When to Cross the Spread? - Trading in Two-Sided Limit Order Books -

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1 When o Cross he Spread? - rading in wo-sided Limi Order Books - Ulrich Hors and Felix Naujoka Insiu für Mahemaik Humbold-Universiä zu Berlin Uner den Linden 6, 0099 Berlin Germany {hors,naujoka}@mah.hu-berlin.de Absrac: In his aricle he problem of opimal rading in illiquid markes is addressed when he deviaions from a given sochasic arge funcion describing, for insance, exernal aggregae clien flow are penalised. Using echniques of singular sochasic conrol, we exend he resuls of NW o a wo-sided limi order marke wih emporary marke impac and resilience, where he bid ask spread is now also conrolled. In addiion o using marke orders, he rader can also submi orders o a dark pool. We firs show exisence and uniqueness of an opimal conrol. In a second sep, a suiable version of he sochasic maximum principle is derived which yields a characerisaion of he opimal rading sraegy in erms of a nonsandard coupled FBSDE. We show ha he opimal conrol can be characerised via buy, sell and no-rade regions. he new feaure is ha we now ge a nondegenerae no-rade region, which implies ha marke orders are only used when he spread is small. his allows o describe precisely when i is opimal o cross he bid ask spread, which is a fundamenal problem of algorihmic rading. We also show ha he conrolled sysem can be described in erms of a refleced BSDE. As an applicaion, we solve he porfolio liquidaion problem wih passive orders. AMS 000 subjec classificaions: 93E0, 9G80. Keywords and phrases: rading in illiquid markes, Dark pool, Sochasic maximum principle, Singular conrol.. Inroducion In modern financial insiuions, due o exernal regulaions, risk managemen requiremens or clien preferences, here are ofen imposed rading arges ha need o be followed. hese can ake he form of a curve giving he desired sock holdings over a given period of ime as a funcion of sochasic marke facors. ypical examples include porfolio liquidaion wih random exernal clien flow and -hedging under marke impac. In an idealised seing he rader would simply say on he arge (such as he aggregae exogenous clien flow or he Black-Scholes ). Prevening his are ofen associaed ransacion coss and limied availabiliy of liquidiy, hus he rader needs o balance he wo conflicing objecives of saying close o he arge and minimizing rading coss. In NW he problem of curve following in an order book model wih insananeous price impac and absoluely coninuous marke orders is solved. In paricular i is assumed ha rades have no lasing impac on fuure prices. However in limi order markes he bes bid and bes ask prices ypically recover only slowly afer large discree rades. In he presen work we exend heir resuls o a wo-sided limi order marke model wih emporary marke impac and resilience, where he price impac of rading decays only gradually. rading sraegies now include infiniesimally small ( coninuous ) as well as block ( discree ) rades, so ha we are in he framework of singular sochasic conrol. he singular naure of he marke order complicaes he analysis. he opimal marke order canno be characerised as he poinwise maximiser of he Hamilonian as in he absoluely coninuous case. Moreover, we now face an opimisaion problem wih consrains, since passive buy and sell orders are modelled separaely and boh are nonnegaive. A previous version of his paper circulaed under he ile: When o Cross he Spread - Curve Following wih Singular Conrol

2 U. Hors, N. Naujoka/When o Cross he Spread? Mehods of singular conrol have been applied in differen fields including he monoone follower problem as in BSW80, he consumpion-invesmen problem wih proporional ransacion coss in DN90 and finie fuel problems as in KOWZ00. Mos of hem rely on he dynamic programming approach or a maringale opimaliy principle. We shall prove a version of he sochasic maximum principle. In conras o dynamic programming, his does no require regulariy of he value funcion and provides informaion on he opimal conrol direcly. Maximum principles for singular sochasic conrol problems can be found in CH94, ØS0 and BM05, among ohers. hese resuls canno direcly be applied o he opimisaion problem under consideraion, since i involves jumps and sae dependen singular cos erms. he recen paper ØS0 provides necessary and sufficien maximum principles for jump diffusions wih parial informaion. Despie being fairly general, heir seup does no cover he paricular model we consider; insead we give a direc proof based on CH94 and he ideas developped in NW. We consider an invesor who wans o minimise he deviaion of his sock holdings from a prespecified arge funcion, which is driven by a vecor of unconrolled sochasic signals. he applicaions we have in mind are index racking, porfolio liquidaion, hedging and invenory managemen in he presence of exernal clien flow. Our problem can be seen as an exension of he monoone follower problem o an order book framework wih marke and passive orders. In our model, he invesor can use marke orders and simulaneously place passive orders. We hink of he passive orders as orders submied o a dark pool. hey generally incur lower rading coss ha marke orders bu heir execuion is uncerain. he rader hus faces a radeoff beween he penaly for deviaing from he arge funcion, he coss of marke order submission and he use of passive orders wih uncerain execuion imes. he invesor s marke orders widen he spread emporarily; he gap hen aracs new limi orders from oher marke paricipans and he spread recovers. he key decision he rader has o ake is he following: If he spread is small, rading is cheap and a marke order migh be beneficial. For large spreads however i migh be beer o sop rading and wai unil he spread recovers. When o cross he spread is a fundamenal quesion of algorihmic rading in limi order markes. An equivalen quesion would be when o conver a limi ino a marke order. o he bes of our knowledge, he problem of when o cross he bid ask spread has no been addressed in he mahemaical finance lieraure on limi order markes. OW05 and PSS0, for insance, consider porfolio liquidaion for a one-sided order book wih iniial spread zero and wihou passive orders; in his case i is opimal never o sop rading. Our order book model is inspired by OW05, a model which has recenly been generalised o arbirary shape funcions by AFS0 as well as PSS0 and sochasic order book heigh in Fru. While he menioned aricles focus on porfolio liquidaion, we consider here he more general problem of curve following and herefore need a wo-sided order book model. In addiion, we allow for passive orders. hese are orders wih random execuion which do no induce liquidiy coss, such as limi orders or orders placed in a dark venue. Our firs mahemaical resul is an a priori esimae on he conrol. For he proof, we reduce he curve following problem o an opimisaion problem wih quadraic penaly and wihou arge funcion and hen use a scaling argumen. his resul provides he exisence and uniqueness of an opimal conrol via a Komlós argumen. Nex we prove a suiable version of he sochasic maximum principle and characerise he opimal rading sraegy in erms of a coupled forward backward sochasic differenial equaion (FBSDE). he proof builds on resuls from CH94 and exends hem o he presen case where we have jumps, sae-dependen singular cos erms and general dynamics for he sochasic signal. Nex we give a second characerisaion of opimaliy in erms of buy, sell and no-rade regions. I urns ou ha here is always a region where he coss of rading are larger han he penaly for deviaing, so ha i is opimal o sop rading when he conrolled sysem is inside his region. his is in conras o NW, where only absoluely coninuous rading sraegies are allowed and a smoohness condiion on he cos funcion is imposed. I was shown herein ha under hese condiions he no-rade region Unlike he classical HJB approach, our proof of exisence does no need smoohness of he value funcion. Moreover, Ponryagins maximum principle provides a more explici characerisaion of he opimal rading sraegies han he more radiional dynamic programming approach.

