Optimal Order Scheduling for Deterministic Liquidity Patterns
OOptimal Order Scheduling for DeterministicLiquidity Patterns
Peter Bank and Antje FruthTechnische Universit¨at BerlinInstitut f¨ur MathematikStraße des 17. Juni 136, 10623 Berlin, Germany([email protected])July 8, 2018
Abstract
We consider a broker who has to place a large order which con-sumes a sizable part of average daily trading volume. The broker’saim is thus to minimize execution costs he incurs from the adverseimpact of his trades on market prices. By contrast to the previousliterature, see, e.g., Obizhaeva and Wang [7], Predoiu et al. [8], weallow the liquidity parameters of market depth and resilience to varydeterministically over the course of the trading period. The resultingsingular optimal control problem is shown to be tractable by methodsfrom convex analysis and, under minimal assumptions, we constructan explicit solution to the scheduling problem in terms of some concaveenvelope of the resilience adjusted market depth.
Keywords:
Order scheduling, liquidity, convexification, singular control,convex analysis, envelopes, optimal order execution
It is well-known that market liquidity exhibits deterministic intraday pat-terns; see, e.g., Chordia et al. [3] or Kempf and Mayston [6] for some em-pirical investigations. The academic literature on optimal order scheduling,1 a r X i v : . [ q -f i n . T R ] O c t owever, mostly considers time-invariant specifications of market depth andresilience; cf. Obizhaeva and Wang [7], Alfonsi et al. [2], Predoiu et al. [8].It thus becomes an issue how to account for time-varying specifications ofthese liquidity parameters when minimizing the execution costs of a tradingschedule.Using dynamic programming techniques and calculus of variations, thisproblem was addressed by Fruth et al. [5]. These authors show that under cer-tain additional assumptions on these patterns there is a time-dependent levelfor the ratio of the number of orders still to be scheduled and the current mar-ket impact which signals when additional orders should be placed. Explicitsolutions are provided for some special cases where the broker is continuallyissuing orders. The thesis [4] discusses conditions under which the order sig-nal structure persists in case of stochastically varying liquidity parameters.Acevedo and Alfonsi [1] use backward induction arguments in discrete timeand then pass to continuous time to compute optimal policies for nonlinearspecifications of market impacts which are scaled by a time-dependent factorsatisfying some strong regularity conditions. In their approach order sched-ules are allowed in principle to sell and buy along the way, regardless of thesign of the desired terminal position, and they proceed to identify conditions(deemed to ensure absence of market manipulation strategies) under whichoptimal schedules will not do so. Optimal schedules are then obtained onlyunder a strong assumption linking resilience and market depth to each otheralong with their time derivatives.By contrast to these approaches, we focus from the outset on pure buyingor selling schedules and show how to reduce our optimization problem to aconvex one. Hence, we do not have to impose conditions ensuring that ordersare scheduled in certain ways at certain times. Instead, optimal order sizesand times are derived endogenously from the structure of market depth andresilience alone. This is made possible by the use of convex analytic first-ordercharacterizations of optimality which we show are intimately related to theconstruction of generalized concave envelopes of a resilience-adjusted formof market depth. Under minimal assumptions, this allows us to characterizewhen optimal schedules exist and, if so, to construct them explicitly in termsof these envelopes. We illustrate our findings by recovering the analyticsolution of Obizhaeva and Wang [7] and we show how optimal schedulesdepend on fluctuations in market depth and the level of resilience. It turnsout that with time-varying market depth optimal order schedules do not haveto consist of big initial and terminal trades with infinitesimal ones in between2s typically found in the previous literature. We also find that lower resiliencewill let optimal schedules focus more on (local) maxima of market depth tothe extent that with no resilience optimal schedules trade only when marketdepth is at its global maximum. We consider a broker who has to place an order of a total number of x > X = ( X t ) t ≥ , a right-continuous increasing process with X − (cid:44)
0, the resulting mark-up evolvesaccording to the dynamics(1) η X − (cid:44) η ≥ , dη Xt = dX t δ t − r t η Xt dt where δ t describes the market’s depth at time t ≥ r t measuresits current resilience. Thus, in our model market impact is taken to be alinear function of order size, the slope at any one time being determined bythe market depth. Moreover, market impact decays over time at the ratespecified by the market’s resilience.Clearly, (1) has the right-continuous solution(2) η Xt (cid:44) (cid:18) η + (cid:90) [0 ,t ] ρ s δ s dX s (cid:19) /ρ t with ρ t (cid:44) exp (cid:18)(cid:90) t r s ds (cid:19) , t ≥ , under Assumption 2.1.
The resilience pattern is given by a strictly positive andlocally Lebesgue-integrable function r : [0 , ∞ ) → (0 , ∞ ).3n the sequel we shall require furthermore Assumption 2.2.
The pattern of market depth δ : [0 , ∞ ) → [0 , ∞ ) isnonnegative, not identically zero, bounded and upper-semicontinuous withlim sup t ↑∞ δ t /ρ t = 0.The broker’s aim is to minimize the cumulative mark-up costs:(3) Minimize C ( X ) (cid:44) (cid:90) [0 , ∞ ) (cid:18) η Xt − + ∆ t X δ t (cid:19) dX t subject to X ∈ X where ∆ t X (cid:44) X t + − X t − and X (cid:44) { ( X t ) t ≥ right-cont., incr. : X − = 0 , X ∞ = x, C ( X ) < ∞} with the notation X ∞ (cid:44) lim t ↑∞ X . Remark .
