] < (cid:27) (cid:116) (cid:26) E n [ Y n +1 ] = 12 (cid:27) =: A n (cid:116) B n . (23)On A n there always exist profitable round trips when we start at time n , while on B n we distinguish between the nonexistence and the existence of profitable round trips onthe basis of whether E n [ β n +1 ] = 1 or E n [ β n +1 ] (cid:54) = 1 holds (Proposition 4.1).A special case, where we obtain an explicit criterion to distinguish between Y n = and Y n < for all n ∈ Z ∩ ( −∞ , N − only in terms of the process β is the case ofprocesses with independent multiplicative increments of Section 3: Corollary 4.4.
Let the assumptions of Corollary 3.1 be in force. We define n = N ∧ inf { n ∈ Z ∩ ( −∞ , N −
1] : E [ β k ] = 1 for all k ∈ Z ∩ [ n + 1 , N ] } ( inf ∅ = ∞ ) and notice that n ∈ ( Z ∪ {−∞} ) ∩ [ −∞ , N ] . Then, for the (deterministic)process Y , we have • Y n < for n ∈ Z ∩ ( −∞ , n ) , • Y n = for n ∈ Z ∩ [ n , N ] .Proof. The result follows from the previous discussion and the fact that, by Corollary 3.1,the process Y is deterministic.The next proposition contains a sufficient condition for existence of profitable roundtrips, which is expressed in different terms.17 roposition 4.5. Under the assumptions of Theorem 2.1 for all n ∈ Z ∩ ( −∞ , N − it holds (cid:26) Y n = 12 (cid:27) ⊆ (cid:26) min k ∈{ n +1 ,...,N } E n ( γ k ) ≥ γ n (cid:27) (equivalently, { min k ∈{ n +1 ,...,N } E n ( γ k ) < γ n } ⊆ { Y n < } ).Proof. While the result can be again inferred from the characterization of the event { Y n = } in Proposition 4.1, the shortest proof is to recall that Y n < on the event { min k ∈{ n +1 ,...,N } E n ( γ k ) < γ n } due to Remark 2.3.We now discuss the inclusion { Y n = } ⊆ { E n [ β n +1 ] = 1 } in more detail. Firstwe present a simple example, where for n = N − this inclusion is strict (cf. withCorollary 4.2). Example 4.6.
We take any deterministic sequences β and γ with β N (cid:54) = 1 and β N − = 1 that satisfy the assumptions of Theorem 2.1. Then the process Y is deterministic.Corollary 4.2 implies that Y N − < . Hence, by Corollary 4.3, Y N − < . We thus have (cid:26) Y N − = 12 (cid:27) = ∅ (cid:40) Ω = { E N − [ β N − ] = 1 } . In other words, for d (cid:54) = 0 , we have profitable round trips when we start at time N − ,although E N − [ β N − ] = 1 . This is not surprising in this example, as we see that profitableround trips are already present when we start at time N − ( Y N − < , which is causedby β N (cid:54) = 1 ). One might, therefore, intuitively expect that here all round trips do notcontain a trade at time N − , but this is not the case! If d (cid:54) = 0 , then we have forthe (here, deterministic) optimal strategy ξ ∗ (0 , d ) of (8) that ξ ∗ N − (0 , d ) (cid:54) = 0 . Indeed,a straightforward calculation using (8) and the fact that β , η , Y are deterministic and β N − = 1 reveals that ξ ∗ N − (0 , d ) = 0 if and only if it holds ( − Y N − )(1 − η N − ) = 0 ,but the latter is not true in this example because Y N − < and η N − = β N − η N − < (recallthe assumptions of Theorem 2.1).Example 4.6 raises the question of whether profitable round trips for d (cid:54) = 0 with start-ing time n ∈ Z ∩ ( −∞ , N − can occur on the event (cid:84) N − k = n { E k [ β k +1 ] = 1 } . Corollary 4.4implies that this is impossible in the framework of (PIMI) (let alone with deterministic β and γ ). But, in general, such a phenomenon is possible, and we present a specificexample after the following lemma. Lemma 4.7.
