Almost-sure hedging with permanent price impact

Transcription

1 Almos-sure hedging wih permanen price impac B. Bouchard and G. Loeper and Y. Zou November 3, 215 Absrac We consider a financial model wih permanen price impac. Coninuous ime rading dynamics are derived as he limi of discree rebalancing policies. We hen sudy he problem of super-hedging a European opion. Our main resul is he derivaion of a quasi-linear pricing equaion. I holds in he sense of viscosiy soluions. When i admis a smooh soluion, i provides a perfec hedging sraegy. Keywords: Hedging, Price impac. AMS 21 Subjec Classificaion: 91G2; 93E2; 49L2 Inroducion Two of he fundamenal assumpions in he Black and Scholes approach for opion hedging are ha he price dynamics are unaffeced by he hedger s behaviour, and ha he can rade unresriced amouns of asse a he insananeous value of he price process. In oher words, i relies on he absence of marke impac and of liquidiy coss or liquidiy consrains. This work addresses he problem of opion hedging under a price dynamics model ha incorporaes direcly he hedger s rading aciviy, and hence ha violaes hose wo assumpions. CEREMADE, Universié Paris Dauphine and CREST-ENSAE. Research suppored by ANR Liquirisk and Invesissemens d Avenir (ANR-11-IDEX-3/Labex Ecodec/ANR- 11-LABX-47). BNP-Paribas and FiQuan - Chaire de finance quaniaive CEREMADE, Universié Paris Dauphine and CREST-ENSAE. 1

2 In he lieraure, one finds numerous sudies relaed o his opic. Some of hem incorporae liquidiy coss bu no price impac, he price curve is no affeced by he rading sraegy. In he seing of 6], his does no affec he super-hedging price because rading can essenially be done in a bounded variaion manner a he marginal spo price a he origine of he curve. However, if addiional resricions are imposed on admissible sraegies, his leads o a modified pricing equaion, which exhibis a quadraic erm in he second order derivaive of he soluion, and renders he pricing equaion fully non-linear, and even no uncondiionally parabolic, see 7] and 2]. Anoher branch of lieraure focuses on he derivaion of he price dynamics hrough clearing condiion. In he papers 9], 16], 15], he auhors work on supply and demand curves ha arise from reference and program raders (i.e. opion hedgers) o esablish a modified price dynamics, bu do no ake ino accoun he liquidiy coss, see also 12]. This approach also leads o non-linear pde s, bu he non-lineariy comes from a modified volailiy process raher han from a liquidiy cos source erm. Finally, he series of papers 17], 19], 14] address he liquidiy issue indirecly by imposing bounds on he gamma of admissible rading sraegies, no liquidiy cos or price impac are modeled explicily. More recenly, 13] and 1] have considered a novel approach in which he price dynamic is driven by he sum of a classical Wiener process and a (locally) linear marke impac erm. The linear marke impac mechanism induces a modified volailiy process, as well as a non rivial average execuion price. However, he rader sars his hedging wih he correc posiion in socks and does no have o unwind his final posiion (his corresponds o covered opions wih delivery). Those combined effecs lead o a fully non-linear pde giving he exac replicaion sraegy, which is no always parabolic depending on he raio beween he insananeous marke impac (liquidiy coss) and permanen marke impac. In his paper we build on he same framework as 13], in he case where he insananeous marke impac equals he permanen impac (no relaxaion effec), and go one sep furher by considering he effec of (possibly) unwinding he porfolio a mauriy, and of building he iniial porfolio. Consequenly he spo jumps a iniial ime when building he hedge porfolio, and a mauriy when unwinding i (depending on he naure of he payoff - delivery can also be made in socks). In his framework, we find ha he opimal super-replicaion sraegy follows a modified quasi-linear Black and Scholes pde. Alhough he underlying model is similar o he one proposed by he second auhor 13], he pricing pde is herefore fundamenally differen (quasi-linear vs fully non-linear). Concerning he mahemaical approach, while in 13] he auhor focused on exhibiing an exac replicaion sraegy by a verificaion approach, in his work we follow a sochasic arge approach and derive he pde from a dynamic program- 2

3 ming principle. The difficuly is ha, because of he marke impac mechanism, he sae process mus be described by he asse price and he hedger s porfolio (i.e. he amoun of risky asse deained by he hedger) and his leads o a highly singular conrol problem. I is overcome by a suiable change of variable which allows one o reduce o a zero iniial posiion in he risky asse and sae a version of he geomeric dynamic programming principle in erms of he pos-porfolio liquidaion asse price process: he price ha would be obained if he rader was liquidaing his posiion immediaely. The paper is organized as follows. In Secion 1, we presen he impac rule and derive coninuous ime rading dynamics as limis of discree ime rebalancing policies. The super-hedging problem is se in Secion 2 as a sochasic arge problem. We firs prove a suiable version of he geomeric dynamic programming and hen derive he corresponding pde in he viscosiy soluion sense. Uniqueness and regulariy are esablished under suiable assumpions. We finally furher discuss he case of a consan impac coefficiens, o provide a beer undersanding of he hedging sraegy. General noaions. Given a funcion φ, we denoe by φ and φ is firs and second order derivaives if hey exis. When φ depends on several argumens, we use he noaions x φ, xxφ 2 o denoe he firs and second order parial derivaives wih respec o is x-argumen, and wrie xyφ 2 for he cross second order derivaive in is (x, y)-argumen. All over his paper, Ω is he canonical space of coninuous funcions on R + saring a, P is he Wiener measure, W is he canonical process, and F = (F ) is is augmened raw filraion. All random variables are defined on (Ω, F, P). L (resp. L 2 ) denoes he space of (resp. square inegrable) R n -valued random variables, while L λ (resp. Lλ 2 ) sands for he collecion of predicable Rn -valued processes ϑ (resp. such ha ϑ L λ := E 2 ϑ s 2 ds] 1 2 ). The ineger n 1 is given by he conex and x denoe he Euclidean norm of x R n. 1 Porfolio and price dynamics This secion is devoed o he derivaion of our model wih coninuous ime rading. We firs consider he siuaion where a rading signal is given by a coninuous Iô process and he posiion in sock is rebalanced in discree ime. In his case, he dynamics of he sock price and he wealh process are given according o our impac rule. A firs coninuous ime rading dynamic is obained by leing he ime beween wo consecuive rades vanish. Then, we incorporae jumps as he limi of coninuous rading on a shor ime horizon. 3

4 We resric here o a single sock marke. This is only for simpliciy, he exension o a muli-dimensional marke is jus a maer of noaions. 1.1 Impac rules We model he impac of a sraegy on he price process hrough an impac funcion f: he price variaion du o buying a (infiniesimal) number δ R of shares is δf(x), if he price of he asse is x before he rade. The cos of buying he addiional δ unis is given by in which δx + 1 δ 2 δ2 f(x) = δ δ 1 (x + f(x)ι)dι δ 1 (x + f(x)ι)dι, δ should be inerpreed as he average cos for each addiional uni. Beween wo imes of rading τ 1 τ 2, he dynamics of he sock is given by he srong soluion of he sochasic differenial equaion All over his paper, we assume ha dx = µ(x )d + σ(x )dw. (1.1) f Cb 2 and is (sricly) posiive, (µ, σ, σ 1 ) is Lipschiz and bounded. (H1) Remark 1.1. a. We resric here o an impac rule which is linear in he size of he order. However, noe ha in he following i will only be applied o order of infiniesimal size (a he limi). One would herefore obain he same final dynamics (1.23)-(1.24) below by considering a more general impac rule δ F (x, δ) whenever is saisfies F (x, )= 2 δδ F (x, ) = and δf (x, ) = f(x). See Remark 1.2 below. Oherwise saed, for our analysis, we only need o consider he value and he slope a δ = of he impac funcion. b. A ypical example of such a funcion is F = x where wih x(x, ) defined as he soluion of x(x, δ) := x(x, δ) x, (1.2) x(x, ) = x + f(x(x, s))ds. (1.3) The curve x has a naural inerpreaion. For an order of small size ι, he sock price jumps from x o x + ιf(x) x(x, ι). Passing anoher order of size ι 4