1. Note that the ∆ t X δ t -term in (3) accounts for the costs anon-infinitesimal order will incur due to its own mark-up effect; cf., e.g.,Alfonsi et al. [2] or Predoiu et al. [8] who in addition show how costsfunctionals as in (3) emerge with stochastic reference prices evolving asmartingales when the broker is risk-neutral. Note also that, since welet X − (cid:44)
0, a value of X > X = X in the order schedule.2. To impose liquidation over a finite time horizon T ≥
0, one merelyhas to let the market depth δ t = 0 for t > T . Indeed, following theconvention that 1 / ∞ in the integration (2), η X and thus the costs C ( X ) will then be infinite for any order schedule X which increasesafter T .3. Strict positivity of r comes without loss of generality since if resilience r = 0 vanishes almost everywhere on an interval [ t , t ] there is no needto trade it off against market depth there and it is optimal to tradewhatever amount is to be traded at the moment(s) when market depth δ attains its maximum over this period; cf. Proposition 4.1. Hence, δ could be assumed to take this maximum value at t and the interval( t , t ] then be removed from consideration.4. The assumption of upper-semicontinuous market depth δ is necessaryto rule out obvious counterexamples for existence of optimal sched-ules. For unbounded upper-semicontinuous δ one can easily show that4nf X C = xη /ρ ∞ , and so there is no optimal schedule. The lim sup-condition is needed to rule out the optimality of deferring part of theorder indefinitely.5. Including a discount factor with locally Lebesgue-integrable discountrate ¯ r = (¯ r t ) ≥ (cid:101) δ t (cid:44) δ t exp( (cid:82) t ¯ r s ds ) and (cid:101) r t (cid:44) r t + ¯ r t , t ≥ δ and r above. The main result of this paper is the solution to problem (3). It describes upto what mark-up level our broker should be willing to place orders at anypoint in time in order to minimize mark-up costs:
Theorem 3.1.
Suppose Assumptions 2.1 and 2.2 hold, let λ t (cid:44) δ t /ρ t , (cid:101) λ t (cid:44) sup u ≥ t λ u and define (4) L ∗ t = inf u>t (cid:101) λ u − (cid:101) λ t (cid:101) λ u /ρ u − (cid:101) λ t /ρ t , t ≥ , where we follow the convention that / (cid:44) .Then the optimal order schedule strategy is to place orders at any time t ≥ if and while the resulting mark-up is no larger than y ∗ L ∗ t /ρ t , i.e., (5) X ∗ t = λ ( y ∗ L ∗ − η ) + + (cid:90) (0 ,t ] λ s d sup ≤ v ≤ s { ( y ∗ L ∗ v ) ∨ η } , t ≥ , provided the constant y ∗ > in (5) can be chosen such that X ∗∞ = x . Thisis the case if and only if the right side of (5) with y ∗ (cid:44) remains boundedas t ↑ ∞ . If this is not the case, we have inf X ∈ X C ( X ) = 0 and the problemdoes not have a solution. The following results outline the proof of this theorem and may be ofindependent interest. Our first auxiliary result provides a mathematicallymore convenient formulation of problem (3):
Proposition 3.2.
Suppose Assumptions 2.1 and 2.2 hold, let λ (cid:44) δ/ρ , κ (cid:44) λ/ρ = δ/ρ and define, for increasing and right-continuous Y = ( Y t ) t ≥ , K ( Y ) (cid:44) (cid:90) [0 , ∞ ) κ t d ( Y t ) . hen (6) Y t = η + (cid:90) [0 ,t ] dX s λ s , Y − (cid:44) η , and X t = (cid:90) [0 ,t ] λ s dY s , X − (cid:44) , t ≥ , define mappings from X to Y (cid:44) (cid:26) ( Y t ) t ≥ right-cont., incr. : Y − (cid:44) η , (cid:90) [0 , ∞ ) λ t dY t = x, K ( Y ) < ∞ (cid:27) and vice versa such that C ( X ) = K ( Y ) . As a result, with these choices of κ and λ , optimization problem (3) isequivalent to the following problem:(7) Minimize K ( Y ) (cid:44) (cid:90) [0 , ∞ ) κ t d ( Y t ) subject to Y ∈ Y . Neither problem (3) nor problem (7) is convex in general:
Proposition 3.3.
For upper-semicontinuous κ , the functional K = K ( Y ) of (7) is (strictly) convex for right-continuous, increasing Y with Y − = η if and only if κ is (strictly) positive and (strictly) decreasing. Convexity can always be arranged for, though, in the following sense:
Theorem 3.4.
Let λ , κ be as in Proposition 3.2. Then optimization prob-lem (7) has the same value as the convex optimization problem (8) Minimize (cid:101) K ( (cid:101) Y ) (cid:44) (cid:90) [0 , ∞ ) (cid:101) κ t d ( (cid:101) Y t ) subject to (cid:101) Y ∈ (cid:102) Y where (cid:101) κ t (cid:44) (cid:101) λ t /ρ t , (cid:101) λ t (cid:44) sup u ≥ t λ u , t ≥ , and (cid:102) Y (cid:44) (cid:26) ( (cid:101) Y t ) t ≥ right-cont., incr. : (cid:101) Y − (cid:44) η , (cid:90) [0 , ∞ ) (cid:101) λ t d (cid:101) Y t = x, (cid:101) K ( (cid:101) Y ) < ∞ (cid:27) . Moreover, any solution (cid:101) Y ∗ to (8) with { d (cid:101) Y ∗ > } ⊂ { (cid:101) λ = λ } will also be asolution to (7) . emark . For an increasing process Y = ( Y t ) t ≥ we say that t is a pointof increase towards the right and write dY t > Y t − < Y u for any u > t .A similar convention applies to decreasing processes and points of decreasetowards the right.The next proposition describes the (necessary and sufficient) first-orderconditions for optimality in problem (8). As one would expect, the broker hasto strike a balance between the impact of current orders on future mark-upcosts (as represented by the left side of (9) below) and the current prospecton future market conditions (as represented by the decreasing envelope (cid:101) λ ofmarket depth over resilience on the right side of that equation): Proposition 3.6.