3 U. Hors, N. Naujoka/When o Cross he Spread? 3 is degenerae, so ha he invesor always rades. In he presen model he no-rade region is defined in erms of a hreshold for he bid ask spread. We show ha spread crossing is opimal if he spread is smaller han or equal o he hreshold. If i is larger, hen no marke orders should be used. he hreshold is given semi-explicily in erms of he FBSDE and as a resul, we can precisely characerise when spread crossing is opimal for a large class of opimisaion problems. We will see ha marke orders are applied such ha he conrolled sysem remains inside (he closure of) he no-rade region a all imes, and ha is rajecory is refleced a he boundary. o make his precise, we show ha he adjoin process ogeher wih he opimal conrol provides he soluion o a refleced BSDE. In general i is difficul o solve he coupled forward backward SDE (or he corresponding Hamilon-Jacobi-Bellman quasi variaional inequaliy) explicily. his is due o he Poisson jumps (leading o nonlocal erms) and he singular naure of he conrol. For quadraic penaly funcion and zero arge funcion hough he soluion can be given in closed form. his corresponds o he porfolio liquidaion problem in limi order markes and exends he resul of OW05 o rading sraegies wih passive orders. he new feaure is ha he opimal sraegy is no deerminisic, bu adaped o passive order execuion, and he rading rae is no consan bu increasing in ime. he remainder of his paper is organised as follows: We describe he marke environmen and he conrol problem in Secion and show in Secion 3 ha a unique opimal conrol exiss. We hen provide wo characerisaions of opimaliy, firs via he sochasic maximum principle in Secion 4 and hen via buy, sell and no-rade regions in Secion 5. he link o refleced BSDEs is presened in Secion 6, and we discuss he applicaion o porfolio liquidaion in Secion 7.. he Model Le (Ω, F, {F(s) : s 0, }, P ) be a filered probabiliy space saisfying he usual condiions of righ coninuiy and compleeness and > 0 be he erminal ime. Assumpion.. he filraion is generaed by he following joinly independen processes, (i) A d-dimensional Brownian Moion W, d. (ii) wo one-dimensional Poisson processes N i wih respecive inensiies λ i for i =,. (iii) A compound Poisson process M on 0, R k wih compensaor m(dθ)ds, s.. m(r k ) <. he compensaed (compound) Poisson processes are denoed Ñi for i =, and M, respcively. In our model rading akes place in a wo sided limi order marke. here are hree price processes: a benchmark price process (D(s)) s which we assume o be a maringale, he bes ask price process which is above he benchmark, he bes bid price process which is always below he benchmark. On he buy side of he order book liquidiy is available for prices higher han he bes ask price, and we assume a block shaped disribuion of available liquidiy wih consan heigh κ > 0. his assumpion is also made in OW05; i is key for he curren approach as i leads o linear dynamics for he bid ask spread. Similarly, liquidiy is available on he sell side for prices lower han he bes bid. We assume a block shaped disribuion of liquidiy available on he sell side wih consan heigh κ > 0. he invesor s rades will have a emporary impac on he bes bid and ask prices. he benchmark price is hypoheical and canno be observed direcly in he marke. I represens he fair price of he underlying or a reference price in he absence of liquidiy coss such as he mid-quoe. We assume ha he benchmark price is unconrolled. A sylised snapsho of he order book and a ypical rajecory of he price processes are ploed in Figure. A fundamenal propery of illiquid markes is ha rades move prices. here is a large body of empirical lieraure on he price impac of rading, we refer he reader o KS7, HLM87, HLM90, BHS95 and AHL05.

4 U. Hors, N. Naujoka/When o Cross he Spread? 4 Volume PRICE Bes ask, bes bid and midquoe prices Bes Bid Price Benchmark Price Bes Ask Price Price IME Fig. (a) his sylised snapsho of he order book shows he bes bid, benchmark and bes ask price as well as liquidiy ha is available (dark) and consumed (ligh). (b) Here we see a ypical evoluion of he price processes over ime. he bes ask (red) is above he benchmark price (dashed black), which is above he bes bid price (blue). Marke buy (resp. sell) orders lead o jumps in he bes ask (resp. bid) price. In he absence of rading, he bes ask and bes bid converge o he benchmark... rading sraegies he invesor can apply marke buy (sell) orders o consume liquidiy on he buy (sell) side of he order book. His cumulaed marke buy (sell) orders are denoed by η (resp. η ). hese are nondecreasing càdlàg processes, and hence we allow for coninuous as well as discree rades and denoe by η i (s) η i (s) η i (s ) 0 for s 0, and i =, he jumps of η i. Such conrol processes are more general han absoluely coninuous rading sraegies and hey seem beer suied o describe real world rading sraegies, which are purely discree. In addiion, he invesor can use passive buy (sell) order volumes u (resp. u ). We assume ha passive orders are fully execued a he benchmark price a random poins in ime. hus a passive order always achieves a beer price han he corresponding marke order, however is execuion is uncerain. We hink of hem as orders placed in a dark pool. he class of admissible conrols is now defined for 0, as { U (η, u) :, Ω R R η i( ) = 0, E η i ( ) η i is nondecreasing, càdlàg and progressively measurable and } u i is predicably measurable, for i =,. u i (r) dr <, Each conrol consiss of he four componens η, η, u, u, each of hem being nonnegaive. In paricular, we face an opimisaion problem wih consrains. We noe ha η (s) (resp. η (s)) denoes he marke buy (resp. sell) orders accumulaed in, s. In conras, u (s) (resp. u (s)) represens he volume placed as a passive buy (resp. sell) order a ime s,... rading coss In our model here will be hree sources of rading coss associaed wih he use marke orders, passive order placemens, and he deviaion of he invesor s sock holding from some predescribed arge funcion as specified below.

5 U. Hors, N. Naujoka/When o Cross he Spread? 5... Marke orders Insead of modelling he bes bid and bes ask price direcly, we find i more convenien o work wih he buy and sell spreads insead. Specifically, we denoe by X he disance of he bes ask price o he benchmark price and call his process he buy spread. As in OW05 and AFS0 we assume exponenial recovery of he buy spread wih resilience parameer ρ > 0. he dynamics of he buy spread are hen given for s, by s X (s) X ( ) = ρ X (r)dr κ dη (r), X ( ) = x 0.,s As a convenion, we wrie,s for inegrals wih respec o he singular processes η i for i =, o indicae ha possible jumps a imes s and are included. Similarly, he sell spread X is defined as he disance of he bes bid price o he benchmark price and i saisfies s X (s) X ( ) = ρ X (r)dr κ dη (r), X ( ) = x 0. An immediae consequence is ha he spreads X and X are nonnegaive and rever o heir lower bound 0. As a consequence, he bes ask price is larger han or equal o he bes bid price. In our model, he invesor s marke buy orders have a emporary impac on he bes ask price, bu no on he bes bid (and vice versa). Passive orders do no move prices. Moreover, he price impac of rading decays over ime (resilience) and in he absence of rading he price processes converge o he benchmark price. Remark.. In he lieraure he bid ask spread is defined as he disance of he bes ask from he bes bid price; in our noaion his process is given by X X. In he seminal paper Kyl85 hree measures of liquidiy are defined, all of which are capured in he model we propose. Deph, he size of an order flow innovaion required o change prices a given amoun, is given by he parameers κ and κ which denoe he inverse order book heigh. Resiliency, he speed wih which prices recover from a random, uninformaive shock, is capured by he resilience parameers ρ and ρ. Finally, ighness, he cos of urning around a posiion over a shor period of ime, can be measured in erms of he bid ask spread X X. he coss of marke order execuion is measured relaive o he benchmark price, so he specific naure of he benchmark price process (D(s)) s 0, is no imporan a his poin. An infiniesimal marke buy order dη (r) is execued a he bes ask price; he coss of crossing he spread are X (r )dη (r). A discree buy order η (r) eas ino he block shaped order book and shifs he spread from X (r ) o X (r ) κ η (r). Is liquidiy coss are herefore given by ( X (r ) κ ) η (r) η (r). Alogeher, we arrive a he following cos funcional for marke order execuion. Definiion.3. he expeced rading coss from using marke orders is E,x X (r ) κ η (r) dη (r) X (r ) κ η (r) dη (r), (.), where he jump par is undersood as, for i =, E η i (r)dη i (r) = E ( η i (r)), r, ( ) E η i (r) E η i ( ) <. r,,s,