5 makes i move again o approximaely x(x(x, ι), ι) = x(x, 2 ι), ec. Passing o he limi ι bu keeping he oal rade size equal o δ provides asympoically a price move equal o x(x, δ). This specific curve will play a cenral role in our analysis, see Secion Discree rebalancing from a coninuous signal and coninuous ime rading limi We firs consider he siuaion in which he number of shares he rader would like o hold is given by a coninuous Iô process Y of he form where Y = Y + b s ds + (a, b) A := k A k, a s dw s, (1.4) A k := {(a, b) L λ : (a, b) k d dp a.e.} for k >. In order o derive our coninuous ime rading dynamics, we consider he corresponding discree ime rebalancing policy se on a ime grid n i := it/n, i =,..., n, n 1, and hen pass o he limi n. If he rader only changes he composiion of his porfolio a he discree imes n i, hen he holds Y n socks on each ime inerval n i i, n i+1 ). Oherwise saed, he number of shares acually held a T is n 1 Y n := Y n i 1 { n i < n i+1 } + Y T 1 {=T } (1.5) i= and he number of purchased shares is δ n := n 1 {= n i }(Y n i Y n ). i=1 Given our impac rule, he corresponding dynamics for he sock price process is X n = X + µ(x n s )ds + σ(x n s )dw s + 5 n 1 n i,t ]δ n nf(xn i n i ), (1.6) i=1

6 in which X is a consan. To describe he porfolio process, we provide he dynamics of he sum V n of he amoun of cash held and he poenial amoun Y n X n associaed o he posiion in socks: V n = cash posiion + Y n X n. (1.7) Observe ha his is no he liquidaion value of he porfolio, excep when Y n =, as he liquidaion of Y n socks will have an impac on he marke and does no generae a gain equal o Y n X n. However, if we keep Y n in mind, he couple (V n, Y n ) gives he exac composiion in cash and socks of he porfolio. By a sligh abuse of language, we call V n he porfolio value or wealh process. Assuming ha he risk free rae is zero (for ease of noaions), is dynamics is given by n V n = V + Ys dx n s n n i,t ] 2 (δn n)2 f(x n i n i ), (1.8) or equivalenly V n = V + + n i=1 i=1 n 1 n,t ]Y n (X n n i X n n ) i=1 1 n i,t ] ] 1 2 (δn n)2 f(x n i n i ) + Y n δ n nf(xn i n i ), (1.9) in which V R. Le us commen his formula. The firs erm on he righhand side corresponds o he evoluion of he porfolio value sricly beween wo rades ; i is given by he number of shares held muliplied by he price incremen. When a rade of size δ n n occurs a ime n i i, he cos of buying he socks is 2 1 (δ n n)2 f(x n i n ) + δn i nxn i n i bu i provides δn n more socks, on op of he i Y n i = Y n unis ha are already in he porfolio. Afer he price s move generaed by he rade, he socks are evaluaed a X n n. The incremen in value du o he i price s move and he addiional posiion is herefore δ n nxn i n + Y i n i (Xn n X n i n ). i Since X n n X n i n i = δn nf(xn i n i ), we obain (1.9), a compac version of which is given in (1.8). Our coninuous ime rading dynamics are obained by passing o he limi n, i.e. by considering faser and faser rebalancing sraegies. Proposiion 1.1. Le Z := (X, Y, V ) where Y is defined as in (1.4) for some (a, b) A, and (X, V ) solves X = X + σ(x s )dw s + f(x s )dy s + 6 (µ(x s ) + a s (σf )(X s ))ds (1.1)

7 and V = V + Y s dx s + 1 a 2 2 sf(x s )ds. (1.11) Le Z n := (X n, Y n, V n ) be defined as in (1.6)-(1.5)-(1.8). Then, here exiss a consan C > such ha for all n 1. sup E Z n Z 2] Cn 1,T ] Proof. This follows sandard argumens and we only provide he main ideas. In all his proof, we denoe by C a generic posiive consan which does no depend on n nor i n, and may change from line o line. We shall use repeaedly (H1) and he fac ha a and b are bounded by some consan k, in he d dp-a.e. sense. a. The convergence of he process Y n is obvious: sup E Y n Y 2] Cn 1. (1.12),T ] For laer use, se X n := X X n and also observe ha he esimae sup E X n 2] E X n n,n i ) n 2] (1 + Cn 1 ) + Cn 1, (1.13) is sandard. We now se where A i,n := B i,n := Since A i,n n i n n + B i,n n i X n := X n + A i,n f(xs n )dy s + a s (σf )(Xs n )ds n + B i,n, n < n i, (Y s Y n )(µf σ2 f )(X n s )ds + = δ n nf(xn i n i ), we have lim n i X n = X n n i. n Se X n := X X n, β 1 := bf + aσf and β 2 := af, so ha d X n 2 = 2 X n (µ + β 1 )(X ) (µ + β 1 )(X n )]d (Y s Y n )(σf )(X n s )dw s. + (σ + β 2 )(X ) (σ + β 2 )(X n ) (Y Y n )(σf )(X n )] 2 d 2 X n (Y Y n )(µf σ2 f )(X n )d + 2 X n (σ + β 2 )(X ) (σ + β 2 )(X n )]dw 2 X n (Y Y n )(σf )(X n )dw. 7

8 In view of (1.12)-(1.13), his implies, for n < n i, E X n 2] ] E X n n 2] + CE ( X s n 2 + X s Xs n 2 + Y s Y n n 2 )ds ] E X n n 2] (1 + Cn 1 ) + CE X n s 2 ds + n 2, and herefore sup E X n 2] E X n n,n i ) n 2] (1 + Cn 1 ) + Cn 2, (1.14) n by Gronwall s Lemma. Since lim Xn n = X n n, his shows ha i i E X n n 2] sup E X n 2] Cn 1 for all i n. i n,n i ) Plugging his inequaliy in (1.13), we hen deduce sup E X n 2] Cn 1 for all i n. (1.15) n,n i ] b. We now consider he difference V V n. I follows from (1.9) ha V n n i = V n n + n i n i + n n n Y n µ(xs n i )ds + Y n n σ(xs n )dw s ) n i ds + Y n n f(xs n )dy s ( 1 2 a2 sf(xs n ) + Y n a s (f σ)(xs n ) n i + αs 1n ds + n n i n αs 2n dw s where, by (1.12), α 1n and α 2n are adaped processes saisfying In view of (1.12)-(1.15), his leads o sup E α 1n 2 + α 2n 2 ] Cn 1. n,n i ) V n n i n = V n + V i n n V i n + γs 1n i ds + γs 2n dw s (1.16) n n 8

9 where γ 1n and γ 2n are adaped processes saisfying Se Ṽ n sup E γ 1n 2 + γ 2n 2 ] Cn 1. (1.17) n,n i ) := V n + V V n + γs 1n ds + γs 2n dw s, n < n i. n n Then, by applying Iô s Lemma o Ṽ n V 2, using (1.17) and Gronwall s Lemma, we obain sup E n,n i ) so ha, by he ideniy limṽ n n i Ṽ n V 2] E V n V n 2] (1 + Cn 1 ) + Cn 2, = V n, recall (1.16), and an inducion, i E V n V i n i 2] Cn 1, i n. We conclude by observing ha E V n V 2] CE V n V n 2 + V n V n 2 + V n V 2] ( C E V n V n 2] + n 1), for n < n i. Remark 1.2. If he impac funcion δf(x) was replaced by a more general Cb 2 one of he form F (x, δ), wih F (x, ) = δδ 2 F (x, ) =, he compuaions made in he above proof would only lead o erms of he from δ F (X, )dy and aσ(x) xδ 2 F (X, ) in place of f(x)dy and a(σf )(X) in he dynamics (1.1). Similarly, he erm a 2 f(x) would be replaced by a 2 δ F (X, ) in (1.11). 1.3 Jumps and large orders spliing We now explain how we incorporae jumps in our dynamics. Le U k denoe he se of random {,, k}-valued measures ν suppored by k, k], T ] ha are adaped in he sense ha ν(a, ]) is adaped for all Borel subse A of k, k]. We se U := k U k. 9