For (cid:101) κ , (cid:101) λ ≥ as in Theorem 3.4, (cid:101) Y ∗ ∈ (cid:102) Y solves (8) ifand only if there is a constant y > such that (9) − (cid:90) [ t, ∞ ) (cid:101) Y ∗ u d (cid:101) κ u ≥ y (cid:101) λ t for t ≥ with ‘ = ’ whenever d (cid:101) Y ∗ t > . Constructing right-continuous increasing (cid:101) Y ∗ ≥ Theorem 3.7.
Under Assumptions 2.1 and 2.2, consider the level passagetimes τ k (cid:44) inf { t ≥ (cid:101) κ t ≤ k } and let (cid:101) Λ k (cid:44) kρ τ k , k ∈ (0 , (cid:101) κ ] and (cid:101) Λ (cid:44) .Then (cid:101) Λ is a continuous increasing map on [0 , (cid:101) κ ] . Its concave envelope (cid:98) Λ is absolutely continuous with a left-continuous, decreasing density ∂ (cid:98) Λ =( ∂ (cid:98) Λ k )
Corollary 3.8.
Under the assumptions of Theorem 3.7 and using its notationwe have the following dichotomy:In case | ∂ (cid:98) Λ | L (cid:44) ( (cid:82) (cid:101) κ ( ∂ (cid:98) Λ k ) dk ) < ∞ we can choose y ∗ > uniquelysuch that (10) X ∗ t (cid:44) λ ( y ∗ ∂ (cid:98) Λ (cid:101) κ − η ) + + (cid:90) (0 ,t ] λ s d (cid:110) ( y ∗ ∂ (cid:98) Λ (cid:101) κ s ) ∨ η (cid:111) , t ≥ , ncreases from X ∗ − (cid:44) to X ∗∞ = x ; this X ∗ ∈ X is an optimal orderschedule for problem (3) . In the special case where η = 0 , y ∗ = x/ | ∂ (cid:98) Λ | L and the minimal costs are given by C ( X ∗ ) = x / (2 | ∂ (cid:98) Λ | L ) .If, by contrast, | ∂ (cid:98) Λ | L = ∞ then we have inf X ∈ X C ( X ) = 0 and prob-lem (3) does not have a solution. Corollary 3.8 reduces the construction of optimal order schedules to the com-putation of a concave envelope. This can often be done in closed form,see, e.g., our treatment in Section 4.1 of the constant parameter case fromObizhaeva and Wang [7]. Alternatively, one can resort to highly efficientnumerical methods from discrete geometry to come up with solutions to es-sentially arbitrary liquidity patterns as we illustrate in Section 4.2.
Let us first show how to recover the solution of Obizhaeva and Wang [7] whoconsider a time horizon
T > δ t ≡ δ [0 ,T ] ( t ) andconstant market resilience r t ≡ r > t ≥
0. In this case we have λ t = (cid:101) λ t = δ e − r t [0 ,T ] ( t ) and κ t = (cid:101) κ t = δ e − r t [0 ,T ] ( t ) . Hence, ρ τ k = (cid:112) δ / ( k ∨ κ T ) and (cid:101) Λ k = (cid:112) δ k ∧ ( (cid:112) δ /κ T k ) , ≤ k ≤ δ . Thus, (cid:101)
Λ is its own concave envelope, i.e., (cid:101)
Λ = (cid:98)
Λ, and its left-continuousdensity is ∂ (cid:98) Λ k = (cid:40) (cid:112) δ /k, k > κ T , (cid:112) δ /κ T = e r T , k ≤ κ T . Obviously ∂ (cid:98) Λ is square integrable (and hence the problem is well-posed) ifand only if
T < ∞ . In that case, we compute (cid:98) Y t (cid:44) ∂ (cid:98) Λ (cid:101) κ t = (cid:40) (cid:112) δ /κ t = e r t , t < T, (cid:112) δ /κ T = e r T , t ≥ T, y > X yt (cid:44) δ (cid:18) y − η (cid:19) + + 12 yδ r ( t ∧ T − τ y ) + + 12 δ y [ T, ∞ ] ( t ) , t ≥ , with τ y (cid:44) (cid:16) r log η y (cid:17) + ∧ T, y > η e − r T ,T, η e − r T ≤ y ≤ η e − r T , ∞ y < η e − r T , is optimal for the total volume it trades. In particular, if η = 0, we find that X yt = yδ (cid:0) r ( t ∧ T ) + 1 [ T, ∞ ] ( t ) (cid:1) , t ≥ . So choosing y ∗ (cid:44) x/ ( δ (1 + r T / X ∗ = X y ∗ with X ∗∞ = x . Wetherefore recover the result of Obizhaeva and Wang [7]: If η = 0, i.e., if therehave been no previous orders, it is optimal to place orders of size y ∗ δ / t = 0 and t = T , and to place orders at the constant rate y ∗ δ r / ∆Λ(cid:142)(cid:61)ΛΚ(cid:142)(cid:61)Κ X (cid:42) T0 Figure 1: Optimal order schedule X ∗ (black) for constant market depth δ (blue), its resilience adjustment λ = (cid:101) λ (red), κ = (cid:101) κ (green) over a finitehorizon T . 