6 U. Hors, N. Naujoka/When o Cross he Spread? 6... Passive orders A jump of he Poisson process N i represens a liquidiy even which execues he passive order u i, for i =,. For simpliciy we consider full execuion only, i.e. only fill-or-kill orders are admissible. his assumpion is also made in Kra and in NW. 3 As a resul, he invesor s sock holdings a ime s s, are given by s s X 3 (s) X 3 ( ) = dη (r) dη (r) u (r)n (dr) u (r)n (dr),,s,s X 3 ( ) = x 3 R. Passive orders are execued a he benchmark price and hence do no incur direc rading coss. However, we allow for adverse selecion coss. Such coss are commonly observed in pracice. hey occur if he benchmark price moves in a favorable direcion afer he passive order is execued, making he execuion look less beneficial in hindsigh. 4 We posulae he following adverse selecion cos srucure. Definiion.4. Le γ i 0 for i =,. he expeced adverse selecion coss from rading passive buy/sell orders are E γ i u i (r)dr (.) 0 A linear penalizaion of passive orders of he above form has previously been used by Kra. herein he performance funcional in coninuous ime is no derived from firs principles, bu aken as he coninuous-ime analogue of he discree-ime case. he following example illusraes how such a linear cos erm can indeed be derived from firs principles. Example.5. Le us assume ha every ime he passive buy (resp. sell) order is execued, he benchmark prices jumps down (resp. up). he upward and downward jumps can modelled by compound Poisson processes M i for i =, whose jump imes agree 5 wih he jump imes of he Poisson processes N i. Denoe he compensaed Poisson maringales by M i (0, s A) M i (0, s A) λ i m i (A)s. Le he benchmark price wih adverse selecion coss be given by D(s) D(s) M (s) M (s). he process D is hen also a maringale which jumps down (up) if our passive buy (sell) order is execued. he key observaion is ha he passive buy order is execued before he jump of he benchmark price, and hus he passive buy orders incur he following coss: E u i (τ i,j ) D(τ i,j ), j where τ i,j denoes he j-h jump of M i for i =, and j N. We noe ha D(τ i,j ) = D(τ i,j ) M i (τ i,j ), so he above equals E j ;τ i,j u i (τ i,j ) D(τi,j ) M i (τ i,j ) = E u i (r) D(r) θ Mi (dθ, dr). 0 3 he assumpion of full execuion is key o our analysis. While an exension o proporional execuion is sraighforward, general parial execuions are no covered by our mehod. 4 See for insance Kra for a deailed analysis in he framework of porfolio liquidaion. 5 In oher words, M i is consruced from N i by replacing he jumps of size one by a sochasic jump size θ > 0 whose disribuion is given by m i (θ), for i =,.

7 U. Hors, N. Naujoka/When o Cross he Spread? 7 I follows ha adverse selecion leads o an addiional loss (relaive o he benchmark price D) of size E u i (r)θm i (dθ, dr). 0 By NW Lemma A.3, he process u i(r)θ M i (dθ, dr) is a maringale, so he expeced losses are of he form (.) wih..3. Deviaions from he arge funcion γ i λ i θm i (dθ). 0 he las and final cos erm capures cosly deviaions from some pre-specified arge funcion. Specifically, we assume ha he rader wans o minimise he deviaion of his sock holdings o a arge funcion α :, R n R. his funcion depends on a vecor of unconrolled sochasic signals Z wih dynamics given for s, by Z(s) Z( ) = s µ(r, Z(r))dr s s σ(r, Z(r ))dw (r) R k γ(r, Z(r ), θ) M(dr, dθ), Z( ) = z R n. where M(0, s A) M(0, s A) m(a)s denoes he compensaed Poisson maringale. Definiion.6. he coss of deviaing from he arge funcions are specified in erms of penaly funcions h, f : R R. Specifically, he expeced cos from curve-following is E,x,z h (X 3 (r) α(r, Z(r))) dr f (X 3 ( ) α(, Z( ))) (.3) In a model where Z represens exogenous clien flow, he rader s ne posiion a ime s, is X 3 (s) Z(s). In his case one migh choose α(r, z) = z and penalize her aggregae holdings by choosing h(x) = f(x) = x. Anoher possible inerpreaion is ha Z is a vecor whose enries represen boh clien flow and he benchmark price and f (X 3 ( ) α(, Z( )) represens eiher he cos of unwinding he erminal posiion a he marke price a ime or he deviaion of he acual sock holding from ha of a perfec hedge he conrol problem Having defined he sae processes and heir respecive dynamics, as well as he various coss of rading, we are now ready o formulae our conrol problem. he performance funcional is defined for (, x, z) 0, R 3 R n and a conrol (η, u) U as J(, x, z, η, u) E,x,z X (r ) κ η (r), γ u (r)dr γ u (r)dr dη (r) X (r ) κ, η (r) dη (r) h (X 3 (r) α(r, Z(r))) dr f (X 3 ( ) α(, Z( ))) and he opimisaion problem under consideraion is: (.4) 6 he problem of opimal hedging under marke impac and resilience has been sudied in a mean-variance seing in AG and in a game-heoreic framework in HN.

8 U. Hors, N. Naujoka/When o Cross he Spread? 8 Problem.7. Minimise J(η, u) over (η, u) U. For (, x, z) 0, R 3 R n he value funcion is defined as v(, x, z) = inf J(, x, z, η, u). (η,u) U Remark.8. Problem.7 is a singular sochasic conrol problem. Maximum principles for singular conrol are derived for insance in CH94, ØS0 and BM05. However, he above problem is no covered by heir resuls for several reasons. Firsly, i involves jumps. Secondly, he singular cos erms Xi (r ) κi, η i(r) dη i (r) for i =, depend on he sae variable and on he jumps of he conrol, which is no he case in he usual formulaion. he sandard seup only allows for cos erms of he form k(s, ω)dη(s). A hird difficuly in he presen model is ha he conrol u (he passive order) does no incur rading coss, so he sandard characerisaion as he poinwise maximiser of he Hamilonian does no apply. he recen aricle ØS0 provides necessary and sufficien maximum principles for he singular conrol of jump diffusions, where he singular cos erm may depend on he sae variable. However, hey do no allow for erms like, η i(r)dη i (r) and heir sufficien condiion is based on a convexiy condiion on he Hamilonian which is no saisfied in our specific case. Insead we give a direc proof based on CH94 and ideas used in NW. o ensure exisence and uniqueness of an opimal conrol, we impose he following assumpions. Here and hroughou, we wrie c for a generic consan, which migh be differen a each occurrence. Assumpion.9. (i) he penaly funcions f, h : R R are sricly convex, coninuously differeniable, nonnegaive and normalised in he sense f(0) = h(0) = 0. (ii) In addiion, f and h have a leas quadraic growh, i.e. here exiss ε > 0 such ha f(x), h(x) ε x for all x R. (iii) he funcions µ, σ and γ are Lipschiz coninuous, i.e. here exiss a consan c such ha for all z, z R n and s,, µ(s, z) µ(s, z ) R σ(s, z) σ(s, n z ) R n d γ(s, z, θ) γ(s, z, θ) R nm(dθ) c z z R n. R k In addiion, hey saisfy sup µ(s, 0) Rn σ(s, γ(s, 0, θ) 0) Rn d R nm(dθ) <. s R k (iv) he arge funcion α has a mos polynomial growh in he variable z uniformly in s, i.e. here exis consans c α, q > 0 such ha for all z R n, sup α(s, z) c α ( z q R n). s (v) he penaly funcions f and h have a mos polynomial growh. Remark.0. Le us briefly commen on hese assumpions. aking f and h nonnegaive is reasonable for penaly funcions. Normalisaion is no loss of generaliy, his may always be achieved by a linear shif of f, h and α. Quadraic growh of f and h is only needed in Lemma 3.4 for an a priori L -norm bound on he conrol, which is hen used for a Komlós argumen. he convexiy condiion leads naurally o a convex coercive problem which hen admis a unique soluion. Once he exisence of an opimal conrol is esablished, we need one furher assumpion. I guaranees he exisence and uniqueness of he adjoin process. Assumpion.. he derivaives f and h have a mos linear growh, i.e. for all x R we have f (x) h (x) c( x ).