Let the assumptions of Theorem 2.1 be in force and let n ∈ Z ∩ ( −∞ , N − .(i) We have (cid:26) Y n = 12 (cid:27) ⊆ N − (cid:92) k = n { E k [ β k +1 ] = 1 } . (24)18 ii) The inclusion in (24) is strict (in the sense that the set difference has positive P -probability) if and only if N − (cid:92) k = n { E k [ β k +1 ] = 1 } / ∈ F n , (25) where F n = σ ( F n ∪ N ) with N = { A ∈ F : P ( A ) = 0 } .Proof. Inclusion (24) follows from Corollary 4.3. Clearly, under (25), the inclusion isstrict, as { Y n = } ∈ F n . It remains to prove that, if there is A n ∈ F n , which is (up toa P -null set) equal to (cid:84) N − k = n { E k [ β k +1 ] = 1 } , then Y n = a.s. on A n .First, Corollary 4.2 yields Y N − = a.s. on A n . In the case n = N − this concludesthe proof. Let n ≤ N − . As A n ∈ F n ⊆ F N − , we get E N − [ Y N − ] = a.s. on A n .Proposition 4.1 now yields Y N − = a.s. on A n . In the case n = N − this concludesthe proof. If n ≤ N − , we obtain the result by iterating the same procedure.We, finally, present a specific example, where for n = N − the inclusion in (24) isstrict, or, in other words, P ( Y N − < , E N − [ β N − ] = E N − [ β N ] = 1) > (recall thediscussion following Example 4.6). Example 4.8.
Take arbitrary a, p ∈ (0 , . Let F n = {∅ , Ω } for n ∈ Z ∩ ( −∞ , N − , F N − = F N = σ ( β N − ) with β N − being distributed according to P ( β N − = 1) =1 − p and P ( β N − = 1 ± a ) = p/ . We set β N = β N − and choose any process γ satisfying the assumptions of Theorem 2.1 (e.g., one can easily take deterministic γ ).Then E N − [ β N − ] = E [ β N − ] = 1 , hence { E N − [ β N − ] = 1 } ∩ { E N − [ β N ] = 1 } = { E N − [ β N ] = 1 } = { β N = 1 } , which is an event of probability − p ∈ (0 , . We thus obtain (25) for n = N − .By Lemma 4.7, the inclusion in (24) for n = N − is strict. As a result, we get P ( Y N − < , E N − [ β N − ] = E N − [ β N ] = 1) > , as required. Proof of Proposition 4.1.
Throughout the proof fix n ∈ Z ∩ ( −∞ , N − . Let ν = − (cid:0) − Y n +1 (cid:1) β n +1 η n +1 . Rewriting the definition of Y n , we obtain Y n = E n [ η n +1 Y n +1 ] − ( E n [ Y n +1 β n +1 ]) − E n [ Y n +1 β n +1 ] E n [ Y n +1 η n +1 ] + ( E n [ Y n +1 β n +1 ]) E n [ ν − Y n +1 β n +1 + Y n +1 η n +1 ]= E n [ ν ] E n [ ν − Y n +1 β n +1 + Y n +1 η n +1 ] − ( E n [ ν − Y n +1 β n +1 ]) E n [ ν − Y n +1 β n +1 + Y n +1 η n +1 ]= 12 − E n (cid:20)(cid:18) − Y n +1 (cid:19) β n +1 η n +1 (cid:21) − γ n a n (cid:18) − E n (cid:20)(cid:18) − Y n +1 (cid:19) β n +1 η n +1 (cid:21) − E n [ Y n +1 β n +1 ] (cid:19) with a n from (38). Since η n +1 , γ n , a n > and Y n +1 ≤ a.s., it now follows that (cid:26) Y n = 12 (cid:27) = (cid:26) E n (cid:20)(cid:18) − Y n +1 (cid:19) β n +1 η n +1 (cid:21) = 0 , E n [ Y n +1 β n +1 ] = 12 (cid:27) . (26)19et C n = (cid:110) E n (cid:104)(cid:0) − Y n +1 (cid:1) β n +1 η n +1 (cid:105) = 0 (cid:111) and denote B n = (cid:8) E n [ Y n +1 ] = (cid:9) as before. Weshow that C n = B n . For the inclusion C n ⊇ B n note first that (cid:90) { E n [ Y n +1 ]= } Y n +1 dP = (cid:90) { E n [ Y n +1 ]= } E n [ Y n +1 ] dP = (cid:90) { E n [ Y n +1 ]= } dP (27)and hence that Y n +1 = on B n . This together with the fact that B n ∈ F n implies B n E n (cid:20)(cid:18) − Y n +1 (cid:19) β n +1 η n +1 (cid:21) = E n (cid:20) B n (cid:18) − Y n +1 (cid:19) β n +1 η n +1 (cid:21) = 0 . To prove C n ⊆ B n , observe that C n ∈ F n and that C n ⊆ (cid:26)(cid:18) − Y n +1 (cid:19) β n +1 η n +1 = 0 (cid:27) = (cid:26) Y n +1 = 12 (cid:27) (by an argument similar to (27)) since β n +1 , η n +1 > and Y n +1 ≤ a.s. It thus holdsthat C n E n [ Y n +1 ] = E n [1 C n Y n +1 ] = 1 C n . From C n = B n together with (26) we obtain (cid:26) Y n = 12 (cid:27) = (cid:26) E n [ Y n +1 ] = 12 , E n [ Y n +1 β n +1 ] = 12 (cid:27) . Furthermore, we have B n E n [ Y n +1 β n +1 ] = E n [1 B n Y n +1 β n +1 ] = 1 B n E n [ β n +1 ] , and hence (cid:26) Y n = 12 (cid:27) = (cid:26) E n [ Y n +1 ] = 12 , E n [ β n +1 ] = 1 (cid:27) . Proof of Corollary 4.3.