9 .2 Time-varying market depth We next illustrate that the above order placement strategy of [7] is indeedstrongly dependent on constant market depth and resilience. Figure 2 belowexhibits how a fluctuating market depth affects the timing of the optimalorder placement as provided by Corollary 3.8. Note that we include a shut-down period for the market over the time period ( t , t ) when market depthvanishes. The corresponding concepts introduced by Theorem 3.7 are illus-trated in Figure 3 below. ∆ΛΛ(cid:142) Κ(cid:142) X (cid:42) T0 t t Λ(cid:142) (cid:72) t (cid:76) x Λ(cid:142) (cid:72) t (cid:43) (cid:76) Figure 2: A specification of market depth δ (blue) with finite horizon T , itsresilience adjustment λ (purple), the corresponding decreasing envelope (cid:101) λ (red) and (cid:101) κ (green) along with an optimal order schedule X ∗ (black).If we decrease the resilience parameter to r = 0, i.e., we assume perma-ment price impact of the broker’s orders, the focus on peaks of market depthsharpens to the extent that eventually only one huge order is placed whenmarket depth reaches its global maximum; see Figure 4. Proposition 4.1. If r ≡ and δ satisfies Assumption 2.2, the solutions tooptimization problem (3) are precisely those order schedules X ∗ ∈ X with { dX ∗ > } ⊂ arg max δ .Proof. When r ≡ ρ ≡ η Xt = η + (cid:82) [0 ,t ] dX s δ s ≥ η + X t max δ , t ≥ (cid:142)(cid:76)(cid:96)(cid:182)(cid:76)(cid:96) Κ(cid:142) (cid:72) T (cid:76) Κ(cid:142) (cid:72) (cid:76) Κ(cid:142) (cid:72) t (cid:43) (cid:76) Κ(cid:142) (cid:72) t (cid:76) Λ(cid:142) (cid:72) t (cid:76) x Λ(cid:142) (cid:72) t (cid:43) (cid:76) Figure 3: The decreasing envelope of resilience adjusted market depth (cid:101)
Λ(red), its concave envelope (cid:98)
Λ (orange) and the density ∂ (cid:98) Λ (black). ∆(cid:61)ΛΛ(cid:142) X (cid:42) T0 t t Figure 4: Optimal order schedule X ∗ (black) without market resilience andtime-varying market depth δ (blue). 11hus, C ( X ) ≥ η x + x δ , X ∈ X , with equality for all X ∗ ∈ X with { dX ∗ > } ⊂ arg max δ .Conversely, with high resilience, orders tend to be spread out more aroundlocal maxima of market depth as illustrated by Figure 5. Figures 2 and 5also show that the precise moments when it is optimal to issue orders wouldbe hard to guess in advance. Hence, an approach via classical calculus ofvariations as in Fruth et al. [5] or via the methods of Acevedo and Alfonsi[1] seems infeasible in these general cases. ∆ X (cid:42) T0 t t Figure 5: Optimal order schedule X ∗ (black) with strong market resiliencefor time-varying market depth δ (blue). We first prove that the original problem (3) can indeed be reformulated as (7)by giving the
Proof of Proposition 3.2
We first observe that for X ∈ X the mappingin (6) defines an increasing right-continuous Y with Y = ρη X . Because C ( X ) < ∞ , η X is dX -integrable and thus finite on { X < x } . Hence, Y is12nite on this set as well and we conclude dX = λ dY . It follows by elementarycalculus that K ( Y ) = C ( X ) and, thus, Y ∈ Y as desired.Conversely, for Y ∈ Y , κ = λ/ρ is d ( Y )-integrable. Since ρ > λ is locally dY -integrable and so X given by (6)is right-continuous and increasing with dX = λ dY . By the same reasoningas above this implies C ( X ) = K ( Y ) as well as X ∈ X .We next characterize when problem (7) is convex: Proof of Proposition 3.3 If κ is upper semi-continuous and decreasing,it is also left-continuous and we can use Fubini’s theorem to write K ( Y ) = 12 (cid:18) κ ∞ ( Y ∞ − η ) − (cid:90) [0 , ∞ ) ( Y t − η ) dκ t (cid:19) for any right-continuous increasing Y with Y − = η . Hence, K = K ( Y ) isobviously convex in such Y with strict convexity holding true on its domainfor strictly decreasing κ .Conversely, consider for 0 ≤ s < t < ∞ the function Y (cid:44) η + a [ s, ∞ ] + b [ t, ∞ ] . Then K ( Y ) = 12 (cid:0) κ s (( a + η ) − η ) + κ t (cid:0) ( a + b + η ) − ( a + η ) (cid:1)(cid:1) = 12 κ s a + κ t ab + 12 κ t b + η ( aκ s + bκ t )is convex in a, b > κ s ≥ κ t ≥
0, with strict inequalitiescorresponding to strict convexity.In order to prepare the proof of Theorem 3.4 let us recall that for anyincreasing Z : [0 , ∞ ) → R we let { dZ > } (cid:44) { t ≥ Z t − < Z u for all u > t } denote the collection of all points of increase towards the right. For a de-creasing Z we let { dZ < } (cid:44) { d ( − Z ) > } . In either case we let supp dZ denote the support of the measure dZ , i.e., the smallest closed set whosecomplement has vanishing dZ -measure. Lemma 5.1.