9 3. Exisence of a Soluion U. Hors, N. Naujoka/When o Cross he Spread? 9 he aim of he presen secion is o show ha he performance funcional is sricly convex and ha i is enough o consider conrols wih a uniform L -norm bound. Combining hese resuls wih a Komlós argumen, we hen prove ha here is a unique opimal conrol. Henceforh we impose Assumpion.9. For he proof of exisence we also assume ha here are no adverse selecion coss, γ i = 0 for i =,. Due o he linear penalizaion of passive orders he exension o he general case wih adverse selecion is sraighforward. We begin wih some growh esimaes for he sae processes. his resul exends NW Lemma 4. o he singular conrol case. Lemma 3.. (i) For every p here exiss a consan c p such ha for every (, x, z) 0, R 3 R n we have E,x,z sup Z(s) p R c n p ( z p R n). s (ii) here exiss a consan c x such ha for any (η, u) U we have ( ) E,x,z sup X η,u (s) R c 3 x E,x,z η( ) R E u(r) R dr. s In paricular, X η,u has square inegrable supremum for all (η, u) U. Proof. he argumen is as in NW Lemma 4.. A firs consequence of he above lemma is ha he zero conrol incurs finie coss. Corollary 3.. he zero conrol incurs finie coss, i.e. for each (, x, z) 0, R 3 R n we have J(, x, z, 0, 0) <. Proof. his resul is a consequence of he polynomial growh of f, h and α ogeher wih Lemma 3.. We now show ha he performance funcional is sricly convex in he conrol, so ha mehods of convex analysis can be applied. Proposiion 3.3. he performance funcional (η, u) J(, x, z, η, u) is sricly convex, for every (, x, z) 0, R 3 R n. Proof. From he definiion of X i for i =, we have dη i (s) = dxi(s)ρixi(s)ds κ i. We use his o rewrie he performance funcional as J(, x, z, η, u) κ ρ X (r) dr κ X ( ) x =E,x,z X ( ) x κ h (X 3 (r) α(r, Z(r))) dr f (X 3 ( ) α(, Z( ))) ρ X (r) dr κ. (3.) he righ hand side is sricly convex in X. Due o he fac ha (η, u) X η,u is affine, i follows ha (η, u) J(, x, z, η, u) is sricly convex. he aim in his secion is o prove exisence and uniqueness of an opimal conrol. For he proof of his resul, we need wo auxiliary lemmaa. We firs show a quadraic growh esimae on he value funcion in Lemma 3.4. his exends Lemma 4. from NW o he singular conrol case. Lemma 3.4. here are consans c,, c, c 3 > 0 such ha for (, x, z) 0, R 3 R n v(, x, z) c, x 3 c ( z c3 R n).

10 U. Hors, N. Naujoka/When o Cross he Spread? 0 Proof. he idea is o use he growh condiions on he penaly funcions o reduce he opimisaion problem o simpler linear-quadraic problem, which can hen be esimaed in erms of x 3. Using he quadraic growh of f and h yields v(, x, z) inf (η,u) U E,x,z, X (r ) κ η (r) dη (r) X (r ) κ, η (r) dη (r) ε (X 3 (r) α(r, Z(r))) dr ε (X 3 ( ) α(, Z( ))). Nex an applicaion of he inequaliy (a b) a b leads o v(, x, z) inf E,x,z (η,u) U, X (r ) κ η (r) dη (r) εe,x,z ε X 3(r) dr ε X 3( ) X (r ) κ, η (r) dη (r) α(r, Z(r)) dr α(, Z( )). he polynomial growh of α coupled wih Lemma 3. provides he exisence of consans c, c 3 > 0 such ha v(, x, z) (3.) inf E,x,z X (r ) κ (η,u) U η (r) dη (r) X (r ) κ η (r) dη (r), ε X 3(r) dr ε X 3( ), c ( z c3 R n). his provides an esimae of he original value funcion in erms of an easier opimisaion problem wih a quadraic penaly funcion and zero arge funcion. Economically, his may be inerpreed as a porfolio liquidaion problem. o coninue he esimae, we define he following value funcion, v (, x) inf (η,u) U E,x, X (r ) κ η (r) dη (r) ε X 3(r) dr ε X 3( ) Monooniciy properies of he value funcion v in he sae variable yield., v (, x) v (, (0, 0, x3 ) ) v (, x 3 ). X (r ) κ η (r) dη (r) Le us denoe by J he performance funcional associaed o he value funcion v. Due o x i = 0 for i =, he mappings (η, u) X η,u i are linear and he mapping (x 3, η, u) X η,u 3 x 3 is also linear. his can be used o show ha J scales quadraically, i.e. for (, x 3 ) 0, R and a scaling facor β > 0 we have As a resul, v scales quadraically as well: J (, βx 3, βη, βu) = β J (, x 3, η, u). v (, βx 3 ) = β v (, x 3 ).

11 U. Hors, N. Naujoka/When o Cross he Spread? Moreover, if x 3 = 0 hen v (, 0) = 0. Choosing now β = x 3 for x 3 0 we ge x 3v (, ), x 3 > 0 v (, x 3 ) = 0, x 3 = 0 x 3v (, ), x 3 < 0, and defining c, min{v (, ), v (, )} leads o v (, x 3 ) c, x 3. Plugging his resul ino (3.) provides he following esimae v(, x, z) c, x 3 c ( z c3 R n). o prove he asserion of he lemma, i remains o show ha he consan c, is sricly posiive and finie for each 0,. he proof of his resul can be found in Nau Lemma A... We are now ready o prove an a priori esimae on he conrol, which will be needed in he Komlós argumen below. his resul exends NW Lemma 4.3 o he singular conrol case. Lemma 3.5. For each (, x, z) 0, R 3 R n here is a consan K(, x, z) such ha any conrol wih E,x,z η( ) R u(r) R dr > K(, x, z) canno be opimal. Proof. We firs consider he marke order η. he dynamics of X i for i =, imply ha for s, we have X i (s) = e ρi(s ) x i κ i e ρi(r ) dη i (r), (3.3) and hus X i ( ) κ i e ρi η i ( ). Combining his wih (3.) yields Xi ( ) J(η, u) E,x,z x x K E,x,z η i ( ) K,x. (3.4) κ i κ κ for consans K, K,x > 0. I follows ha if E,x,z η i ( ) > J(0,0)K,x K hen η canno be opimal. We have J(0, 0) < due o Corollary 3.. he esimae in erms of he passive order u is slighly more involved. Le τ i denoe he firs jump ime of he Poisson process N i afer for i =,, an exponenially disribued random variable wih parameer λ i, and se τ τ τ. A he jump ime τ he sae process jumps from X(τ ) o 0 X(τ ) N X(τ) X(τ ) 0. u (τ) {τ<τ } u (τ) {τ<τ } We use he definiion of he cos funcional and he fac ha he cos erms are nonnegaive o ge J(η, u) =E,x,z X (r ) κ,τ) η (r) dη (r) X (r ) κ,τ) η (r) dη (r) τ h (X 3 (r) α(r, Z(r))) dr J (τ, X(τ ) N X(τ), Z(τ), η, u) (3.5),s E,x,z J(τ, X(τ ) N X(τ), Z(τ), η, u)