We fix n ∈ Z ∩ ( −∞ , N − .1. The claim follows from (cid:26) Y n = 12 (cid:27) ⊆ (cid:26) E n [ Y n +1 ] = 12 (cid:27) ⊆ (cid:26) Y n +1 = 12 (cid:27) , where the first inclusion is immediate from Proposition 4.1 and the second onefollows from the facts that Y n +1 ≤ a.s. and (27).2. Due to Proposition 4.1 only the inclusion { E n [ β n +1 ] ≥ } ⊆ { E n [ η n +1 ] > } needsto be proved. By the Cauchy-Schwarz inequality and the assumption E n (cid:104) β n +1 η n +1 (cid:105) < a.s. we get ( E n [ β n +1 ]) ≤ E n (cid:20) β n +1 η n +1 (cid:21) E n [ η n +1 ] < E n [ η n +1 ] a.s. , which implies the claim. 20 . Closing the position in one go Let the assumptions of Theorem 2.1 be in force. Let n ∈ Z ∩ ( −∞ , N − . We nowstudy when ξ ∗ n ( x, d ) = − x for all x, d ∈ R , i.e., when it is optimal to close the wholeposition at time n < N .Recall that, for each x, d ∈ R , a version of the optimal trade ξ ∗ n ( x, d ) (which is definedup to a P -null set) is given by the right-hand side of (8). We choose the versions in sucha way that the random field ( x, d ) (cid:55)→ ξ ∗ n ( x, d ) is continuous (the most natural choice inview of (8)). Then we have { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } = { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ Q } = (cid:92) x,d ∈ Q { ξ ∗ n ( x, d ) = − x } , hence { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } is an F n -measurable event (as a countable intersectionof such events). Lemma 5.1.
Let n ∈ Z ∩ ( −∞ , N − . Under the assumptions of Theorem 2.1 we have { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } = (cid:26) E n (cid:20)(cid:18) Y n +1 − (cid:19) β n +1 η n +1 − Y n +1 β n +1 + 12 (cid:21) = 0 (cid:27) , (28) up to a P -null set.Proof. The result follows from (8) via a straightforward calculation.The next result presents a relation between the previously studied question of nonex-istence of profitable round trips for d (cid:54) = 0 and the currently studied question of closingthe position in one go. Proposition 5.2.
Let n ∈ Z ∩ ( −∞ , N − . Under the assumptions of Theorem 2.1 wehave1. { Y n = } ⊆ { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } .2. { Y n = } = { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } ∩ { E n [ Y n +1 ] = } . It is worth noting that the inclusion in part 1 can be strict in the sense that the setdifference can be non-negligible, i.e., with positive probability there are profitable roundtrips at time n for d (cid:54) = 0 and still it is optimal to close the whole position at time n (seeExample 5.5 below). Proof.
1. Recall that by Proposition 4.1 and Corollary 4.3 we have (cid:26) Y n = 12 (cid:27) = (cid:26) E n [ Y n +1 ] = 12 , E n [ β n +1 ] = 1 (cid:27) ⊆ (cid:26) Y n +1 = 12 (cid:27) . In particular, on the event { Y n = } ∈ F n it holds Y n +1 = and E n [ β n +1 ] = 1 , whichimplies that on the event { Y n = } ∈ F n we have E n (cid:20)(cid:18) Y n +1 − (cid:19) β n +1 η n +1 − Y n +1 β n +1 + 12 (cid:21) = 0 . ⊆ ” follows from the previous part together with Proposition 4.1. Toprove the reverse inclusion “ ⊇ ” we first note that (cid:26) E n [ Y n +1 ] = 12 (cid:27) ⊆ (cid:26) Y n +1 = 12 (cid:27) (29)because Y n +1 ≤ a.s. It follows from (28) and (29) that on the F n -measurable set A n := { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } ∩ (cid:26) E n [ Y n +1 ] = 12 (cid:27) it holds E n [ β n +1 ] = E n [ Y n +1 β n +1 ] = , i.e., E n [ β n +1 ] = 1 . Hence, A n ⊆ (cid:26) E n [ Y n +1 ] = 12 , E n [ β n +1 ] = 1 (cid:27) = (cid:26) Y n = 12 (cid:27) , where the set equality is again Proposition 4.1. This concludes the proof. Corollary 5.3.