For upper-semicontinuous, bounded λ : [0 , ∞ ) → R , we havethat (cid:101) λ t (cid:44) sup u ≥ t λ u is left-continuous and decreasing with (11) { d (cid:101) λ < } ⊂ { (cid:101) λ = λ } . oreover, we have the partition (12) R = { d (cid:101) λ < } ∪ (cid:91) n ∈ N [ l n , r n ) ∪ (cid:91) n ∈ N ( l n , r n ) where ( l n , r n ) , n ∈ N , are the disjoint open intervals forming R \ supp d (cid:101) λ andwhere N = (cid:110) n ∈ N : l n ≥ , ∆ l n (cid:101) λ = 0 (cid:111) and N = N \ N .Proof. Left-continuity of (cid:101) λ and relation (11) are immediate. Note next that { d (cid:101) λ < } ⊂ supp d (cid:101) λ and therefore R \{ d (cid:101) λ < } ⊃ (cid:83) n ∈ N ( l n , r n ). Hence, todeduce partition (12) it suffices to observe that for n ∈ N we have l n (cid:54)∈ { d (cid:101) λ < } and that for t ≥ (cid:101) λ t = (cid:101) λ u for some u > t we have ( t, u ) ⊂ ( l n , r n )for some n ∈ N , and thus t ∈ ( l n , r n ) or t = l n with ∆ l n (cid:101) λ = 0.The main tool in the proof of Theorem 3.4 is the following Lemma 5.2.
Under the conditions of Theorem 3.4, we can find for anyincreasing, right-continuous Y ≥ η an increasing, right-continuous (cid:101) Y ≥ η such that (cid:101) Y ≤ Y and(i) (cid:82) [0 , ∞ ) λ t dY t = (cid:82) [0 , ∞ ) λ t d (cid:101) Y t ,(ii) { d (cid:101) Y > } ⊂ { d (cid:101) λ < } ,(iii) K ( Y ) ≥ K ( (cid:101) Y ) = (cid:101) K ( (cid:101) Y ) .Proof. We let I n , n ∈ N , denote the disjoint intervals of Lemma 5.1 formingthe complement of { d (cid:101) λ < } and we will use l n , r n to denote their respectiveboundaries. For the one interval I n whose left bound is l n = −∞ we nowredefine, for simplicity of notation, l n (cid:44) r n >
0; if, bycontrast, this I n is just the negative half line we can and shall remove itfrom consideration in the sequel. Similarly, if r n = ∞ for some n ∈ N , itfollows from Assumption 2.2 that δ t = λ t = κ t ≡ I n which thus can bedisregarded as well.Observe then that(13) sup I n λ = λ r n , by upper semi-continuity of λ and our choice when to include l n in I n andwhen not. 14et, for t ≥ (cid:101) Y t (cid:44) η + (cid:90) [0 ,t ] { d (cid:101) λ< } ( s ) dY s + (cid:88) n ∈ N,r n ≤ t (cid:90) I n λ s λ r n dY s . We first note that (cid:101) Y ≤ Y . Indeed Y t − (cid:101) Y t = (cid:90) [0 ,t ] R \{ d (cid:101) λ< } ( s ) ( dY s − d (cid:101) Y s )= (cid:88) n ∈ N,l n ≤ t (cid:18)(cid:90) I n ∩ [0 ,t ] dY s − [ r n , ∞ ) ( t ) (cid:90) I n λ s λ r n dY s (cid:19) is nonnegative because of (13).Assertion (i) is readily checked using the partition given by (12). Forassertion (ii) it suffices to observe that all r n , n ∈ N , are contained in { d (cid:101) λ < } . In order to prove assertion (iii), we first note that K ( (cid:101) Y ) = (cid:101) K ( (cid:101) Y ) is animmediate consequence of (ii) and (11). To establish K ( Y ) − K ( (cid:101) Y ) ≥ { d (cid:101) λ < }\ { r n : n ∈ N } we collect12 (cid:90) [0 , ∞ ) ∩ ( { d (cid:101) λ< }\{ r n : n ∈ N } ) κ t (cid:104) d ( Y t ) − d ( (cid:101) Y t ) (cid:105) = (cid:90) [0 , ∞ ) ∩ ( { d (cid:101) λ< }\{ r n : n ∈ N } ) κ t (cid:20)(cid:18) Y t − + 12 ∆ t Y (cid:19) dY t − (cid:18) (cid:101) Y t − + 12 ∆ t (cid:101) Y (cid:19) d (cid:101) Y t (cid:21) which is nonnegative because Y ≥ (cid:101) Y and because dY t = d (cid:101) Y t for t ∈ { d (cid:101) λ < }\ { r n : n ∈ N } by construction.From I n ∪ { r n } , n ∈ N , we get the contribution12 (cid:40)(cid:90) I n ∪{ r n } κ t d ( Y t ) − κ r n (cid:34)(cid:18) (cid:101) Y r n − + (cid:90) I n ∪{ r n } λ s λ r n dY s (cid:19) − (cid:101) Y r n − (cid:35)(cid:41) . . . ]-part can be written as12 (cid:34)(cid:18) (cid:101) Y r n − + (cid:90) I n ∪{ r n } λ s λ r n dY s (cid:19) − (cid:101) Y r n − (cid:35) = 12 (cid:18)(cid:90) I n ∪{ r n } λ s λ r n dY s (cid:19) + (cid:101) Y r n − (cid:90) I n ∪{ r n } λ s λ r n dY s = (cid:90) I n ∪{ r n } (cid:90) ( I n ∪{ r n } ) ∩ [ l n ,t ) λ s λ r n λ t λ r n dY s dY t + 12 (cid:88) ∆ t Y (cid:54) =0 ,t ∈ I n ∪{ r n } (cid:18) λ t λ r n (cid:19) (∆ t Y ) + (cid:101) Y r n − (cid:90) I n ∪{ r n } λ s λ r n dY s . Hence, using (13) again, we obtain with y n (cid:44) Y l n − if l n ∈ I n and y n (cid:44) Y l n otherwise that12 [ . . . ] ≤ (cid:90) I n ∪{ r n } ( Y t − − y n ) λ t λ r n dY t + 12 (cid:88) ∆ t Y (cid:54) =0 ,t ∈ I n ∪{ r n } λ t λ r n (∆ t Y ) + (cid:101) Y r n − (cid:90) I n ∪{ r n } λ s λ r n dY s ≤ (cid:90) I n ∪{ r n } ( Y t − − y n ) λ t λ r n dY t + 12 (cid:88) ∆ t Y (cid:54) =0 ,t ∈ I n ∪{ r n } λ t λ r n (∆ t Y ) + y n (cid:90) I n ∪{ r n } λ s λ r n dY s = 12 (cid:90) I n ∪{ r n } λ t λ r n d ( Y t )where the second estimate holds since (cid:101) Y r n − = (cid:101) Y l n ≤ y n because of (ii). Since ρ = λ/κ is increasing by assumption, we have λ t λ r n = ρ t ρ r n κ t κ r n ≤ κ t κ r n and thus 12 κ r n [ . . . ] ≤ (cid:90) I n ∪{ r n } κ t d ( Y t )as remained to be shown.With the preceding policy improvement lemma it is now easy to give the16 roof of Theorem 3.4 By Lemma 5.2 and using its notation, we can findfor any Y ∈ Y a (cid:101) Y ∈ (cid:102) Y ∩ Y such that (cid:101) K ( Y ) ≥ K ( Y ) ≥ K ( (cid:101) Y ) = (cid:101) K ( (cid:101) Y ) . As a result, inf Y K = inf (cid:101) Y (cid:101) K . Moreover, if (cid:101) Y ∗ ∈ (cid:102) Y attains the latterinfimum we can apply Lemma 5.2 to (cid:101) λ and (cid:101) K instead of λ and K to obtainanother optimal (cid:101) Y ∗∗ ∈ (cid:102) Y which satisfies in addition { d (cid:101) Y ∗∗ > } ⊂ { d (cid:101) λ < } .By Lemma 5.1, the latter set is contained in { λ = (cid:101) λ } = { κ = (cid:101) κ } and thusthis (cid:101) Y ∗∗ is also contained in Y and optimal for (7) as well.Let us next derive the first-order conditions of the convexified problem (8)in the Proof of Proposition 3.6
Recalling that (cid:101) κ ∞ = 0, we obtain by Fubini’stheorem(14) (cid:101) K ( Y ) = − (cid:90) [0 , ∞ ) ( Y t − η ) d (cid:101) κ t . For necessity, we observe that for any Y ∈ (cid:102) Y and 0 < ε ≤ ≤ (cid:101) K ( εY + (1 − ε ) Y ∗ ) − (cid:101) K ( Y ∗ )= − ε (cid:90) [0 , ∞ ) ( Y t − Y ∗ t ) Y ∗ t d (cid:101) κ t − ε (cid:90) [0 , ∞ ) ( Y t − Y ∗ t ) d (cid:101) κ t which, upon division by ε > ε ↓
0, yields that Y ∗ also solvesthe linear problem(15) Minimize − (cid:90) [0 , ∞ ) Y ∗ t Y t d (cid:101) κ t subject to Y ∈ (cid:102) Y . Equivalently, due to Fubini’s theorem, Y ∗ is a solution to the problem:(16) Minimize (cid:90) [0 , ∞ ) (cid:18) − (cid:90) [ t, ∞ ) Y ∗ u d (cid:101) κ u (cid:19) dY t subject to Y ∈ (cid:102) Y . As a consequence, Y ∗ can solve (15) only if dY ∗ t > t ≥ − (cid:82) [ t, ∞ ) Y ∗ u d (cid:101) κ u / (cid:101) λ t attains its infimum over { (cid:101) λ > } . Hence, thisinfimum is actually a minimum and is thus strictly positive. Denoting it by y > Y ∈ (cid:102) Y : (cid:101) K ( Y ) − (cid:101) K ( Y ∗ ) = − (cid:90) [0 , ∞ ) (( Y t ) − ( Y ∗ t ) ) d (cid:101) κ t ≥ − (cid:90) [0 , ∞ ) Y ∗ t ( Y t − Y ∗ t ) d (cid:101) κ t . The last term is nonnegative if Y ∗ solves (15), which due to the equivalenceof (15) and (16) amounts to our first-order condition (9).The construction of solutions to the first order conditions given in Theo-rem 3.7 can now be established: Proof of Theorem 3.7 (cid:101)
Λ is continuous on [0 , (cid:101) κ ] since so is k (cid:55)→ τ k becauseof the strict monotonicity of ρ and, thus, of (cid:101) κ on { (cid:101) κ > } . (cid:101) Λ is increasingbecause, along with (cid:101) κ t , also (cid:101) Λ (cid:101) κ t = (cid:101) κ t ρ t = (cid:101) λ t is decreasing in t ≥
0. Absolutecontinuity of the concave envelope (cid:98)
Λ follows from the continuity of (cid:101)