12 U. Hors, N. Naujoka/When o Cross he Spread? E,x,z v(τ, X(τ ) N X(τ), Z(τ)), where J in he above is evaluaed a conrols on he sochasic inerval 7 τ,. Combining his wih Lemma 3.4 we ge In view of Lemma 3. we have J(η, u) E,x,z c, X 3 (τ ) N X 3 (τ) c ( Z(τ) c3 R n). E,x,z Z(τ) c3 R n E,x,z and hus here is a consan c,z 0 such ha sup Z(s) c3 R n s, c ( z c3 R n), J(η, u) c,z c, E X 3 (τ ) N X 3 (τ). (3.6) By definiion, he sock holdings direcly afer a jump of he Poisson process are given by X 3 (τ ) N X 3 (τ) = x 3 η (τ ) η (τ ) u (τ) {τ<τ } u (τ) {τ<τ }, and an applicaion of he inequaliy (a b) a b leads o X 3 (τ ) N X 3 (τ) ( u (τ ) {τ<τ } u (τ ) {τ<τ }) (x3 η (τ ) η (τ )) ( ( u (τ ) {τ<τ } u (τ ) {τ<τ }) 3 x3 η (τ ) η (τ ) ) ( ) ( u (τ ) {τ<τ } u (τ ) {τ<τ } 3 x3 η ( ) η ( ) ). (3.7) Combining (3.6) and (3.7) we ge (u c ),E,x,z (τ ) {τ<τ } u (τ ) {τ<τ } ( J(η, u) c,z 3c, x3 E,x,z η ( ) η ( ) ). Due o equaion (3.4) we have for i =, E,x,z η i ( ) K,x K K J(η, u), so combining he las wo displays and relabelling consans provides (u ) E,x,z (τ ) {τ<τ } u (τ ) {τ<τ } c,,x,z c, J(η, u). (3.8) We shall now compue he erm on he lef hand side of inequaliy (3.8). he jump imes τ and τ are independen and exponenially disribued wih parameer λ and λ, respecively. We hus have (u ) E,x,z (τ ) {τ<τ } u (τ ) {τ<τ } = λ e λ(r ) λ e λ(r ) ( u (r ) {r<r } u (r ) {r<r }) dr dr λ e λ(r ) λ e λ(r ) u (r ) dr dr, 7 More precisely, we spli he inerval, ino he subinervals, τ and (τ,. By definiion of he cos funcional, he singular order on he second subinerval (τ, includes a possible jump a he lef endpoin τ, so his jump mus be excluded from he firs subinerval, τ. For his reason, he sae process direcly afer he Poisson jump in (3.5) is given by X(τ ) N X(τ) and no by X(τ ) N X(τ) ηx(τ) = X(τ).

13 U. Hors, N. Naujoka/When o Cross he Spread? 3 where we have used he nonnegaiviy of he inegrand in he las line and resriced inegraion o (r, r ),, ). We now compue (u ) E,x,z (τ ) {τ<τ } u (τ ) {τ<τ } λ e λ(r ) dr λ e λ(r ) E,x,z u (r ) dr =e λ( ) λ e λ(r ) E,x,z u (r ) dr e λ( ) λ e λ( ) E,x,z u (r ) dr. Combining his wih equaion (3.8) and relabelling consans we ge E,x,z u (r) dr c,,x,z c, J(η, u). In paricular if E,x,z u (r) dr c,,x,z c, J(0, 0), hen we see ha J(η, u) > J(0, 0) and he conrol (η, u) is clearly no opimal. A similar esimae holds for he passive sell order u. heorem 3.6. here is a unique opimal conrol (ˆη, û) U for Problem.7. Proof. Le (η n, u n ) n N U be a minimising sequence, i.e. lim n J(ηn, u n ) = inf J(η, u). (η,u) U Due o he uniform L -norm bound of Lemma 3.5 we can hen apply he Komlós heorem for singular sochasic conrol given in Kab99 Lemma 3.5. I saes he exisence of a subsequence (also indexed by n) ha is Cesaro-convergen along wih all is furher subsequences. In paricular, i provides adaped, non-decreasing and righ-coninuous processes ˆη :, Ω R such ha ˆη(, ) L and he sequence of random measures η n n converges weakly o ˆη in he sense ha for almos all ω Ω he measures η n (ω) on, converge weakly o ˆη(ω). Similarly, he same argumens as in he proof of heorem 3. in NW yield a predicable process û :, Ω R such ha he sequence ū n n and all is subsequences converges weakly o û. Having esablished he appropriae measurabiliy properies, we deduce ha (ˆη, û) is admissible. We now prove opimaliy of he limiing process. Weak convergence of he process ( η n ) n N coupled wih equaion (3.3) implies ha for s, such ha ˆη(s) = 0 as well as for s = we have for i =, ha e ρ(s ) x κ i lim X ηn,ū n n i (s) = lim n n i= n i= η i u i e ρi(r ) d η i n (r),s

14 In paricular, U. Hors, N. Naujoka/When o Cross he Spread? 4 =e ρ(s ) x i κ i e ρi(r ) dˆη i (r) = X ˆη,û i (s) P-a.s.,s lim X ηn,ū n n i = X ˆη,û i P λ-a.s. From he definiion of X ηn ū n i for i =, we have d η i n η dxi (s) = inegraion-by-pars formula for Sieljes inegrals we obain E,x,z, =E,x,z X ηn,ū n i (r ) κ i η i(r) d η i n (r) X ηn,ū n i ( ) x i ρ κ i A similar equaion holds in he limi wih X ηn,ū n i by Faou s lemma E,x,z, lim inf E,x,z n X ˆη,û i, X η i κ i (r ) κ i ˆη i(r) dˆη i (r) X ηn,ū n i n,ū n nūn (s)ρ ix ηn ū n i (s)ds κ i (r) dr.. Using he replaced by X ˆη,û and η n replaced by ˆη. Hence, (r ) κ i ηn i (r) d η i n (r), i =,. Le us now consider he coss of deviaing from he arge funcion. he d η i n -inegrals in he definiion of he process X ηn,ū n 3 again converge P λ-a.s. he Poisson inegral erms are as in NW. In he proof of heir heorem 3.6 i is shown ha s E ū n i (r) û(r) dn i (r) = 0 sup s for i =,. Passing o a subsequence (again denoed {( η n, ū n )}) if necessary we may hus assume lim sup n s s ū n i (r) û(r) dn i (r) = 0 P-a.s. o obain P λ-almos sure convergence of X ηn,ū n 3 o X ˆη,û 3. We can hen again apply Faou s lemma o obain ( ) ( ) E h X ˆη,û 3 (r) α(r, Z(r)) dr f X ˆη,û 3 ( ) α(, Z( )) lim inf n E ) h (X ηn,ū n 3 (r) α(r, Z(r)) dr f (X ηn,ū n 3 ( ) α(, Z( ))). Combining he preceding argumens wih he convexiy of J gives J(ˆη, û) lim inf n J( ηn, ū n ) lim inf n n n J(η i, u i ) = inf J(η, u). (η,u) U Uniqueness of he opimal sraegy is due o he sric convexiy of (η, u) J(η, u). hroughou, we denoe by (ˆη, û) he opimal conrol and by ˆX = X ˆη,û he opimal sae rajecory. i=