Under the assumptions of Theorem 2.1 it holds (cid:26) Y N − = 12 (cid:27) = { ξ ∗ N − ( x, d ) = − x ∀ x, d ∈ R } . Proof.
This follows from part 2 of Proposition 5.2 because Y N = .We now provide more details for the case of processes with independent multiplicativeincrements of Section 3. We recall that in this case the process Y is deterministic.Notice, however, that the trades ξ ∗ n ( x, d ) are still, in general, random because of therandomness in γ n , see (8). Proposition 5.4.
Let n ∈ Z ∩ ( −∞ , N − . Under the assumptions of Corollary 3.1 itholds:1. { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } is either Ω or ∅ .2. The following statements are equivalent:(i) { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } = Ω .(ii) There exist x, d ∈ R with P ( γ n x (cid:54) = d ) > such that { ξ ∗ n ( x, d ) = − x } = Ω .(iii) It holds that E [ β n +1 ] = 1 + (cid:16) − E (cid:104) β n +1 η n +1 (cid:105)(cid:17) (cid:0) − Y n +1 (cid:1) Y n +1 . (30)
3. Under (30) we have that E [ β n +1 ] ≥ and, if Y n +1 < , even that E [ β n +1 ] > . β and γ ), closing the position in one go is never optimal in the(usual) framework, where the resilience process β is assumed to be (0 , -valued.This raises the question of whether closing the position in one go can be optimal ingeneral (that is, beyond (PIMI)) with the resilience process β taking values in (0 , . Inour setting the answer is affirmative (see Example 5.6 below). It is worth noting that inthe related setting, where trading is constrained only in one direction and the process β is (0 , -valued, the answer is negative, i.e., closing the position in one go is neveroptimal (see Proposition A.3 in [14] and Proposition 5.6 in [13]). Proof.
1. Since Y is deterministic and η n +1 and β n +1 are independent of F n , Lemma 5.1yields { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } = (cid:26)(cid:18) Y n +1 − (cid:19) E (cid:20) β n +1 η n +1 (cid:21) − Y n +1 E [ β n +1 ] + 12 = 0 (cid:27) , (31)which can be either Ω or ∅ .2. The equivalence between (i) and (ii) is a direct calculation using (8) and thefact that the factor in front of ( x − dγ n ) on the right-hand side of (8) is deterministicunder our assumptions. The equivalence between (i) and (iii) follows from (31) via astraightforward calculation.3. The last statement is clear.We close the section with two examples announced above. Example 5.5.
Consider the processes β and γ satisfying the assumptions of Corol-lary 3.1 (in particular, (PIMI)) and, moreover, E [ β N ] (cid:54) = 1 and E [ β N − ] = 1 + (cid:16) − E (cid:104) β N − η N − (cid:105)(cid:17) (cid:0) − Y N − (cid:1) Y N − . (32)Below we present a specific choice of the parameters such that (32) is satisfied.As we are in the framework of (PIMI), the process Y is deterministic. Moreover, since E [ β N ] (cid:54) = 1 , we have Y N − ∈ (0 , ) (see Corollary 4.2). Recall that on { Y N − < } ( = Ω ,up to a P -null set) there exist profitable round trips when we start at time N − with d (cid:54) = 0 . In particular, P ( ξ ∗ N − (0 , d ) (cid:54) = 0) = 1 whenever d (cid:54) = 0 . (33)That is, even without an open position we trade at time N − as soon as d (cid:54) = 0 . For completeness we mention the explicit formula ξ ∗ N − (0 , d ) = E [ β N ] − E [ η N − β N + 1] dγ N − , which can be obtained from (8) via a direct calculation and yields an alternative proof of (33). { ξ ∗ N − ( x, d ) = − x ∀ x, d ∈ R } = Ω . (34)To summarize, the optimal strategy in this example is to close the position at time N − ,to build up a new position at time N − (at least if D ( N − − = ( d − γ N − x ) β N − ) (cid:54) = 0 )and to close this position at time N . Interestingly, such a phenomenon can only occur if E [ β N − ] > , and hence it cannot happen in the (usual) framework, where the resilienceprocess β is assumed to take values in (0 , .We, finally, remark that in this example the inclusion in part 1 of Proposition 5.2 fortime n = N − is strict (cf. (34) with the fact that { Y N − = } = ∅ , where the latterfollows from Y N − < and part 1 of Corollary 4.3). It remains to explain how we can satisfy (32). An easy specific example, where therequirements on β and γ listed above are satisfied, can be constructed with deterministicsequences β and γ . For instance, choose arbitrary deterministic γ N , γ N − > and β N ∈ (0 , √ η N ) \{ } . These inputs yield a deterministic Y N − ∈ (0 , ) (see Corollary 4.2).Take a sufficiently small a > such that aY N − − Y N − ∈ (0 , . Finally, set β N − = 1 + a and choose γ N − > to satisfy aY N − − Y N − = 1 − (1 + a ) η N − (recall that η N − = γ N − γ N − ). This choice gives us (32) together with β N − η N − < . Example 5.6.