Λ.The monotonicity of (cid:101) Y ∗ is obvious from the monotonicity of (cid:101) κ and ∂ (cid:98) Λ.For its right-continuity note that lim t ↓ t (cid:101) Y ∗ t = ( y∂ (cid:98) Λ (cid:101) κ t ) ∨ η by left-continuityof ∂ (cid:98) Λ and its definition at 0. Hence, our assertion amounts to ∂ (cid:98) Λ k = ∂ (cid:98) Λ k where k (cid:44) (cid:101) κ t + and k (cid:44) (cid:101) κ t ≥ k . If k = k there is nothing to show.In case k < k , τ k = τ k for k ∈ [ k , k ) and, thus, (cid:101) Λ is linear with slope ρ τ k on this interval. As a consequence, (cid:98) Λ is linear there as well and, thus, ∂ (cid:98) Λ k = ∂ (cid:98) Λ k + by left-continuity of ∂ (cid:98) Λ. Hence, it suffices to show that thereis no downward jump in ∂ (cid:98) Λ at k . If there was such a jump then, by theproperties of concave envelopes, necessarily (cid:98) Λ k = (cid:101) Λ k and ∂ (cid:98) Λ k + ≥ ρ τ k .Hence, for k ≤ k we would have kρ τ k ≤ (cid:101) Λ k ≤ (cid:98) Λ k ≤ (cid:98) Λ k + ∂ (cid:98) Λ k + ( k − k ) ≤ kρ τ k , where the first estimate is due to the monotonicity of ρ , the second is theenvelope property of (cid:98) Λ, the third follows from its concavity and the last is aconsequence of the just derived properties of (cid:98)
Λ and ∂ (cid:98) Λ at k . We would thushave equality everywhere in the above estimates and in particular ∂ (cid:98) Λ k = ρ τ k ≤ ∂ (cid:98) Λ k + . This is a contradiction to the presumed downward jump of ∂ (cid:98) Λ at k .To verify that (cid:101) Y ∗ satisfies the first oder condition (9), let us first arguethat − (cid:90) [ t, ∞ ) (cid:101) Y ∗ u d (cid:101) κ u ≥ − y (cid:90) [ t, ∞ ) ∂ (cid:98) Λ (cid:101) κ u d (cid:101) κ u = y (cid:90) (cid:101) κ t ∂ (cid:98) Λ k dk = y (cid:98) Λ (cid:101) κ t ≥ y (cid:101) Λ (cid:101) κ t = y (cid:101) κ t ρ t = y (cid:101) λ t . (cid:101) Y ∗ . Thefirst identity follows by the change-of-time formula for Lebesgue-Stieltjes-integrals: just observe that (cid:101) κ ∞ = 0 by Assumption 2.2 and that ∂ (cid:98) Λ is con-stant on those intervals contained in [0 , (cid:101) κ ] which (cid:101) κ jumps across because (cid:101) Λ is linear on such intervals. The second identity follows from the absolutecontinuity of (cid:98)
Λ and because (cid:98) Λ = (cid:101) Λ = 0, again by Assumption 2.2. Thesecond estimate holds because (cid:98) Λ ≥ (cid:101) Λ by definition of concave envelopes andfor the last identity we note that τ (cid:101) κ t = t if (cid:101) κ t > (cid:101) κ t = (cid:101) λ t = 0 otherwise.Finally, we observe that d (cid:101) Y ∗ t > y∂ (cid:98) Λ (cid:101) κ has increasedabove η which ensures equality in the first of the above estimates. Equalityin the second holds for such t as well because if ∂ (cid:98) Λ (cid:101) κ increases at time t , ∂ (cid:98) Λmust decrease at (cid:101) κ t , and so (cid:101) Λ coincides with its concave envelope (cid:98)
Λ at thispoint.We are now in a position to wrap up and give the
Proof of Corollary 3.8
Let (cid:98) Y t (cid:44) ∂ (cid:98) Λ (cid:101) κ t , (cid:98) Y − (cid:44) Y yt (cid:44) ( y (cid:98) Y t ) ∨ η , t ≥ Y y − (cid:44) η .As a first step we check that(17) d (cid:98) Y t > t ≥ (cid:101) λ t = λ t . In fact, we will show that d (cid:101) λ t < t . If ∆ t (cid:101) κ <
0, this is obvious.So let us suppose that (cid:101) κ t + = (cid:101) κ t and assume that there is t > t such that (cid:101) λ t = (cid:101) λ t for t ∈ [ t , t ]. In that case, (cid:101) Λ is constant on the interval ( (cid:101) κ t , (cid:101) κ t ].Because d (cid:98) Y t >
0, the density ∂ (cid:98) Λ must decrease at k (cid:44) (cid:101) κ t + = (cid:101) κ t and so theenvelope (cid:98) Λ coincides with (cid:101)
Λ at this point. Concavity and monotonicity of (cid:98)
Λthen imply, however, that ∂ (cid:98) Λ = 0 around k , a contradiction to its decreasethere.Let us next prove that | ∂ (cid:98) Λ k | L < ∞ if and only if (cid:101) λ is d (cid:98) Y -integrable. Tosee this we argue that with ∂ (cid:98) Λ (cid:101) κ − (cid:44) (cid:90) [0 , ∞ ) λ t d (cid:98) Y t = (cid:90) [0 , ∞ ) (cid:101) Λ (cid:101) κ t d ( ∂ (cid:98) Λ (cid:101) κ t ) = (cid:90) [0 , ∞ ) (cid:98) Λ (cid:101) κ t d ( ∂ (cid:98) Λ (cid:101) κ t )= (cid:90) (cid:101) κ ∂ (cid:98) Λ l ∂ (cid:98) Λ (cid:101) κ τl dl = (cid:90) (cid:101) κ ( ∂ (cid:98) Λ l ) dl . Indeed, the first identity is just (17) and the definition of (cid:98) Y and (cid:101) Λ. Thesecond identity holds because (cid:101)
Λ = (cid:98)