15 4. he Sochasic Maximum Principle U. Hors, N. Naujoka/When o Cross he Spread? 5 In he preceding secion we showed ha Problem.7 admis a unique soluion under Assumpion.9. We shall now prove a version of he sochasic maximum principle which yields a characerisaion of he opimal conrol in erms of he adjoin equaion. In he sequel, we impose Assumpion. and we wrie E insead of E,x,z. he adjoin equaion is defined as he following BSDE on,, P (s) P ( ) P (s) P ( ) = P 3 (s) P 3 ( ) s P ( ) P ( ) = P 3 ( ) s ρ P (r) ρ P (r) dr Q (r) Q (r) dw (r) h ( ˆX 3 (r) α(r, Z(r))) Q 3 (r) R,(r) s R, (r) Ñ (dr) R,(r) R, (r) Ñ (dr) R,3 (r) R,3 (r) R 3,(r, θ) R 3, (r, θ) M(dr, dθ) R k R 3,3 (r, θ) s s,s 0 0 dˆη (r),s 0 0 f ( ˆX 3 ( ) α(, Z( ))) 0 dˆη (r), (4.) 0 Remark 4.. Noe ha he opimal conrol ˆη now eners he adjoin equaion, which is no he case in he usual formulaion of singular conrol problems, see e.g. CH94. We will show in Secion 6 ha he soluion o he BSDE defined above provides he soluion o a refleced BSDE, where he bid ask spread plays he role of he reflecing barrier. he adjoin process is hen a riple of processes (P, Q, R) defined on, (resp., R k ) by P (s) Q (s) R, (s) R, (s) R,3 (s) P (s) P (s), Q(s) Q (s) and R(s, θ) R, (s) R, (s) R,3 (s), P 3 (s) Q 3 (s) R 3, (s, θ) R 3, (s, θ) R 3,3 (s, θ) which saisfy for i =,, 3 P i :, Ω R, Q i :, Ω R d, R,i :, Ω R, R,i :, Ω R, R 3,i :, R k Ω R and which also saisfy he dynamics (4.) where P is adaped and Q, R are predicable. Proposiion 4.. he BSDE (4.) admis a unique soluion which saisfies for i =,, 3 E sup P i (s) E Q i (r) R ddr E R,i (r) dr s E R,i (r) dr E R 3,i (r, θ) m(dθ)dr <. R k I is unique among riples (P, Q, R) saisfying he above inegrabiliy crierion. Proof. he backward equaion (4.) is a linear BSDE, so sandard argumens imply ha is soluion can be given in closed form as P (s) = E (s, e ρ(r s) dˆη (r) Fs, P (s) = E (s, e ρ(r s) dˆη (r) Fs, P 3 (s) = E ( ) s h ˆX3 (r) α(r, Z(r)) dr f ( ˆX3 ( ) α(, Z( )) ) (4.) Fs..

16 U. Hors, N. Naujoka/When o Cross he Spread? 6 We refer he reader o Nau Proposiion.4. for deails. he characerisaion of he opimal conrol we shall derive explois an opimaliy condiion in erms of he Gâeaux derivaive of J. Given conrols (η, u), ( η, ū) U, i is defined as J ( η, ū), (η, u) = lim J ( η εη, ū εu) J( η, ū). ε 0 ε In our paricular case, he Gâeaux derivaive can be compued explicily. his is he conen of he following lemma. Lemma 4.3. he performance funcional J : (η, u) J(η, u) is Gâeaux differeniable. Is derivaive is given by, for conrols (η, u), ( η, ū) U, J ( η, ū), (η, u) =E X η,u (r ) e ρ(r ) x κ, η (r) d η (r) X η,u (r ) e ρ(r ) x κ, η (r) d η (r) X η,ū (r ) κ η (r) dη (r),, γ u (r) γ u (r) X η,u 3 (r)h ( X η,ū 3 (r) α(r, Z(r)) ) dr X η,u 3 ( )f ( X η,ū 3 ( ) α(, Z( )) ). X η,ū (r ) κ η (r) dη (r) Proof. he erms involving γ i are sraighforward. hose involving h and f can be reaed exacly as in NW Lemma 5.3, so i is enough o compue he Gâeaux derivaive of J (η, u) E X (r ) κ η (r) dη (r)., From equaion (3.3) i follows ha he map (η, u) X η,u is affine, so for s,, ε 0, and (η, u), ( η, ū) U we have X ηεη,ūεu (s) =X η,ū (s) εκ We can now compue,s J ( η, ū), (η, u) = lim ε 0 ε J ( η εη, ū εu) J ( η, ū) = lim ε 0 ε E, X η,ū (r ) κ, η (r) d η (r) = lim ε 0 ε E,, e ρ(r ) dη (r) = X η,ū (s) ε ( X η,u (s) e ρ(s ) x ). X ηεη,ūεu (r ) κ η (r) ε κ η (r) X η,ū (r ) ε ( X η,u (r ) e ρ(r ) x ) d ( η (r) εη (r)) κ η (r) ε κ η (r) d η (r) X η,ū (r ) κ, η (r) d η (r) ε X η,ū (r ) ε ( X η,u (r ) e ρ(r ) ) κ x η (r) ε κ η (r) dη (r)

17 =E,, his complees he proof. U. Hors, N. Naujoka/When o Cross he Spread? 7 X η,u (r ) e ρ(r ) x κ X η,ū (r ) κ η (r) dη (r) η (r) d η (r). Our version of he maximum principle is based on an opimaliy condiion on he Gâeaux derivaive. As a prerequisie for some algebraic manipulaions of he Gâeaux derivaive, le us now compue d(p X) for a fixed conrol (η, u) U. Using inegraion by pars, we have for s, = P (s)x(s) P ( )X( ) s s X 3 (r )h ( ˆX3 (r) α(r, Z(r)) ) dr s s s s λ u (r) ( P 3 (r) R,3 (r) ) λ u (r) ( P 3 (r) R,3 (r) ) dr X (r )Q (r) X (r )Q (r) X 3 (r )Q 3 (r) dw (r) X (r )R, (r) X (r )R, (r) X 3 (r )R,3 (r) u (r) ( P 3 (r ) R,3 (r) ) Ñ (dr) X (r )R, (r) X (r )R, (r) X 3 (r )R,3 (r) u (r) ( P 3 (r ) R,3 (r) ) Ñ (dr),s,s X (r )R 3, (r, θ) X (r )R 3, (r, θ) X 3 (r )R 3,3 (r, θ) M(dr, dθ) R k κ P (r) P 3 (r) dη (r) κ P (r) P 3 (r) dη (r),s X (r )dˆη (r) X (r )dˆη (r).,s his can be wrien as Y η,u (s) = P ( )X( ) L η,u (s), (4.3) where we define he local maringale par L η,u for s, by L η,u (s) s s X (r )Q (r) X (r )Q (r) X 3 (r )Q 3 (r) dw (r) X (r )R, (r) X (r )R, (r) X 3 (r )R,3 (r) s s u (r) ( P 3 (r ) R,3 (r) ) Ñ (dr) X (r )R, (r) X (r )R, (r) X 3 (r )R,3 (r) u (r) ( P 3 (r ) R,3 (r) ) Ñ (dr) R k X (r )R 3, (r, θ) X (r )R 3, (r, θ) X 3 (r )R 3,3 (r, θ) M(dr, dθ), and he non-maringale par Y η,u for s, by Y η,u (s) P (s)x(s) s ( ) X 3 (r)h ˆX3 (r) α(r, Z(r)) dr