In this example we consider a version of our model with three tradingperiods N − , N − and N , where the resilience process β is (0 , -valued and still it isoptimal at time N − to close the position in one go. To this end assume that F N − = {∅ , Ω } and F N − = σ ( γ N − ) and that we can specify the positive random variables γ N − , γ N and the (0 , -valued random variable β N in such a way that E N − (cid:104) β N η N (cid:105) < , (cid:16) − E N − (cid:104) β N η N (cid:105)(cid:17) − ∈ L ∞− and that Y N − and γ N − are strictly negatively correlated,i.e., E (cid:20) Y N − γ N − (cid:21) − E [ Y N − ] E (cid:20) γ N − (cid:21) < . (35) More generally, the inclusion in part 1 of Proposition 5.2 is strict whenever on a set of positiveprobability we have the phenomenon described in the previous paragraph. Indeed, an event, wheresuch a phenomenon happens, is a subset of { ξ ∗ n ( x, d ) = − x ∀ x, d ∈ R } \ { Y n = } because on { Y n = } we have Y n = Y n +1 = . . . = Y N − = (part 1 of Corollary 4.3) and hence ξ ∗ k ( x, d ) = − x for all x, d ∈ R and k ∈ { n, n + 1 , . . . , N − } (part 1 of Proposition 5.2), in particular, ξ ∗ k (0 , d ) = 0 for all such k and d ∈ R . β N − ∈ E (cid:104) Y N − γ N − (cid:105) E [ Y N − ] E (cid:104) γ N − (cid:105) , (36)and then define γ N − = E (cid:2) − Y N − β N − (cid:3) E (cid:104)(cid:0) − Y N − (cid:1) β N − γ N − (cid:105) . (37)Note that, indeed, β N − ∈ (0 , and γ N − > . Next, we verify that E (cid:104) β N − η N − (cid:105) < .By (36) it holds E [ β N − Y N − ] E (cid:104) γ N − (cid:105) > E (cid:104) Y N − γ N − (cid:105) . This implies E (cid:20) − β N − Y N − (cid:21) E (cid:20) β N − γ N − (cid:21) < E (cid:20)(cid:18) − Y N − (cid:19) β N − γ N − (cid:21) and hence γ N − = E (cid:2) − β N − Y N − (cid:3) E (cid:104)(cid:0) − Y N − (cid:1) β N − γ N − (cid:105) < E (cid:104) β N − γ N − (cid:105) . Since γ N − is deterministic and η N − = γ N − γ N − , we get E (cid:104) β N − η N − (cid:105) < .From (37) we obtain that E (cid:20)(cid:18) Y N − − (cid:19) β N − η N − − Y N − β N − + 12 (cid:21) = 0 . Therefore, it follows from Lemma 5.1 that for all x, d ∈ R it holds that ξ ∗ N − ( x, d ) = − x ,i.e., it is optimal to close the whole position at time N − .It remains to specify γ N − , γ N and β N such that E N − (cid:104) β N η N (cid:105) < , (cid:16) − E N − (cid:104) β N η N (cid:105)(cid:17) − ∈ L ∞− and that (35) is satisfied. To this end let γ N − be { , } -valued with P ( γ N − = 1) = p ∈ (0 , and P (cid:0) γ N − = (cid:1) = 1 − p . Define γ N = γ N − and β N = γ N − .Note that β N is (0 , -valued, γ N − , γ N > and η N = γ N − . Observe further that E N − [ β N ] = γ N − < γ N − and hence E N − (cid:104) β N η N (cid:105) = γ N − ≤ < and (cid:16) − E N − (cid:104) β N η N (cid:105)(cid:17) − ∈ L ∞− . By definition of β N − , we have E N − [ β N ] = γ N − . It therefore holds Y N − = 12 E N − [ η N ] − ( E N − [ β N ]) − E N − [ β N ] + E N − [ η N ] = 12 (cid:18) γ N − − γ N − (cid:19) . Since E [ γ N − ] = p + 12 (1 − p ) , E (cid:2) γ N − (cid:3) = p + 14 (1 − p ) and E (cid:20) γ N − (cid:21) = p + 2(1 − p ) ,