Λ at points where ∂ (cid:98) Λ changes; the third19dentity follows from an application of Fubini’s theorem after writing (cid:98) Λ (cid:101) κ t = (cid:82) (cid:101) κ t ∂ (cid:98) Λ l dl and the last equality holds since ∂ (cid:98) Λ is left-continuous and constantover intervals that (cid:101) κ jumps across.So if | ∂ (cid:98) Λ | L < ∞ , then X yt (cid:44) (cid:82) [0 ,t ] λ dY y , t ≥
0, is real-valued, right-continuous and increasing in t . Moreover, X y ∞ is increasing in y ≥ X ∞ = 0 and X y ∞ ≥ X y → ∞ as y ↑ ∞ . In fact, X y ∞ = y (cid:82) [0 , ∞ ) λ d (cid:98) Y for y ≥ η / (cid:98) Y (where 0 / (cid:44) ∞ ) and, for y ∈ [ η / (cid:98) Y ∞ , η / (cid:98) Y ], X y ∞ = y (cid:82) [ τ y , ∞ ) λ d (cid:98) Y = (cid:82) [ τ y + , ∞ ) λ d (cid:98) Y where τ y (cid:44) inf (cid:110) t ≥ y > η / (cid:98) Y t (cid:111) . Hence, X y ∞ is in fact con-tinuous and strictly increasing from 0 to ∞ in y ≥ η / (cid:98) Y ∞ and we thus obtainexistence and uniqueness of y ∗ > X y ∗ ∞ = x . Hence, we can concludethat X ∗ (cid:44) X y ∗ is contained in X (and that thus the corresponding Y ∗ = Y y ∗ of (6) is contained in Y ) once we have established that K ( Y ∗ ) < ∞ . Forthis it suffices to observe that K ( Y ∗ ) ≤ ( y ∗ ) K ( (cid:98) Y ) and that by the samearguments as in our previous calculation of (cid:82) [0 , ∞ ) λ d (cid:98) Y we have K ( (cid:98) Y ) = (cid:101) K ( (cid:98) Y ) = 12 (cid:90) [0 , ∞ ) (cid:101) κ t d (cid:16) ( ∂ (cid:98) Λ (cid:101) κ t ) (cid:17) = 12 (cid:90) (cid:101) κ ( ∂ (cid:98) Λ l ) dl < ∞ . We next show that X ∗ and Y ∗ are optimal, respectively, for problem (3)and problems (7) and (8). In fact, due to Theorem 3.7, Y ∗ = ( y ∗ ∂ (cid:98) Λ (cid:101) κ ) ∨ η satisfies the first order condition (9) and, by Proposition 3.6, is thus optimalfor the convexified problem (8) provided that Y ∗ is also contained in (cid:102) Y . Tosee that even (cid:82) [0 , ∞ ) (cid:101) λ dY ∗ = x and to deduce the optimality of Y ∗ also forproblem (7) (and thus, by Proposition 3.2, optimality of X ∗ for the originalproblem (3)) it suffices by Theorem 3.4 to check { dY ∗ > } ⊂ { λ = (cid:101) λ } which, in fact, is immediate from (17).The formula for the minimal costs when η = 0 is an immediate conse-quence of our above computations for (cid:98) Y . It thus remains to show that ouroptimization problems do not have a solution if | ∂ (cid:98) Λ | L = ∞ . To see this notethat in this case there is, for any sufficiently large 0 ≤ S < T < ∞ , a schedule X S,T ∈ X which is optimal for δ S,T (cid:44) δ [ S,T ] instead of δ when η = 0. Thisfollows from our earlier results once we note that the corresponding concaveenvelope (cid:98) Λ S,T always has a bounded density because
T < ∞ , and thus a solu-tion to this finite time horizon problem exists provided its market depth doesnot vanish identically. This latter condition clearly holds for δ S,T when T ischosen sufficiently large, for otherwise δ ≡ S which would20ule out the presumed explosion of ∂ (cid:98) Λ at 0. Note that we can futhermorechoose
S, T ↑ ∞ such that (cid:98) Λ (cid:101) κ coincides with (cid:101) Λ (cid:101) κ at these points. This ensuresthat (cid:98) Λ = (cid:98) Λ S,T on [ (cid:101) κ T , (cid:101) κ S ] and hence | ∂ (cid:98) Λ S,T | L → (cid:82) (cid:101) κ ( ∂ (cid:98) Λ k ∧ (cid:101) κ S ) dk = ∞ as T ↑ ∞ .Now because C ( X S,T ) = (cid:90) [0 , ∞ ) η ρ t dX S,Tt + C ( X S,T ) ≤ η xρ S + C ( X S,T )where C ( X ) denotes the cost of any X ∈ X when η = 0, we obtaininf X C ≤ η xρ S + x | ∂ (cid:98) Λ S,T | L where we used our formula for the optimal costs C ( X S,T ). Because of ourspecial choice of
S, T , the second term vanishes for any fixed S as T ↑ ∞ .The first term vanishes for S ↑ ∞ because ρ has to be unbounded for ∂ (cid:98) Λ k toincrease to ∞ as k ↓
0. Indeed: ∂ (cid:98) Λ = sup k> (cid:101) Λ k /k = sup k> ρ τ k .Finally let us show how Theorem 3.1 follows from Corollary 3.8: Proof of Theorem 3.1
In view of Corollary 3.8 it suffices to show thatsup ≤ t ≤ s L ∗ t = ∂ (cid:98) Λ (cid:101) κ s , s ≥
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