10 Noe ha an elemen ν of U can be wrien in he form ν(a,, ]) = k 1 {(δj,τ j ) A,]} (1.18) j=1 in which τ 1 < < τ k T are sopping imes and each δ j is a real-valued F τj -random variable. Then, given (a, b, ν) A U, we define he rading signal as Y = Y + b s ds + a s dw s + δν(dδ, ds), (1.19) where Y R. For laer use, we le Y c denoe is coninuous par, i.e. Y c := Y δν(dδ, ds). In view of he previous secions, we assume ha he dynamics of he sock price and porfolio value processes are given by (1.1)-(1.11) when Y has no jump. We incorporae jumps by assuming ha he rader follows he naural idea of spliing a large order δ j ino small pieces on a small ime inerval. This is a curren pracice which aims a avoiding having a oo large impac, and paying a oo high liquidiy cos. Given he asympoic already derived in he previous secion, we can reduce o he case where his is done coninuously a a consan rae δ j /ε on τ j, τ j +ε], for some ε >. We denoe by (X, V ) he iniial price and porfolio values. Then, he number of socks in he porfolio associaed o a sraegy (a, b, ν) A k U k is given by k Y ε = Y + 1 τj,t ] δj + ε 1 δ j ( (τ j + ε) τ j ) ], (1.2) j=1 and he corresponding sock price and porfolio value dynamics are X ε = X + V ε = V + σ(x ε s)dw s + Y ε s dx ε s f(x ε s)dy ε s + (µ(x ε s) + a s (σf )(X ε s))ds (1.21) a 2 sf(x ε s)ds. (1.22) When passing o he limi ε, we obain he convergence of Z ε := (X ε, Y ε, V ε ) o Z = (X, Y, V ) wih (X, V ) defined in (1.23)-(1.24) below. In he following, we only sae he convergence of he erminal values, see he proof for a more complee descripion. I uses he curve x defined in (1.3) above, recall also (1.2). 1

11 Proposiion 1.2. Given (a, b, ν) A U, le Z = (X, Y, V ) be defined by (1.19) and X = X + + V = V + + σ(x s )dw s + f(x s )dy c s + (µ(x s ) + a s (σf )(X s ))ds x(x s, δ)ν(dδ, ds) (1.23) Y s dx c s a 2 sf(x s )ds (Y s x(x s, δ) + I(X s, δ)) ν(dδ, ds) (1.24) where X c := X x(xs, δ)ν(dδ, ds) and I(x, z) := z sf(x(x, s))ds, for x, z R. (1.25) Se Z ε := (X ε, V ε, Y ε ). Then, here exiss a consan C > such ha for all ε (, 1). Moreover, E ZT ε +ε Z T 2] C(ε + Psup ν(r,, + ε]) 2] 1 2 ), T lim Psup ν(r,, + ε]) 2] =. ε T Proof. In all his proof, we denoe by C a generic posiive consan which does no depend on ε, and may change from line o line. Here again, we shall use repeaedly (H1) and he fac ha a and b are bounded by some consan k, in he d dp-a.e. sense. Le ν be of he form (1.18) for some k and noe ha he las claim simply follows from he fac ha {τ j+1 τ j ε} Ω up o a P-null se for all j k. Sep 1. We firs consider he case where τ j+1 τ j + ε for all j 1. Again, he esimae on ZT ε +ε Z T follows from simple observaions and sandard esimaes, and we only highligh he main ideas. We will indeed prove ha for 1 j k + 1 E ] sup Z Z ε 2 + sup E Z τj +s Zτ ε j +ε 2 Cε, (1.26) τ j 1 +ε,τ j ) s ε where we use he convenion τ = and τ k+1 = T. The resul is rivial for (Y ε, Y ) since hey are equal on each inerval τ j 1 + ε, τ j ) and (a, b) is bounded. 11

12 a. We firs prove a sronger resul for (X ε, X). Fix p {2, 4}. Le x ε be he soluion of he ordinary differenial equaion x ε δ j = X τj + ε f(xε s)ds. Se X ε := X ε x ε τ j. Iô s Lemma leads o d( X ε ) p = p( X ε ) p 1 α 1,ε d + + p( X ε ) p 1 α 2,ε dw p(p 1) 2 + p δ j ε ( Xε ) p 1 (f(x ε ) f(x ε τ j ))d ( X ε ) p 2 (α 2,ε ) 2 d on τ j, τ j +ε], in which α 1,ε and α 2,ε are bounded processes. The inequaliy x p 1 x p 2 + x p, he Lipschiz coninuiy of f and Gronwall s Lemma hen imply sup E Xτ ε j + x ε p] ε ] CE Xτ ε j X τj p + Xτ ε j +s x ε s p 2 ds. ε We now use a simple change of variables o obain in which x is defined in (1.3), while x ε ε = x(x τj, δ j ) = X τj, sup E X τj + X τj p] Cε p 2. ε Since X and X ε have he same dynamics on τ j + ε, τ j+1 ), his shows ha ] E sup X X ε p CE X τj +ε X ε τj +ε p] τ j +ε,τ j+1 ) For p = 2, his provides E CE x ε ε X ε τj +ε p + X τj +ε X τj p] CE X ε τ j X τj p + ε X ε τ j +s x ε s p 2 ds + ε p 2 ] sup X X ε p + sup E X τj +s Xτ ε j +ε p Cε p 2, τ j 1 +ε,τ j ) s ε ]. 12

13 by inducion over j, and he case p = 4 hen follows from he above. For laer use, noe ha he esimae sup E Xτ ε j + x ε 4] Cε 2 (1.27) ε is a by-produc of our analysis. b. The esimae on V V ε is proved similarly. We inroduce v ε δ 2 j δ j := V τj + ε 2 sf(xε s)ds + Y τj ε f(xε s)ds = V τj + Ys ε δ j ε f(xε s)ds, and obain a firs esimae by using (1.27): E Vτ ε j + v ε 2] CE Vτ ε j V τj 2 + ε + ] CE Vτ ε j V τj 2 + ε, for ε. Then, we observe ha while ( ε v ε ε = V τj + I(X τj, δ j ) + Y τj x(x τj, δ j ) = V τj, sup E V τj + V τj 2] Cε. ε ) ] 2 ε 1 Yτ ε j +sδ j Xτ ε j +s x ε s ds By using he esimae on X X ε obained in a., we hen show ha ] ] E sup V V ε 2 CE V τj +ε Vτ ε j +ε 2 + ε, τ j +ε,τ j+1 ) and conclude by using an inducion over j. Sep 2. We now consider he general case. We define τj+1 ε = (ε + τj ε ) τ j+1, δj+1 ε = δν(dδ, d), j 1, (τj ε,τ j+1 ε ] where (τ ε 1, δε 1 ) = (τ 1, δ 1 ). On E ε := {min j k 1 (τ j+1 τ j ) ε}, (τ ε j, δε j ) j 1 = (τ j, δ j ) j 1. Hence, i follows from Sep 1. ha E ZT ε +ε Z T 2] Cε + CE Z T ε +ε 4 + Z T 4] 1 2 PEε] c 1 2, in which Z ε sands for he dynamics associaed o (τj ε, δε j ) j 1. I now follows from sandard esimaes ha ( Z T ε +ε ) <ε 1 and Z T are bounded in L 4. 13

