Almost-sure hedging with permanent price impact
aa r X i v : . [ q -f i n . P R ] M a r Almost-sure hedging with permanent priceimpact
B. Bouchard ∗ and G. Loeper † and Y. Zou ‡ October 17, 2018
Abstract
We consider a financial model with permanent price impact. Con-tinuous time trading dynamics are derived as the limit of discrete re-balancing policies. We then study the problem of super-hedging aEuropean option. Our main result is the derivation of a quasi-linearpricing equation. It holds in the sense of viscosity solutions. When itadmits a smooth solution, it provides a perfect hedging strategy.
Keywords:
Hedging, Price impact.
AMS 2010 Subject Classification:
Introduction
Two of the fundamental assumptions in the Black and Scholes approach for optionhedging are that the price dynamics are unaffected by the hedger’s behaviour,and that he can trade unrestricted amounts of asset at the instantaneous valueof the price process. In other words, it relies on the absence of market impactand of liquidity costs or liquidity constraints. This work addresses the problemof option hedging under a price dynamics model that incorporates directly thehedger’s trading activity, and hence that violates those two assumptions. ∗ CEREMADE, Université Paris Dauphine and CREST-ENSAE. Research supported byANR Liquirisk and Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047). † BNP-Paribas and FiQuant - Chaire de finance quantitative ‡ CEREMADE, Université Paris Dauphine and CREST-ENSAE). n the literature, one finds numerous studies related to this topic. Some of themincorporate liquidity costs but no price impact, the price curve is not affected bythe trading strategy. In the setting of [6], this does not affect the super-hedgingprice because trading can essentially be done in a bounded variation manner at themarginal spot price at the origine of the curve. However, if additional restrictionsare imposed on admissible strategies, this leads to a modified pricing equation,which exhibits a quadratic term in the second order derivative of the solution,and renders the pricing equation fully non-linear, and even not unconditionallyparabolic, see [7] and [20]. Another branch of literature focuses on the derivationof the price dynamics through clearing condition. In the papers [9], [16], [15],the authors work on supply and demand curves that arise from “reference” and“program” traders (i.e. option hedgers) to establish a modified price dynamics,but do not take into account the liquidity costs, see also [12]. This approach alsoleads to non-linear pde’s, but the non-linearity comes from a modified volatilityprocess rather than from a liquidity cost source term. Finally, the series of papers[17], [19], [14] address the liquidity issue indirectly by imposing bounds on the“gamma” of admissible trading strategies, no liquidity cost or price impact aremodeled explicitly.More recently, [13] and [1] have considered a novel approach in which the pricedynamic is driven by the sum of a classical Wiener process and a (locally) linearmarket impact term. The linear market impact mechanism induces a modifiedvolatility process, as well as a non trivial average execution price. However, thetrader starts his hedging with the correct position in stocks and does not haveto unwind his final position (this corresponds to “covered” options with delivery).Those combined effects lead to a fully non-linear pde giving the exact replicationstrategy, which is not always parabolic depending on the ratio between the instan-taneous market impact (liquidity costs) and permanent market impact.In this paper we build on the same framework as [13], in the case where theinstantaneous market impact equals the permanent impact (no relaxation effect),and go one step further by considering the effect of (possibly) unwinding the portfo-lio at maturity, and of building the initial portfolio. Consequently the spot “jumps”at initial time when building the hedge portfolio, and at maturity when unwindingit (depending on the nature of the payoff - delivery can also be made in stocks).In this framework, we find that the optimal super-replication strategy follows amodified quasi-linear Black and Scholes pde. Although the underlying model issimilar to the one proposed by the second author [13], the pricing pde is thereforefundamentally different (quasi-linear vs fully non-linear).Concerning the mathematical approach, while in [13] the author focused onexhibiting an exact replication strategy by a verification approach, in this work wefollow a stochastic target approach and derive the pde from a dynamic program- ing principle. The difficulty is that, because of the market impact mechanism,the state process must be described by the asset price and the hedger’s portfolio(i.e. the amount of risky asset detained by the hedger) and this leads to a highlysingular control problem. It is overcome by a suitable change of variable whichallows one to reduce to a zero initial position in the risky asset and state a versionof the geometric dynamic programming principle in terms of the post-portfolio liq-uidation asset price process: the price that would be obtained if the trader wasliquidating his position immediately.The paper is organized as follows. In Section 1, we present the impact ruleand derive continuous time trading dynamics as limits of discrete time rebalancingpolicies. The super-hedging problem is set in Section 2 as a stochastic target prob-lem. We first prove a suitable version of the geometric dynamic programming andthen derive the corresponding pde in the viscosity solution sense. Uniqueness andregularity are established under suitable assumptions. We finally further discussthe case of a constant impact coefficients, to provide a better understanding of the“hedging strategy”. General notations.