18 s U. Hors, N. Naujoka/When o Cross he Spread? 8,s,s λ u (r) ( P 3 (r) R,3 (r) ) λ u (r) ( P 3 (r) R,3 (r) ) dr κ P (r) P 3 (r) dη (r),s X (r )dˆη (r) X (r )dˆη (r).,s Le us now check ha L is indeed a maringale. κ P (r) P 3 (r) dη (r) Lemma 4.4. For each (η, u) U, he process L η,u is a maringale saring in 0. Proof. We firs consider he process X (r )Q (r)dw (r). o prove ha i is a rue maringale i is enough o check ha s E sup X (r )Q (r)dw (r) <. s, An applicaion of he Burkholder-Davis-Gundy and Hölder inequaliies yields s ( ) E sup X (r )Q (r)dw (r) ce X (r )Q (r) R ddr s, ce sup X (r) E r, Q (r) R ddr. he las expression is finie due o Lemma 3. and Proposiion 4.. Now consider he process X R k (r )R (r, θ) M(dr, dθ). A Hölder argumen as above shows ha E X (r)r (r, θ) m(dθ)dr <. R k he maringale propery now follows from NW Lemma A.3. he remaining erms of L η,u can be reaed similarly. We are now in a posiion o formulae our second main resul, he sochasic maximum principle in inegral form. heorem 4.5. A conrol (ˆη, û) U is opimal if and only if for each (η, u) U we have E ˆX (r) κ, P (r) P 3 (r) d (η (r) ˆη (r)) 0, E ˆX (r) κ, P (r) P 3 (r) d (η (r) ˆη (r)) 0, (4.4) E u (r) û (r) R,3 (r) P 3 (r) γ λ dr 0, E u (r) û (r) R,3 (r) P 3 (r) γ λ dr 0. Proof. We proceed as in CH94 heorem 4.. We are minimising he convex funcional J over U, so by E99 Proposiion.. a necessary and sufficien condiion for opimaliy of (ˆη, û) is ha J (ˆη, û), (η ˆη, u û) 0 for each (η, u) U. Due o Lemma 4.4 we know ha L η,u is a maringale saring in zero for each (η, u) U. In paricular from equaion (4.3) we have ha EY η,u ( ) Y ˆη,û ( ) = 0. he definiion of Y η,u ogeher wih he erminal condiion (4.) for he adjoin equaion allows us o wrie his as ( ) 0 =E f ˆX3 ( ) α(, Z( )) X 3 ( ) ˆX 3 ( )

19 U. Hors, N. Naujoka/When o Cross he Spread? 9 ( ) h ˆX3 (r) α(r, Z(r)) X 3 (r) ˆX 3 (r) dr P 3 (r) κ P (r) d(η (r) ˆη (r)) P 3 (r) κ P (r) d(η (r) ˆη (r)),, λ, X (r ) ˆX (r ) dˆη (r) X (r ) ˆX (r ) dˆη (r), u (r) û (r) P 3 (r) R,3 (r) dr λ u (r) û (r) P 3 (r) R,3 (r) dr. Combining his wih he explici formula for he Gâeaux derivaive given in Proposiion 4.3 yields J (ˆη, û), (η ˆη, u û) κ =E, η κ (r) ˆη (r) dˆη (r), η (r) ˆη (r) dˆη (r) ˆX (r ) κ, ˆη (r) P 3 (r) κ P (r) d (η (r) ˆη (r)) ˆX (r ) κ, ˆη (r) P 3 (r) κ P (r) d (η (r) ˆη (r)) λ u (r) û (r) P 3 (r) R,3 (r) γ dr λ λ u (r) û (r) P 3 (r) R,3 (r) γ dr. λ Noe ha for i =, we have, using he noaion from equaion (.), κ i E, η i(r) ˆη i (r) dˆη i (r) ˆXi (r ) κ i, ˆη i(r) d(η i (r) ˆη i (r)) =E ˆX i (r )d(η i (r) ˆη i (r)), κ i =E =E,, r, η i (r) ˆη i (r) ˆη i (r) ˆη i (r) η i (r) ˆη i (r) ˆXi (r ) κ i ˆη i (r) d(η i (r) ˆη i (r)) ˆX i (r)d(η i (r) ˆη i (r)). Combining he above wo displays leads o J (ˆη, û), (η ˆη, u û) = E,, λ λ ˆX (r) κ P (r) P 3 (r) d (η (r) ˆη (r)) ˆX (r) κ P (r) P 3 (r) d (η (r) ˆη (r)) u (r) û (r) P 3 (r) R,3 (r) γ λ dr u (r) û (r) P 3 (r) R,3 (r) γ dr. λ We conclude ha (ˆη, û) is opimal if and only if for all (η, u) U equaion (4.4) holds.

20 5. Buy, Sell and No-rade Regions U. Hors, N. Naujoka/When o Cross he Spread? 0 In he preceding secion we derived a characerisaion of opimaliy in erms of all admissible conrols. his condiion is no always easy o verify. herefore, we derive a furher characerisaion in he presen secion, his ime in erms of buy, sell and no-rade regions. As a byproduc, his resul shows ha spread crossing is opimal if and only if he spread is smaller han some hreshold. We sar wih he main resul of his secion, which provides a necessary and sufficien condiion of opimaliy in erms of he rajecory of he conrolled sysem ( s, ˆX(s), P (s) ) s,. he proof builds on argumens from CH94 heorem 4. and exends hem o he presen framework where we have jumps and sae-dependen singular cos erms. heorem 5.. A conrol (ˆη, û) U is opimal if and only if i saisfies ( ) P ˆX (s) κ P (s) P 3 (s) 0 s, =, ( ) P ˆX (s) κ P (s) P 3 (s) 0 s, =, (5.) as well as ) P(, { ˆX dˆη (r) κ P (r) P 3(r)>0} (r) = 0 ) P(, { ˆX dˆη (r) κ P (r)p 3(r)>0} (r) = 0 =, =, (5.) and ds dp a.e. on, Ω R,3 P 3 γ λ R,3 P 3 γ λ 0 and 0 and ( ) R,3 P 3 γ λ û = 0, ( ) R,3 P 3 γ λ û = 0. (5.3) Proof. Firs, le (ˆη, û) be opimal and define he sopping ime { } ν(ω) inf s, : ˆX (s) κ P (s) P 3 (s) < 0, wih he convenion inf. Consider he conrol defined by u = û, η = ˆη and η (s, ω) ˆη (s, ω) ν(ω), (s). hen η is equal o ˆη excep for an addiional jump of size one a ime ν. I also is càdlàg and increasing on,. An applicaion of heorem 4.5 yields 0 E ˆX (r) κ P (r) P 3 (r) d (η (r) ˆη (r)), ( ) =E ˆX (ν) κ P (ν) P 3 (ν) {ν } 0, which implies ha P (ν = ) =. his proves he firs line of (5.), he second line follows by similar argumens. Now consider he conrol defined by u = û, η = ˆη and { η ( ) = 0, dη (s, ω) { ˆX(s,ω) κ P (s,ω) P 3(s,ω) 0} dˆη (s, ω).