14 We conclude his secion wih a proposiion collecing some imporan properies of he funcions x and I which appear in Proposiion 1.1. They will be used in he subsequen secion. Proposiion 1.3. For all x, y, ι R, (i) x(x(x, ι), y ι) = x(x, y), (ii) f(x) x x(x, y) = y x(x, y) = f(x(x, y)), (iii) I(x(x(x, ι), y ι), y + ι) I(x(x, y), y) = y x(x, ι) + I(x, ι), (iv) f(x) x I(x, y) + x(x, y) = y I(x, y) = yf(x(x, y)). Proof. (i) is an immediae consequence of he Lipschiz coninuiy of he funcion f, which ensures uniqueness of he ODE defining x in (1.3). More generally, i has he flow propery, which we shall use in he following argumens. The asserion (ii) is an immediae consequence of he definiion of x: x(x(x, ι), y ι) = x(x, y) for ι > and y x(x, ) = f(x), so ha differeniaing a ι = provides (ii). The ideniy in (iii) follows from direc compuaions. As for (iv), i suffices o wrie ha I(x(x, ι), y ι) = y ι ( ι)f(x(x, ))d for ι >, and again o differeniae a ι =. Remark 1.3. I follows from Proposiion 1.3 ha our model allows round rips a (exacly) zero cos. Namely, if x is he curren sock price, v he wealh, and y he number of shares in he porfolio, hen performing an immediae jump of size δ makes (x, y, v) jump o (x(x, δ), y + δ, v + y x(x, δ) + I(x, δ)). Passing immediaely he opposie order, we come back o he posiion (x(x(x, δ), δ), y + δ δ, v + y x(x, δ) + I(x, δ) + (y + δ) x(x(x, δ), δ) + I(x(x, δ), δ)) = (x, y, v), by Proposiion 1.3(i)-(iii). This is a desirable propery if one wans o have a chance o hedge opions perfecly, or more generally o obain a non-degeneraed super-hedging price. 2 Super-hedging of a European claim We now urn o he super-hedging problem. From now on, we define he admissible sraegies as he Iô processes of he form Y = y + b s ds + a s dw s + δν(dδ, ds) (2.1) in which y R, (a, b, ν) A U and Y is essenially bounded. If Y k and (a, b, ν) A k U k, hen we say ha (a, b, ν) Γ k, k 1, and we le Γ := k 1 Γ k. 14

15 We will commen in Remark 2.1 below he reason why we resric o bounded conrols. Given (, z) D :=, T ] R R R, we define Z,z,γ := (X,z,γ, Y,z,γ, V,z,γ ) as he soluion of (1.23)-(2.1)-(1.24) on, T ] associaed o γ Γ and wih iniial condiion Z,z,γ = z. 2.1 Super-hedging price A European coningen claim is defined by is payoff funcion, a measurable map x R (g, g 1 )(x) R 2. The firs componen is he cash-selemen par, i.e. he amoun of cash paid a mauriy, while g 1 is he delivery par, i.e. he number of unis of socks o be delivered. An admissible sraegy γ Γ allows o super-hedge he claim associaed o he payoff g, saring from he iniial condiions z a ime if Z,z,γ T G where G := {(x, y, v) R R R : v yx g (x) and y = g 1 (x)}. (2.2) Recall ha V sands for he fricionless liquidaion value of he porfolio, i is he sum of he cash componen and he value Y X of he socks held wihou aking he liquidaion impac ino accoun. We se G k (, z) := {γ Γ k : Z,z,γ T and define he super-hedging price as G}, G(, z) := k 1 G k (, z), w(, x) := inf k 1 w k (, x) where w k (, x) := inf{v : G k (, x,, v) }. For laer use, le us make precise wha are he T -values of hese funcions. Proposiion 2.1. Define G k (x) := inf{yx(x, y) + g (x(x, y)) I(x, y) : y k s.. y = g 1 (x(x, y))}, x R, and G := inf k 1 G k. Then, w k (T, ) = G k and w(t, ) = G. (2.3) 15

16 Proof. Se z = (x,, v) and fix γ = (a, b, ν) Γ. By (1.23)-(1.24), we have Z T,z,γ T = (x(x, y), y, v + I(x, y)) wih y := δν(dδ, {T }). In view of (2.2), Z T,z,γ T G is hen equivalen o v + I(x, y) yx(x, y) g (x(x, y)) and y = g 1 (x(x, y)). By definiion of w (resp. w k ), we have o compue he minimal v for which his holds for some y R (resp. y k). Remark 2.1. Le us conclude his secion wih a commen on our choice of he se of bounded conrols Γ. a. Firs, his ensures ha he dynamics of X, Y and V are well-defined. This could obviously be relaxed by imposing L 2 λ bounds. However, noe ha he bound should anyway be uniform. This is crucial o ensure ha he dynamic programming principle saed in Secion 2.2 is valid, as i uses measurable selecion argumens: ω ϑω] L λ 2 does no imply E ϑ ] L λ 2 ] <. See Remark 2.2 below for a relaed discussion. b. In he proof of Theorem 2.1, we will need o perform a change of measure associaed o a maringale of he form dm = Mχ a dw in which χ a may explode a a speed a 2 if a is no bounded. See Sep 1. of he proof of Theorem 2.1. In order o ensure ha his local maringale is well-defined, and is acually a maringale, one should impose very srong inegrabiliy condiions on a. In order o simplify he presenaion, we herefore sick o bounded conrols. Many oher choices are possible. Noe however ha, in he case f, a large class of opions leads o hedging sraegies in our se Γ, up o a sligh payoff smoohing o avoid he explosion of he dela or he gamma a mauriy. This implies ha, alhough he perfec hedging sraegy may no belong o Γ, a leas i is a limi of elemens of Γ and he super-hedging prices coincide. 2.2 Dynamic programming Our conrol problem is a sochasic arge problem as sudied in 18]. The aim of his secion is o show ha i saisfies a version of heir geomeric dynamic programming principle. However, he value funcion w is no amenable o dynamic programming per se. The reason is ha i assumes a zero iniial sock holding a ime, while he posiion Y will (in general) no be zero a a laer ime. I is herefore a priori no possible o compare he laer wealh process V wih he corresponding super-hedging price w(, X ). 16

17 Sill, a version of he geomeric dynamic programming principle can be obained if we inroduce he process ˆX,z,γ := x(x,z,γ, Y,z,γ ) (2.4) which represens he value of he sock immediaely afer liquidaing he sock posiion. We refer o Remark 2.2 below for he reason why par (ii) of he following dynamic programming principle is saed in erms of (w k ) k 1 insead of w. Proposiion 2.2 (GDP). Fix (, x, v), T ] R R. (i) If v > w(, x) hen here exiss γ Γ and y R such ha V,z,γ w(,,z,γ ˆX ) + I(,z,γ ˆX, Y,z,γ ), for all sopping ime, where z := (x(x, y), y, v + I(x, y)). (ii) Fix k 1. If v < w 2k+2 (, x) hen we can no find γ Γ k, y k, k] and a sopping ime such ha V,z,γ > w k (,,z,γ ˆX ) + I(,z,γ ˆX, Y,z,γ ) wih z := (x(x, y), y, v + I(x, y)). Proof. Sep 1. In order o ransform our sochasic arge problem ino a ime consisen one, we inroduce he auxiliary value funcion corresponding o an iniial holding y in socks: ŵ(, x, y) := inf k 1 ŵ k (, x, y) where ŵ k (, x, y) := inf{v : G k (, x, y, v) }. Noe ha w k+1 (, x) inf{v : y k, k] s.. G k (, x(x, y), y, v + I(x, y)) }. This follows from (1.23)-(1.24). Since x(x(x, y), y) = x, see Proposiion 1.3, his implies ha ŵ k (, x, y) w k+1 (, x(x, y)) + I(x(x, y), y), (2.5) for y k. Similarly, since I(x, y) + y x(x, y) = I(x(x, y), y) by Proposiion 1.3, we have ŵ k+1 (, x, y) w k (, x(x, y)) + I(x(x, y), y). (2.6) Sep 2. a. Assume ha v > w(, x). The definiion of w implies ha we can find y R and γ G(, z) where z := (x(x, y), y, v + I(x, y)). By he argumens of 17