25e obtain (35): E (cid:20) Y N − γ N − (cid:21) − E [ Y N − ] E (cid:20) γ N − (cid:21) = 12 (cid:18) − E [ γ N − ] (cid:19) − (cid:18) E [ γ N − ] − E (cid:2) γ N − (cid:3)(cid:19) E (cid:20) γ N − (cid:21) = 12 516 (cid:0) p − p (cid:1) < . For completeness, we notice that the assumptions of Theorem 2.1 which were not ex-plicitly discussed above (e.g., β n , γ n , γ n ∈ L ∞− ) are trivially satisfied.Finally, Table 1 summarizes several mentioned qualitative effects and compares ourfindings with the literature. In this table, the term “one-directional trading” refers tosettings, where the trading is constrained in one direction, and the term “two-directionaltrading” refers to settings, where, like in the present paper, trading in both directions isallowed.Table 1: We compare different settings from the viewpoint of whether premature closureis possible. Columns 2–4 briefly describe the settings, while columns 5–6 providethe answers and references to the proofs. It is worth noting that setting 1 isstudied in [10] and [13], setting 2 in [14] and setting 3 in [2] (although thequestion of closing the position in one go is not explicitly considered in [2],hence the reference to our paper in the last column).One- or two-directionaltrading? β and γ de-terministic orstochastic? β (0 , -valued? Prematureclosurepossible? Reason1 one-directional deterministic yes no Proposition 5.6in [13]2 one-directional stochastic yes no Proposition A.3in [14]3 two-directional deterministic yes no Proposition 5.4in this paper4 two-directional deterministic no yes Example 5.5 inthis paper5 two-directional stochastic yes yes Example 5.6 inthis paper6 two-directional stochastic no yes trivial (followsfrom 4 or 5) A. Proof of Theorem 2.1
Proof.
We first prove (7) and (8) by backward induction on n ∈ Z ∩ ( −∞ , N ] . For thebase case n = N note that for all x, d ∈ R it holds that V N ( x, d ) = − ( d − γ N x ) x = N (cid:16) dγ N − x (cid:17) − d γ N . In particular, it holds that Y N = > . Besides that, we have thatfor all x, d ∈ R , ξ ∗ N ( x, d ) = − x is the unique element of A N ( x ) and hence optimal.Consider now the induction step Z ∩ ( −∞ , N ] (cid:51) n + 1 → n ∈ Z ∩ ( −∞ , N − . Forall x, d ∈ R let a n = γ n E n (cid:20) Y n +1 η n +1 ( β n +1 − η n +1 ) + 12 (cid:18) − β n +1 η n +1 (cid:19)(cid:21) ,b n ( x, d ) = E n (cid:20) d (cid:18) − β n +1 η n +1 (cid:19) + 2 Y n +1 (cid:18) β n +1 η n +1 − (cid:19) ( β n +1 d − γ n +1 x ) (cid:21) ,c n ( x, d ) = E n (cid:20) Y n +1 γ n +1 ( β n +1 d − γ n +1 x ) − d β n +1 γ n +1 (cid:21) . (38)Note that for all x, d ∈ R the random variables a n , b n ( x, d ) and c n ( x, d ) are well-definedand finite because all factors and summands are in L ∞− due to the assumption thatfor all k ∈ Z ∩ ( −∞ , N ] , it holds β k , γ k , γ k ∈ L ∞− , and the induction hypothesis
0) = 1 γ n E n [ Y n +1 γ n +1 ] = E n [ η n +1 Y n +1 ] , γ n b n (1 ,
0) = 1 γ n E n (cid:20) − Y n +1 (cid:18) β n +1 η n +1 − (cid:19) γ n +1 (cid:21) = − E n [ Y n +1 ( β n +1 − η n +1 )] . (40)This together with the induction hypothesis Y n +1 ≤ implies that Y n = 1 γ n (cid:18) c n (1 , − b n (1 , a n (cid:19) ≤ γ n (cid:32) c n (1 , − b n (1 , a n + a n (cid:18) b n (1 , a n − (cid:19) (cid:33) = 1 γ n ( a n − b n (1 ,