21 U. Hors, N. Naujoka/When o Cross he Spread? hen η is càdlàg and increasing on,, since ˆη is. Due o heorem 4.5 we have 0 E ˆX (r) κ P (r) P 3 (r) d (η (r) ˆη (r)) =E and in paricular,, ˆX (r) κ P (r) P 3 (r) { ˆX(r) κ d ( ˆη P (r) P 3(r)>0} (r)) 0, 0 = E { ˆX(r) κ dˆη P (r) P 3(r)>0} (r),, which proves he firs par of (5.), he second par follows by similar argumens. I remains o prove (5.3). Again by heorem 4.5 we have for every conrol (η, u) U 0 E (u (r) û (r))(r,3 (r) P 3 (r) γ /λ )dr. Choosing he conrol (ˆη, u) wih u = û and u (r, ω) = û (r, ω) {R,3(r,ω)P 3(r,ω) γ /λ >0} we firs noe ha u is predicable and we ge 0 E {R,3(r)P 3(r) γ /λ >0}(R,3 (r) P 3 (r) γ /λ )dr 0, which shows ha R,3 P 3 γ /λ 0 ds dp a.e. Recall ha we also have û 0 by definiion. We now wan o show a leas one of he processes R,3 P 3 γ /λ and û is zero. o his end, consider he conrol (ˆη, u) whose passive orders are defined by u = û and u = û. We hen ge 0 E (u (r) û (r))(r,3 (r) P 3 (r) γ /λ )dr = E û(r)(r,3 (r) P 3 (r) γ /λ )dr 0. I follows ha ds dp a.e. we have u (R,3 P 3 γ /λ ) = 0. he argumen for he second line in (5.) is similar. his proves he only if par of he asserion. In order o prove he if par, le condiions (5.), (5.) and (5.3) be saisfied. We hen have for each (η, u) U E ˆX (r) κ P (r) P 3 (r) d (η (r) ˆη (r)) =E,, E E ˆX (r) κ P (r) P 3 (r) dη (r),, ˆX (r) κ P (r) P 3 (r) { ˆX(r) κ d ( ˆη P (r) P 3(r)>0} (r)) (5.4) (5.5) ˆX (r) κ P (r) P 3 (r) { ˆX(r) κ d ( ˆη P (r) P 3(r) 0} (r)). (5.6) he inegrand of (5.4) is nonnegaive due o condiion (5.), so (5.4) is nonnegaive. he erm (5.5) is zero due o condiion (5.). he erm (5.6) has a nonposiive inegrand and a decreasing inegraor and is herefore also nonnegaive. In conclusion, we have E ˆX (r) κ P (r) P 3 (r) d (η (r) ˆη (r)) 0,,

22 and by a similar argumen E, U. Hors, N. Naujoka/When o Cross he Spread? ˆX (r) κ P (r) P 3 (r) d (η (r) ˆη (r)) 0. Sill for arbirary (η, u) U we have using (5.3) and u 0 E u (r) û (r) R,3 (r) P 3 (r) γ /λ dr = E u (r) R,3 (r) P 3 (r) γ /λ dr 0. By a similar argumen E u (r) û (r) R,3 (r) P 3 (r)γ /λ dr 0. An applicaion of heorem 4.5 now shows ha (ˆη, û) is indeed opimal. he preceding heorem gives an opimaliy condiion in erms of he conrolled sysem (P, ˆX). We now show how heorem 5. can be used o describe he opimal marke order quie explicily in erms of buy, sell and no-rade regions. Definiion 5.. We define he buy, sell and no-rade regions (wih respec o marke orders) by R buy { (s, x, p), R 3 R 3 x κ p p 3 < 0 }, R sell { (s, x, p), R 3 R 3 x κ p p 3 < 0 }, R n { (s, x, p), R 3 R 3 x κ p p 3 > 0 and x κ p p 3 > 0 }. Moreover, we define he boundaries of he buy and sell regions by R buy { (s, x, p), R 3 R 3 x κ p p 3 = 0 }, R sell { (s, x, p), R 3 R 3 x κ p p 3 = 0 }. Le us emphasise ha each of he hree regions defined above is open. We remark ha he ime variable s is included ino he definiion of he buy, sell and no-rade regions such ha saemens like he rajecory of he process ( s, ˆX(s), P (s) ) under he opimal conrol is inside he no-rade region make sense. Specifically, we now show ha he opimal conrol remains inside he closure of he no-rade region a all imes, i.e. i is eiher inside he no-rade region or on he boundary of he buy or sell region. Moreover, as long as he conrolled sysem is inside he no-rade region, marke orders are no used, i.e. ˆη i does no increase for i =,. Proposiion 5.3. (i) If ( s, ˆX(s), P (s) ) is in he no-rade region, i is opimal no o use marke orders, i.e. for i =, E {(r, ˆX(r),P dˆη (r)) Rn} i(r), = 0. (ii) he opimal rajecory remains a.s. inside he closure of he no-rade region, ( (s, ) ) P ˆX(s), P (s) Rn s, =. In paricular, i spends no ime inside he buy and sell regions. Proof. Iem () is a direc consequence of (5.), while () follows from (5.). Example 5.4. he paricular case of porfolio liquidaion is solved in Secion 7. In his case, he opimal sraegy is composed of discree sell orders a imes = 0, and a rading rae in (0, ). Specifically, hese are chosen such ha he process ( s, ˆX(s), P (s) ) remains on he boundary of he sell region unil he passive order is execued and all remaining shares are sold.

23 U. Hors, N. Naujoka/When o Cross he Spread? 3 he above proposiion shows ha he conrolled sysem remains inside he closure of he no-rade region and marke orders are no used inside he no-rade region. his suggess ha markes orders are only used on he boundary, and we shall now make his more precise. o his end, we firs noe ha for i =, he nondecrasing process ˆη i induces a measure on, Ω by he following map, s A E A dˆη i (r).,s Proposiion 5.5. (i) We have ( P ˆη (s) =,s ) {(r, ˆX(r),P (r)) Rbuy } dˆη (r) s, =. In paricular, he suppor of he measure induced by ˆη is a subse of ( r, ˆX(r), ) P (r) R buy, i.e. marke buy orders are only used if he conrolled sysem is on he boundary of he buy region. (ii) Similarly, we have ( ) P ˆη (s) = {(r, ˆX(r),P (r)) Rsell } dˆη (r) s, =.,s In paricular, he suppor of he measure induced by ˆη is a subse of ( r, ˆX(r), ) P (r) R sell, i.e. marke sell orders are only used if he conrolled sysem is on he boundary of he sell region. Proof. We only show he firs asserion. For s, we have using ˆη ( ) = 0 ˆη (s) = dˆη (r) =,s,s { ˆX(r) κ P (r) P 3(r)<0} { ˆX (r) κ P (r) P 3(r)>0} { ˆX dˆη (r) κ P (r) P 3(r)=0} (r). We shall show ha erms in he second line vanish a.s. By Proposiion 5.3 () we have ( (r, ) ) P ˆX(r), P (r) / Rbuy r, = i.e. he opimal rajecory spends no ime in he buy region, so ha a.s. for each s, { ˆX(r) κ dˆη P (r) P 3(r)<0} (r) = {(r, ˆX(r),P (r)) Rbuy } dˆη (r) = 0.,s,s Due o equaion (5.) we have a.s. for each s, 0 = { ˆX(r) κ dˆη P (r) P 3(r)>0} (r), so ha a.s. for each s,,s,s { ˆX(r) κ P (r) P 3(r)>0} dˆη (r) = 0. his shows ha a.s. for each s, we have ˆη (s) = { ˆX(r) κ dˆη P (r) P 3(r)=0} (r) =,s { ˆX(r) κ P (r) P 3(r)>0} dˆη (r) 0,,s {(r, ˆX(r),P (r)) Rbuy } dˆη (r).