18 18, Sep 1 proof of Theorem 3.1], V,z,γ ŵ(, X,z,γ, Y,z,γ ), for all sopping ime. Then, (2.5) applied for k provides (i). b. Assume now ha we can find γ Γ k, y k, k] and a sopping ime such ha V,z,γ,z,γ > (w k + I)(, ˆX, Y,z,γ ), where z := (x(x, y), y, v + I(x, y)). By (2.4)-(2.6), V,z,γ > ŵ k+1 (, X,z,γ, Y,z,γ ), and i follows from 18, Sep 2 proof of Theorem 3.1] and Corollary A.1 ha v + I(x, y) ŵ 2k+1 (, x(x, y), y). We conclude ha (ii) holds by appealing o (2.5) and he ideniies x(x(x, y), y) = x and I(x(x(x, y), y), y) = I(x, y), see Proposiion 1.3. We conclude his secion wih purely echnical consideraions ha jusify he form of he above dynamic programming principle. They are of no use for he laer developmens bu may help o clarify our approach. Remark 2.2. Par (ii) of Proposiion 2.2 can no be saed in erms of w. The reason is ha measurable selecion echnics can no be used wih he se Γ. Indeed, if ω γω] Γ, hen he corresponding bounds depend on ω and are no uniform: a measurable family of conrols {γω], ω Ω} does no permi o consruc an elemen in Γ. Par (i) of Proposiion 2.2 only requires o use a condiioning argumen, which can be done wihin Γ. Remark 2.3. A version of he geomeric dynamic programming principle also holds for (ŵ k ) k 1, his is a by-produc of he above proof. I is herefore emping o ry o derive a pde for he funcion ŵ. However, he fac ha he conrol b appears linearly in he dynamics of (X, Y, V ) makes his problem highly singular, and sandard approaches do no seem o work. We shall see in Lemma 2.1 ha his singulariy disappears in he parameerizaion x(x, Y ) used in Proposiion 2.2. Moreover, hedging implies a conrol on he diffusion par of he dynamics which ranslaes ino a srong relaion beween Y and he space gradien Dŵ(, X, Y ). This would lead o a pde se on a curve on he coordinaes (, x, y) depending on Dŵ (he soluion of he pde). 2.3 Pricing equaion In order o undersand wha is he parial differenial equaion ha w should solve, le us sae he following key lemma. Alhough he conrol b appears linearly in he dynamics of (X, Y, V ), he following shows ha he singulariy his may creae does indeed no appear when applying Iô s Lemma o V (ϕ + I)(, ˆX, Y ), recall (2.4), i is absorbed by he funcions x and I (compare wih Remark 2.3). The proof of his Lemma is posponed o Secion

19 Lemma 2.1. Fix (, x, y, v) D, z := (x, y, v), γ = (a, b, ν) Γ. Then, ˆX,z,γ = x(x, y) + + ˆµ( ˆσ( ˆX,z,γ s, Y,z,γ s,z,γ ˆX s, Ys,z,γ )dw s. ) + ( x xµ 1 2 xxa 2 sff )(X,z,γ s, Ys,z,γ )]ds Given ϕ C b, se E,z,γ := V,z,γ (ϕ + I)(, ˆX,z,γ, Y,z,γ ). Then, in which and E,z,γ E,z,γ = Y,z,γ s + + Y,z,γ s ˆF ϕ(s, ˇY,z,γ s ](µ f fa 2 s/2)(xs,z,γ )ds ˇY,z,γ s ˆX,z,γ s ˇY,z,γ := Y,z,γ + ˆX,z,γ X,z,γ f(x,z,γ ) ]σ(xs,z,γ )dw s, Ys,z,γ )ds + x ϕ(, ˆX,z,γ ) f( ˆX,z,γ ) f(x,z,γ ) where for (x, y ) R R ˆF ϕ := ϕ ˆµ x ϕ + I] 1 2 ˆσ2 2 xxϕ + I], ˆµ(x, y ) := xxxσ 2 ](x(x, y ), y ) and ˆσ(x, y ) := (σ x x)(x(x, y ), y ). Le us now appeal o Proposiion 2.2 and apply Lemma 2.1 o ϕ = w, assuming ha w is smooh and ha Proposiion 2.2(i) is valid even if we sar from v = w(, x), i.e. assuming ha he inf in he definiion of w is a min. Wih he noaions of he above lemma, Proposiion 2.2(i) formally applied o = + leads o de,z,γ = (y ŷ) { µ ff a 2 /2)(x(x, y))]d + σ(x(x, y))dw } + ˆF w(, ˆx, y)d in which ŷ = y + ˆx x(x, y) f(x(x, y)) + f(ˆx) xw(, ˆx) f(x(x, y)) 19 and ˆx = x(x(x, y), y) = x.

20 Remaining a a formal level, his inequaliy canno hold unless y = ŷ, because σ, and ˆF w(, x, ŷ) = ˆF w(, ˆx, y). This means ha w should be a super-soluion of where, for a smooh funcion ϕ, F ϕ(, x) := ˆF ϕ(, x, ŷϕ](, x)) = (2.7) ŷϕ](, x) := x 1 (x, x + f(x) x ϕ(, x)) and x 1 denoes he inverse of x(x, ). From (ii) of Proposiion 2.2, we can acually (formally) deduce ha he above inequaliy should be an equaliy, and herefore ha w should solve (2.7). In order o give a sense o he above, we assume ha { x(x, ) is inverible for all x R (x, z) R R x 1 (x, z) is C 2 (H2). In view of (2.3), we herefore expec w o be a soluion of F ϕ1,t + (ϕ G)1 {T } = on, T ] R. (2.8) Since w may no be smooh and (ii) of Proposiion 2.2 is saed in erms of w k insead of w, we need o consider he noion of viscosiy soluions and he relaxed semi-limis of (w k ) k 1. We herefore define w (, x) := lim inf w k(, x ) and w (, x) := lim sup w k (, x ), (,x,k) (,x, ) (,x,k) (,x, ) in which he limis are aken over < T, as usual. Noe ha w acually coincides wih he lower-semiconinuous envelope of w, his comes from he fac ha w = inf k 1 w k = lim k w k, by consrucion. We are now in posiion o sae he main resul of his secion. In he following, we assume ha { G is coninuous and Gk G uniformly on compac ses. w and w (H3) are finie on, T ] R. The firs par of (H3) will be used o obain he boundary condiion. The second par is naural since oherwise our problem would be ill-posed. 2

21 Theorem 2.1 (Pricing equaion). The funcions w and w are respecively a viscosiy super- and a subsoluion of (2.8). If hey are bounded and inf f >, hen w = w = w and w is he unique bounded viscosiy soluion of (2.8). If in addiion G is bounded and C 2 wih G, G, G Hölder coninuous, hen w C 1,2 (, T ) R) C (, T ] R). The proof is repored in Secion 2.5. Le us now discuss he verificaion counerpar. Remark 2.4 (Verificaion). Assume ha ϕ is a smooh soluion of (2.8) and ha we can find (a, b) A such ha he following sysem holds on, T ): X = x + x(x, ŷϕ](, x)) + + Y = ŷϕ](, x) + σ(x s )dw s + f(x s )dy c s (µ(x s ) + a s (σf )(X s ))ds + x(x T, Y T )1 {T } b s ds + a s dw s Y T 1 {T } = x 1 ( ˆX, ˆX + (f x ϕ)(, ˆX)) Y T 1 {T } ˆX := x(x, Y ) V = ϕ(, x) + I(x, ŷϕ](, x)) + Y s dxs c + 1 a 2 2 sf(x s )ds + (Y T x(x T, Y T ) + I(X T, Y T ))1 {T }. a. Noe ha ˆX = x(x, Y ) = x(x(x, ŷϕ](, x)), ŷϕ](, x)) = x, recall Proposiion 1.3(i), so ha Y = ŷϕ](, x) = x 1 ( ˆX, ˆX + (f x ϕ)(, ˆX )). We herefore need o find (a, b) such ha X = x( ˆX, Y ) = ˆX + (f x ϕ)(, ˆX). This amouns o solving: σ(x) + f(x)a = ˆσ( ˆX, Y ) x ψ(, ˆX) f(x)b + (µ + aσf )(X) = (ˆµ( ˆX, Y ) + ( x xµ 1 2 xxa 2 sff )(X, Y )) x ψ(, X) ˆσ2 ( ˆX, Y ) 2 xxψ(, ˆX) where ψ(, x) := x + (f x ϕ)(, x). Since f >, his sysem has a soluion. Under addiional smoohness and boundedness assumpion, (a, b) A. b. Le ˇY be as in Lemma 2.1 for he above dynamics. Since X = x( ˆX, Y ) = ˆX + (f x ϕ)(, ˆX) on, T ) by consrucion, we have ˇY = Y on, T ). Then, i follows from Lemma 2.1 and (2.7)-(2.8) ha V T = ϕ(t, ˆX T ) + I( ˆX T, Y T ) = G( ˆX T ) + I( ˆX T, Y T ). 21