0) + c n (1 , E n (cid:20) β n +1 η n +1 (cid:18) Y n +1 − (cid:19)(cid:21) ≤ . (41)27et S n be the set of all real-valued F n -measurable random variables ξ ∈ L . Thedynamic programming principle and the induction hypothesis ensure for all x, d ∈ R that V n ( x, d )= ess inf ξ ∈S n (cid:104)(cid:16) d + γ n ξ (cid:17) ξ + E n [ V n +1 ( x + ξ, ( d + γ n ξ ) β n +1 )] (cid:105) = ess inf ξ ∈S n (cid:20)(cid:16) d + γ n ξ (cid:17) ξ + E n (cid:20) Y n +1 γ n +1 (( d + γ n ξ ) β n +1 − γ n +1 ( x + ξ )) − ( d + γ n ξ ) β n +1 γ n +1 (cid:21)(cid:21) = ess inf ξ ∈S n (cid:34)(cid:16) d + γ n ξ (cid:17) ξ + E n (cid:34) γ n +1 Y n +1 (cid:18)(cid:18) β n +1 γ n γ n +1 − (cid:19) ξ + dβ n +1 γ n +1 − x (cid:19) − ( d + γ n ξ ) β n +1 γ n +1 (cid:35)(cid:35) = ess inf ξ ∈S n (cid:2) a n ξ + b n ( x, d ) ξ + c n ( x, d ) (cid:3) . (42)For all x, d ∈ R we find ξ ∗ n ( x, d ) = − b n ( x,d )2 a n to be the unique minimizer of ξ (cid:55)→ a n ξ + b n ( x, d ) ξ + c n ( x, d ) . Observe further that for all x, d ∈ R it holds that b n ( x, d ) = 2 da n γ n − E n [ Y n +1 ( β n +1 γ n − γ n +1 )] (cid:18) x − dγ n (cid:19) , (43)which yields the representation of ξ ∗ n ( x, d ) in (8). Clearly, for all x, d ∈ R the randomvariable ξ ∗ n ( x, d ) is F n -measurable. It remains to verify that for all x, d ∈ R we have ξ ∗ n ( x, d ) ∈ L ∞− .To show this we verify first that E n [ Y n +1 ( β n +1 − η n +1 )] E n (cid:104) Y n +1 η n +1 ( β n +1 − η n +1 ) + (cid:16) − β n +1 η n +1 (cid:17)(cid:105) ∈ L ∞− . (44)We have η n +1 ∈ L ∞− as η n +1 is the product of the two L ∞− -variables γ n +1 and γ n .Furthermore, we have that β n +1 ∈ L ∞− by assumption and that Y n +1 is bounded due tothe induction hypothesis. Hence, by the Minkowski inequality, it holds that ( E [ | Y n +1 ( β n +1 − η n +1 ) | p ]) p ≤ ( E [ | Y n +1 β n +1 | p ]) p + ( E [ | Y n +1 η n +1 | p ]) p < ∞ (45)for every p ∈ [1 , ∞ ) , so that E n [ Y n +1 ( β n +1 − η n +1 )] ∈ L ∞− . (46)Next we recall that α n ∈ L ∞− , where α n = 1 − E n (cid:104) β n +1 η n +1 (cid:105) , which implies E n (cid:104) Y n +1 η n +1 ( β n +1 − η n +1 ) + (cid:16) − β n +1 η n +1 (cid:17)(cid:105) ∈ L ∞− , (47) Note that our assumption that for all k ∈ Z ∩ ( −∞ , N ] it holds β k , γ k , γ k ∈ L ∞− and the fact that Y n +1 is bounded ensure that all conditional expectations in (42) are well-defined and that we canmove any ξ ∈ S n , γ n and γ n outside the conditional expectations. This reasoning also applies toother calculations in this proof.
28s the random variable in (47) is positive and smaller than α n . Together with (46) thisestablishes (44). Now (8) and (44) imply that ξ ∗ n ( x, d ) ∈ L ∞− for all x, d ∈ R , as x and d are deterministic and γ n ∈ L ∞− .By inserting the optimal trade size ξ ∗ n ( x, d ) = − b n ( x,d )2 a n into (42), we obtain for all x, d ∈ R that V n ( x, d ) = − b n ( x, d ) a n + c n ( x, d ) . (48)The dynamic programming principle ensures for all x, d, h ∈ R that V n ( x, d ) − (cid:16) d + γ n h (cid:17) h = ess inf ξ ∈S n (cid:104)(cid:16) d + γ n ξ (cid:17) ξ − (cid:16) d + γ n h (cid:17) h + E n [ V n +1 ( x + ξ, ( d + γ n ξ ) β n +1 )] (cid:105) = ess inf ξ ∈S n (cid:104)(cid:16) d + γ n ξ + h ) (cid:17) ( ξ − h ) + E n [ V n +1 ( x + ξ, ( d + γ n ξ ) β n +1 )] (cid:105) = ess inf ˜ ξ ∈S n (cid:104)(cid:16) d + γ n h + γ n ξ (cid:17) ˜ ξ + E n (cid:104) V n +1 ( x + h + ˜ ξ, ( d + γ n ( h + ˜ ξ )) β n +1 ) (cid:105)(cid:105) = V n ( x + h, d + γ n h ) . (49)This implies for all x, d ∈ R that ( ∂ x V n )( x, d ) + γ n ( ∂ d V n )( x, d ) ← V n ( x + h, d + γ n h ) − V n ( x, d ) h = − (cid:16) d + γ n h (cid:17) → − d (50)as h → . In particular, we obtain that ( ∂ xx V n )(0 ,
0) + γ n ( ∂ dx V n )(0 ,