22 Since X T = ˆX T and Y T x(x T, Y T ) + I(X T, Y T ) + I( ˆX T, Y T ) =, see Proposiion 1.3, his implies ha V T = G(X T ). Hence, he hedging sraegy consiss in aking an iniial posiion is socks equal o Y = ŷϕ](, x) and hen o use he conrol (a, b) up o T. A final immediae rade is performed a T. In paricular, he number of socks Y is coninuous on (, T ). 2.4 An example: he fixed impac case In his secion, we consider he simple case of a consan impac funcion f: f(x) = λ > for all x R. This is cerainly a oo simple model, bu his allows us o highligh he srucure of our resul as he pde simplifies in his case. Indeed, for we have x(x, y) = x + yλ and I(x, y) = 1 2 y2 λ, ˆµ(x, y) =, ˆσ(x, y) := σ(x + yλ), ŷϕ] := x ϕ. The pricing equaion is given by a local volailiy model in which he volailiy depends on he hedging price iself, and herefore on he claim (g, g 1 ) o be hedged: = ϕ(, x) 1 2 σ2 (x + x ϕλ) 2 xxϕ(, x). As for he process Y in he verificaion argumen of Remark 2.4, i is given by Y = x ϕ(, ˆX) = x ϕ(, X λy ). This shows ha he hedging sraegy (if i is well-defined) consiss in following he usual -hedging sraegy bu for a = x ϕ compued a he value of he sock ˆX which would be obained if he posiion in socks was liquidaed. Noe ha we obain he usual hea equaion when σ is consan. This is expeced, showing he limiaion of he fixed impac model. To explain his, le us consider he simpler case g 1 = and use he noaions of Remark 2.4. We also se µ = for ease of noaions. Since σ is consan, he sraegy Y does no affec he coefficiens in he dynamics of X, i jus produces a shif λdy each ime we buy or sell. Since Y T = (afer he final jump), and Y =, he oal impac is null: 22

23 X T = X + σ(w T W ). As for he wealh process, we have T T V T = ϕ(, x) Y 2 λ + Y s dxs c T = ϕ(, x) + Y s σdw s λ(y 2 YT 2 ) + = ϕ(, x) + T Y s σdw s. a 2 sλds Y 2 T λ Y 2 T λ T λy s dys c + 1 T a 2 2 sλds Oherwise saed, he liquidaion coss are cancelled: when buying, he rader pays a cos bu moves he price up, when selling back, he pays a cos again bu sell a a higher price. If here is no effec on he underlying dynamics of X and f is consan, his perfecly cancels. However, he hedging sraegy is sill affeced: Y = x ϕ(, X λy ) on, T ). 2.5 Proof of he pde characerizaion The key lemma We firs provide he proof of our key resul. Proof of Lemma 2.1. To alleviae he noaions, we omi he super-scrips. a. We firs observe from Proposiion 1.3(i) ha x(x, Y ) has coninuous pahs, while Proposiion 1.3(ii) implies ha f x x y x = (and herefore f x x+f 2 xxx 2 xyx = ). Using Iô s Lemma, his leads o dx(x s, Y s ) = (µ 1 2 a2 sff )(X s ) x x(x s, Y s )ds + σ(x s ) x x(x s, Y s )dw s σ 2 2 xxx a 2 sf 2 xyx + a 2 s 2 yyx ] (X s, Y s )ds. We now use he ideniy f 2 xyx 2 yyx =, which also follows from Proposiion 1.3(ii), o simplify he above expression ino dx(x s, Y s ) = x x(µ 1 2 a2 sff ) xxxσ 2 ](X s, Y s )ds + (σ x x)(x s, Y s )dw s. b. Similarly, i follows from Proposiion 1.3(iii) ha V I( ˆX, Y ) has coninuous pahs, and so does E by a. Before applying Iô s lemma o derive he dynamics of E, le us observe ha y I(x(x, y), y) = yf(x(x(x, y), y)) = yf(x) 23

24 and ha 2 yyi(x(x, y), y) = y(ff )(x) + f(x). Also noe ha ˆσ(x(x, y), y) = σ(x) x x(x, y). Then, using he dynamics of ˆX derived above, we obain de s =(Y s ˇY s )σ(x s )dw s + (Y s ˇY s )µ 1 2 a2 s(ff )](X s )ds + ˆF ϕ(s, ˆX s, Y s )ds + a s σ(x s )Y s f (X s ) x x(x s, Y s ) 2 xyi( ˆX s, Y s )]ds, where ˇY := x (ϕ + I)(, ˆX, Y ) x x(x, Y ). By Proposiion 1.3(ii)(iv), f(x) 2 xyi(x, y) = y yf(x(x, y)) x(x, y)] = y(f f)(x(x, y)). Since x x(x, y) = f(x(x, y))/f(x), see Proposiion 1.3(ii), i follows ha which implies x x(x, Y ) 2 xyi(x(x, Y ), Y ) = Y f (X), de s = (Y s ˇY s )σ(x s )dw s + (Y s ˇY s )µ 1 2 a2 s(ff )](X s )ds + ˆF ϕ(s, ˆX s, Y s )ds. We now deduce from Proposiion 1.3 ha x I( ˆX, Y ) = x( ˆX, Y ) + Y f(x( ˆX, Y )) f( ˆX) x x(x, Y ) = f( ˆX)/f(X), = ˆX X + Y f(x) f( ˆX) so ha ˇY = x ϕ(, ˆX) f( ˆX) f(x) + ˆX X f(x) + Y Super- and subsoluion properies We now prove he super- and subsoluion properies of Theorem 2.1. Supersoluion propery. We firs prove he supersoluion propery. I follows from similar argumens as in 5]. Le ϕ be a Cb funcion, and ( o, x o ), T ] R be a sric (local) minimum poin of w ϕ such ha (w ϕ)( o, x o ) =. a. We firs assume ha o < T and F ϕ( o, x o ) <, and work owards a conradicion. In view of (2.7), ˆF ϕ(, x, y) < if (, x) B and y ŷϕ](, x) ε, 24

25 for some open ball B, T R which conains ( o, x o ), and some ε >. Since x 1 is coninuous, his implies ha ˆF ϕ(, x, y) < if (, x) B and x + x ϕ(, x)f(x) x(x, y) εf(x(x, y)), (2.9) afer possibly changing B and ε. Le ( n, x n ) n be a sequence in B ha converges o ( o, x o ) and such ha w( n, x n ) w ( o, x o ) (recall ha w coïncides wih he lower-semiconinuous envelope of w). Se v n := w( n, x n ) + n 1. I follows from Proposiion 2.2(i) ha we can find (a n, b n, ν n ) = γ n Γ and y n R such ha V n,zn,γn n w( n, n,zn,γn ˆX n ) + I( n,zn,γn ˆX, Y n,zn,γn n ), (2.1) where z n := (x(x n, y n ), y n, v n + I(x n, y n )) and n is he firs exi ime afer n of (, ˆX n,zn,γn n,zn,γn ) from B (noe ha ˆX n = x(x(x n, y n ), y n ) = x n ). In he following, we use he simplified noaions X n, ˆX n, V n and Y n for he corresponding quaniies indexed by ( n, z n, γ n ). Since ( o, x o ) reaches a sric minimum w ϕ, his implies V n n ϕ( n, ˆX n n ) + I( ˆX n, Y n n ) + ι, (2.11) for some ι >. Le ˇY n be as in Lemma 2.1 and observe ha Se ˇY n Y n = ˆX n + x ϕ(, ˆX n )f( ˆX n ) x( ˆX n, Y n ) f(x( ˆX. (2.12) n, Y n )) χ n := (µ f f(a n s ) 2 /2)(X n ) σ(x n ) and consider he measure P n defined by dp n dp = M n n + ˆF ϕ(, ˆX n, Y n ) (Y n ˇY n )σ(x n ) 1 Y n ˇY n ε n where M n = 1 Ms n χ n s dw s. n Then, i follows from (2.11), Lemma 2.1, (2.9) and (2.12) ha ι E Pn V n n (ϕ + I)( n, ˆX n n, Y n n )] v n + I(x n, y n ) (ϕ + I)( n, x(x(x n, y n ), y n ), y n ) = v n ϕ( n, x n ). The righ-hand side goes o, which is he required conradicion. b. We now explain how o modify he above proof for he case o = T. Afer possibly replacing (, x) ϕ(, x) by (, x) ϕ(, x) T, we can assume ha 25

26 ϕ(, x) as T, uniformly in x on each compac se. Then (2.9) sill holds for B of he form T η, T ) B(x o ) in which B(x o ) is an open ball around x o and η > small. Assume ha ϕ(t, x o ) < G(x o ). Then, afer possibly changing B(x o ), we have ϕ(t, ) G ι 1 on B(x o ), for some ι 1 >. Then, wih he noaions of a., we deduce from (2.3)-(2.1) ha V n n ϕ( n, ˆX n n ) + I( ˆX n, Y n n ) + ι 1 ι 2, in which ι 2 := min{(w ϕ)(, x) : (, x) o η, T ) B(x o )} > and n is now he minimum beween T and he firs ime afer n a which ˆX n exiss B(x o ). The conradicion is hen deduced from he same argumens as above. Subsoluion propery. We now urn o he subsoluion propery. Again he proof is close o 5], excep ha we have o accoun for he specific form of he dynamic programming principle saed in Proposiion 2.2(ii). Le ϕ be a Cb funcion, and ( o, x o ), T ] R be a sric (local) maximum poin of w ϕ such ha (w ϕ)( o, x o ) =. By 2, Lemma 4.2], we can find a sequence (k n, n, x n ) n 1 such ha k n, ( n, x n ) is a local maximum poin of wk n ϕ and ( n, x n, w kn ( n, x n )) ( o, x o, w ( o, x o )). a. As above, we firs assume ha o < T. Se ϕ n (, x) := ϕ(, x) + n 2 + x x n 4 and assume ha F ϕ( o, x o ) >. Then, F ϕ n > on a open neighborhood B of ( o, x o ) which conains ( n, x n ), for all n large enough. Since we are going o localize he dynamics, we can modify ϕ n, σ, µ and f in such a way ha hey are idenically equal o ouside a compac A B. I hen follows from Remark 2.4 a. ha, afer possibly changing n 1, we can find (b n, a n ) A kn such ha he following admis a srong soluion: X n = x n + x(x n, ŷϕ n ]( n, x n )) + + Y n = ŷϕ n ]( n, x n ) + n (µ(x s ) + a n s (σf )(X n s ))ds n b n s ds + = x 1 ( ˆX n, ˆX n + (f x ϕ n )(, ˆX n )) ˆX n := x(x n, Y n ) V n = v n + I(x n, ŷϕ n ]( n, x n )) + n σ(x n s )dw s + n a n s dw s n Y n s dx n,c s f(xs n )dys n,c n n (a n s ) 2 f(x n s )ds. In he above, we have se v n := w kn ( n, x n ) n 1. Observe ha he consrucion of Y n ensures ha i coincides wih he corresponding process ˇY n of Lemma 2.1. Also noe ha ˆX n n = x(x(x n, y n ), y n ) = x n, and le n be he firs ime afer 26

27 n a which (, ˆX n ) exiss B. By applying Iô s Lemma, using Lemma 2.1 and he fac ha F ϕ n on B, we obain V n n (ϕ n + I)( n, ˆX n n, Y n n ) + v n ϕ n ( n, x n ). Le 2ε := min{ o 2 + x x o 4, (, x) B}. For n large enough, he above implies V n n (w kn 1 + I)( n, ˆX n n, Y n n ) + ε + ι n, where ι n := (ϕ n w kn 1 )( n 1, x n 1 ) + v n ϕ n ( n, x n ) converges o. Hence, we can find n such ha V n n > (w kn 1 + I)( n, ˆX n n, Y n n ). Now observe ha we can change he subsequence (k n ) n 1 in such a way ha k n 2k n Then, v n = w kn ( n, x n ) n 1 < w 2kn 1 +2( n, x n ), which leads o a conradicion o Proposiion 2.2(ii). b. I remains o consider he case o = T. As in Sep 1., we only explain how o modify he argumen used above. Le (v n, k n, n, x n ) be as in a. We now se ϕ n (, x) := ϕ(, x) + T + x x n 4. Since ϕ n (, x) as T, we can find n large enough so ha F ϕ n on n, T ) B(x o ) in which B(x o ) is an open ball around x o. Assume ha ϕ(t, x o ) > G(x o ) + η for some η >. Then, afer possibly changing B(x o ), we can assume ha ϕ n (T, ) G + η on B(x o ). We now use he same consrucion as in a. bu wih n defined as he minimum beween T and he firs ime where ˆX n exiss B(x o ). We obain V n n (ϕ n + I)( n, ˆX n n, Y n n ) + v n ϕ n ( n, x n ). Le 2ε := min{ x x o 4, x B(x o )}. For n large enough, he above implies V n n w kn 1 ( n, ˆX n n )1 n<t + G( ˆX n n )1 n=t + I( ˆX n n, Y n n ) + ε η + ι n, where ι n converges o. By (2.3) and (H3), V n n > w kn 1 ( n, ˆX n n ) + I( ˆX n n, Y n n ), for n large enough. We conclude as in a Comparison In all his secion, we work under he addiional condiion inf f >. (2.13) 27

28 Direc compuaions (use (2.7) and Proposiion 1.3) show ha ˆF ϕ is of he form ˆF ϕ = ϕ B(, f x ϕ) x ϕ 1 2 A2 (, f x ϕ) xx ϕ L(, f x ϕ) (2.14) where A, B and L : (, x, p), T ] R R R are Lipschiz coninuous funcions. Le Φ be a soluion of he ordinary differenial equaion Φ () = f(φ()), R. (2.15) Then, Φ is a bijecion on R (as f is Lipschiz and 1/f is bounded) and he following is an immediae consequence of he definiion of viscosiy soluions. Lemma 2.2. Le v be a supersoluion (resp. subsoluion) of (2.8). Fix ρ >. Then, ṽ defined by ṽ(, x) = e ρ v(, Φ(x)), is a supersoluion (resp. subsoluion) of = ρϕ ϕ B(Φ, e ρ x ϕ)/f(φ) 12 ] A2 (Φ, e ρ x ϕ)f (Φ)/f(Φ) 2 x ϕ 1 2 A2 (Φ, e ρ x ϕ) xx ϕ/f(φ) 2 e ρ L(Φ, e ρ x ϕ) (2.16) wih he erminal condiion ϕ(t, ) = e ρt G(Φ). (2.17) To prove ha comparison holds for (2.8), i suffices o prove ha i holds for (2.16)-(2.17). For he laer, his is a consequence of he following resul. I is raher sandard bu we provide he complee proof by lack of a precise reference. Theorem 2.2. Le O be an open subse of R, u (resp. v) be a upper-semiconinuous subsoluion (resp. lower-semiconinuous supersoluion) on, T ) O of: ρϕ ϕ B(, e ρ x ϕ) x ϕ 1 2Ā2 (, e ρ x ϕ) xx ϕ e ρ L(, e ρ x ϕ) = (2.18) where ρ > is consan, Ā, B and L : (, x, p), T ] O R R are Lipschiz coninuous funcions. Suppose ha u and v are bounded and saisfy u v on he parabolic boundary of, T ) O, hen u v on he closure of, T ] O. 28