0) = 0 and ( ∂ xd V n )(0 ,
0) + γ n ( ∂ dd V n )(0 ,
0) = − . (51)It follows from (48) and (38) that, for almost all ω , V n is a quadratic function in ( x, d ) ∈ R with V n (0 ,
0) = 0 . This together with (51) proves that V n ( x, d ) = ( ∂ xx V n )(0 , x + [( ∂ dx V n )(0 , xd + ( ∂ dd V n )(0 , d = ( ∂ xx V n )(0 , (cid:18) dγ n − x (cid:19) − d γ n . (52)Moreover, it follows from (48) that ( ∂ xx V n )(0 , E n [ γ n +1 Y n +1 ] − ( E n [ Y n +1 ( β n +1 γ n − γ n +1 )]) a n = γ n Y n . (53)This together with (52) proves that V n ( x, d ) = Y n γ n ( d − xγ n ) − d γ n for all x, d ∈ R .In the remainder of the proof we show that for all n ∈ Z ∩ ( −∞ , N − , x, d ∈ R theprocess ξ ∗ = ( ξ ∗ k ) k ∈{ n,...,N } recursively defined by (9) is in A n ( x ) . To this end we showby (forward) induction on k ∈ { n, . . . , N } that ξ ∗ k is F k -measurable and belongs to L for all k ∈ { n, . . . , N } . 29or the base case k = n we have ξ ∗ n = ξ ∗ n ( x, d ) which is already known to be in S n forall x, d ∈ R , i.e., ξ ∗ n is F n -measurable and ξ ∗ n ∈ L .Continue with the induction step { n, . . . , N − } (cid:51) k − → k ∈ { n + 1 , . . . , N − } .Now, the optimal trade size ξ ∗ k at time k depends on the current value of the positionpath X ∗ k − = x + (cid:80) k − i = n ξ ∗ i and the current deviation D ∗ k − . By induction on k , it holdsthat ξ ∗ i is in L and F i -measurable for all i ∈ { n, . . . , k − } . This yields that X ∗ k − belongs to L and is F k -measurable. Furthermore, the fact that ξ ∗ i ∈ L for all i ∈ { n, . . . , k − } allows us to use Remark 1.1 to obtain that D ∗ k − ∈ L as well. Besidesthat, it can be seen from (4) that D ∗ k − is F k -measurable given that ξ ∗ i is F i -measurablefor all i ∈ { n, . . . , k − } and β and γ are adapted processes. Hence, ξ ∗ k ( X ∗ k − , D ∗ k − ) = E k [ Y k +1 ( β k +1 − η k +1 )] E k (cid:104) Y k +1 η k +1 ( β k +1 − η k +1 ) + (cid:16) − β k +1 η k +1 (cid:17)(cid:105) (cid:18) X ∗ k − − D ∗ k − γ k (cid:19) − D ∗ k − γ k (54)is F k -measurable. To prove that ξ ∗ k ( X ∗ k − , D ∗ k − ) ∈ L , note that by the Minkowskiinequality, it suffices to show that each summand is in L . To begin with, it holds that D ∗ k − γ k ∈ L due to Lemma B.2 and γ k ∈ L ∞− . It further follows with (44) and LemmaB.2 that E k [ Y k +1 ( β k +1 − η k +1 )] E k (cid:104) Y k +1 η k +1 ( β k +1 − η k +1 ) + (cid:16) − β k +1 η k +1 (cid:17)(cid:105) D ∗ k − γ k ∈ L . (55)Similarly, E k [ Y k +1 ( β k +1 − η k +1 )] E k (cid:104) Y k +1 η k +1 ( β k +1 − η k +1 ) + (cid:16) − β k +1 η k +1 (cid:17)(cid:105) X ∗ k − ∈ L . (56)This finishes the induction step { n, . . . , N − } (cid:51) k − → k ∈ { n + 1 , . . . , N − } .Finally, it follows that for all x, d ∈ R it also holds true that ξ ∗ N = − X ∗ N − = − x − (cid:80) N − i = n ξ ∗ i is in L and F N -measurable. As a result, ξ ∗ ∈ A n ( x ) for all x, d ∈ R .The proof of Theorem 2.1 is thus completed. B. Integrability
Lemma B.1.
Let
X, Y ∈ L ∞− . Then, XY also belongs to L ∞− .Proof. Let p ∈ [1 , ∞ ) . The Cauchy-Schwarz inequality yields E [ | XY | p ] = E [ | X | p | Y | p ] ≤ (cid:0) E (cid:2) | X | p (cid:3)(cid:1) · (cid:0) E (cid:2) | Y | p (cid:3)(cid:1) < ∞ (57)since X, Y ∈ L p . Therefore, XY ∈ L p . This is true for every p ∈ [1 , ∞ ) , hence XY ∈ L ∞− . Lemma B.2.
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