29 Proof. Suppose o he conrary ha and define, for n >, Θ n := sup (,x,y),t ) O 2 sup (u v) >,,T ] O ( u(, x) v(, y) n 2 x y 2 1 ) 2n x 2. Then, here exiss ι >, such ha Θ n ι for n large enough. Since u and v are bounded and u v on he parabolic boundary of he domain, we can find ( n, x n, y n ), T ) O 2 which achieves he above supremum. As usual, we apply Ishii s Lemma combined wih he sub- and super-soluion properies of u and v, and he Lipschiz coninuiy of Ā, B and L o obain, wih he noaion p n := n(x n y n ), ρ(u( n, x n ) v( n, y n )) B(x n, e ρn (p n + 1 n x n)) B(y n, e ρn p n )]p n + 1 n x n B(x n, e ρn (p n + 1 n x n)) + 3n 2 Ā(x n, e ρn (p n + 1 n x n)) Ā(y n, e ρn p n )] 2 + 2nĀ2 1 (x n, e ρn (p n + 1 n x n)) +e ( L(x ρn n, e ρn (p n + 1 ) n x n)) L(y n, e ρn p n ) ( C n(x n y n ) 2 + x n y n + 1 n x2 n + 1 ) n for some consan C which does no depend on n. In view of Lemma 2.3 below, and since ρ > and u( n, x n ) v( n, y n ) Θ n ι, he above leads o a conradicion for n large enough. We conclude wih he proof of he echnical lemma ha was used in our argumens above. Lemma 2.3. Le Ψ be a bounded upper-semiconinuous funcion on, T ] R 2, and Ψ i, i = 1, 2, be wo non-negaive lower-semiconinuous funcions on R such ha {Ψ 1 = } = {}. For n >, se Θ n := sup (,x,y),t ] R 2 ( Ψ(, x, y) nψ 1 (x y) 1 n Ψ 2(x) 29 )

30 and assume ha here exiss (ˆ n, ˆx n, ŷ n ), T ] R 2 such ha: Θ n = Ψ(ˆ n, ˆx n, ŷ n ) nψ 1 (ˆx n ŷ n ) 1 n Ψ 2(ˆx n ). Then, afer possibly passing o a subsequence, (i) (ii) lim nψ 1 1(ˆx n ŷ n ) = and lim n n n Ψ 2(ˆx n ) =. lim Θ n = sup Ψ(, x, x). n (,x),t ] O Proof. For laer use, se R := R { } { } and noe ha we can exend Ψ as a bounded upper-semiconinuous funcion on, T ] R 2. Se M := Ψ(, x, x), and selec a sequence ( n, x n ) n 1 such ha sup (,x),t ] R lim Ψ( 1 n, x n, x n ) = M and lim n n n Ψ 2(x n ) =. Le C be a upper-bound for Ψ. Then, C nψ 1 (ˆx n ŷ n ) 1 n Ψ 2(ˆx n ) Ψ(ˆ n, ˆx n, ŷ n ) nψ 1 (ˆx n ŷ n ) 1 n Ψ 2(ˆx n ) Ψ( n, x n, x n ) 1 n Ψ 2(x n ) M ε n where ɛ n. Since Ψ 1 and Ψ 2 are non-negaive, leing n in he above inequaliy leads o lim n Ψ 1(ˆx n ŷ n ) = which implies lim n (ˆx n ŷ n ) = by he assumpion {Ψ 1 = } = {}. Afer possibly passing o a subsequence, we can hen assume ha lim n ˆx n = lim n ŷ n = ˆx R and ha lim n ˆ n = ˆ, T ]. Since Ψ is upper semiconinuous, he above leads o ( M lim inf nψ 1 (ˆx n ŷ n ) + 1 ) n n Ψ 2(ˆx n ) ( Ψ(ˆ, ˆx, ˆx) lim inf nψ 1 (ˆx n ŷ n ) 1 ) n n Ψ 2(ˆx n ) ( lim sup Ψ(ˆ n, ˆx n, ŷ n ) nψ 1 (ˆx n ŷ n ) 1 ) n n Ψ 2(ˆx n ) M, and our claim follows. 3

31 Remark 2.5. I follows from he above ha, whenever hey are bounded, e.g. if G is bounded, hen w w. Since by consrucion w w w, he hree funcions are equal o he unique bounded viscosiy soluion of (2.8) Smoohness We conclude here he proof of Theorem 2.1 by showing ha exisence of a smooh soluion holds when inf f >, G is bounded and C 2 wih G, G, G Hölder coninuous. (2.19) Noe ha he assumpions inf f > and (H1) imply ha Φ 1 is C 2, recall (2.15). Hence, by he same argumens as in Secion 2.5.3, exisence of a C 1,2 (, T ) R) C (, T ] R) soluion o (2.16)-(2.17) implies he exisence of a C 1,2 (, T ) R) C (, T ] R) soluion o (2.8). As for (2.16)-(2.17), his is a consequence of 11, Thm 14.24], under (H1) and (2.19). I remains o show ha he soluion can be aken bounded, hen he comparison resul of Secion will imply ha w is his soluion. Again, i suffices o work wih (2.16)-(2.17). Le ϕ be a C 1,2 (, T ) R) C (, T ] R) soluion of (2.16)- (2.17). Le S,x be defined by where S,x s = x + s µ S (s, S,x s )ds + s σ S (s, S,x s )dw s, s, µ S := B(Φ, e ρ x ϕ)/f(φ) 1 2 A2 (Φ, e ρ x ϕ)f (Φ)/f(Φ) 2 σ S := A(Φ, e ρ x ϕ)/f(φ). Noe ha he coefficiens of he sde may only be locally Lipschiz. However, hey are bounded (recall (H1) and (2.19)), which is enough o define a soluion by a sandard localizaion procedure. Since σ S is bounded, Iô s Lemma implies ha ϕ(, x)e ρ = E T G(Φ(S,x T )) + L(Φ(X,x s ), e ρs x ϕ(s, X,x s ] ))ds. Since G and L are bounded, by (H1) and (2.19), ϕ is bounded as well. Remark 2.6. We refer o 1] for condiions under which addiional smoohness of he soluion can be proven. 31

32 A Appendix We repor here he measurabiliy propery ha was used in he course of Proposiion 2.2. In he following, A k is viewed as a closed subse of he Polish space L λ 2 endowed wih he usual srong norm opology L λ. 2 We consider an elemen ν U k as a measurable map ω Ω ν(ω) M k where M k denoes he se of non-negaive Borel measures on R, T ] wih oal mass less han k, endowed wih he opology of weak convergence. This opology is generaed by he norm m M := sup{ l(δ, s)m(dδ, ds) : l Lip 1 }, R,T ] in which Lip 1 denoes he class of 1-Lipschiz coninuous funcions bounded by 1, see e.g. 4, Proposiion and Theorem 8.3.2]. Then, U k is a closed subse of he space M k,2 of M k -valued random variables. M k,2 is made complee and separable by he norm ν M2 := E ν 2 ] 1 2 M. See e.g. 8, Chap. 5]. We endow he se of conrols Γ k wih he naural produc opology γ L λ 2 M 2 := ϑ L λ + ν M2, for γ = (ϑ, ν). 2 As a closed subse of he Polish space L λ 2 M k,2, Γ k is a Borel space, for each k 1. See e.g. 3, Proposiion 7.12]. The following sabiliy resul is proved by using sandard esimaes. In he following, we use he noaion Z = (X, Y, V ). Proposiion A.1. For each k 1, here exiss a real consan c k > such ha ( ) Z 1,z 1,γ 1 T Z 2,z 2,γ 2 T L2 c k z1 z 2 + γ 1 γ 2 L λ 2 M 2, for all ( i, z i, γ i ) D Γ k, i = 1, 2. A direc consequence is he coninuiy of (, z, γ) D Γ k Z,z,γ T, which is herefore measurable Corollary A.1. For each k 1, he map (, z, γ) D Γ k Z,z,γ T Borel-measurable. L 2 is 32

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