Given a function φ , we denote by φ ′ and φ ′′ its first andsecond order derivatives if they exist. When φ depends on several arguments, weuse the notations ∂ x φ , ∂ xx φ to denote the first and second order partial derivativeswith respect to its x -argument, and write ∂ xy φ for the cross second order derivativein its ( x, y ) -argument.All over this paper, Ω is the canonical space of continuous functions on R + starting at , P is the Wiener measure, W is the canonical process, and F = ( F t ) t ≥ is its augmented raw filtration. All random variables are defined on (Ω , F ∞ , P ) . L (resp. L ) denotes the space of (resp. square integrable) R n -valued randomvariables, while L λ (resp. L λ ) stands for the collection of predictable R n -valuedprocesses ϑ (resp. such that k ϑ k L λ := E [ R ∞ | ϑ s | ds ] ). The integer n ≥ is givenby the context and | x | denote the Euclidean norm of x ∈ R n .Given a stochastic process ξ , we shall always denote by ξ c its continuous part. This section is devoted to the derivation of our model with continuous time trading.We first consider the situation where a trading signal is given by a continuous Itôprocess and the position in stock is rebalanced in discrete time. In this case, thedynamics of the stock price and the wealth process are given according to ourimpact rule. A first continuous time trading dynamic is obtained by letting thetime between two consecutive trades vanish. Then, we incorporate jumps as the imit of continuous trading on a short time horizon.We restrict here to a single stock market. This is only for simplicity, theextension to a multi-dimensional market is just a matter of notations. We model the impact of a strategy on the price process through an impact function f : the price variation du to buying a (infinitesimal) number δ ∈ R of shares is δf ( x ) ,if the price of the asset is x before the trade. The cost of buying the additional δ units is given by δx + 12 δ f ( x ) = δ Z δ δ ( x + f ( x ) ι ) dι, in which Z δ δ ( x + f ( x ) ι ) dι should be interpreted as the average cost for each additional unit. Between twotimes of trading τ ≤ τ , the dynamics of the stock is given by the strong solutionof the stochastic differential equation dX t = µ ( X t ) dt + σ ( X t ) dW t . All over this paper, we assume that f ∈ C b and is (strictly) positive, ( µ, σ, σ − ) is Lipschitz and bounded. (H1) Remark 1.1. a. We restrict here to an impact rule which is linear in the size ofthe order. However, note that in the following it will only be applied to order ofinfinitesimal size (at the limit). One would therefore obtain the same final dynamics (1.22) - (1.23) below by considering a more general impact rule δ F ( x, δ ) wheneveris satisfies F ( x, ∂ δδ F ( x,
0) = 0 and ∂ δ F ( x,
0) = f ( x ) . See Remark 1.2 below.Otherwise stated, for our analysis, we only need to consider the value and the slopeat δ = 0 of the impact function.b. A typical example of such a function is F = ∆x where ∆x( x, δ ) := x( x, δ ) − x , (1.1) with x( x, · ) defined as the solution of x( x, · ) = x + Z · f (x( x, s )) ds. (1.2) he curve x has a natural interpretation. For an order of small size ∆ ι , the stockprice jumps from x to x + ∆ ιf ( x ) ≃ x( x, ∆ ι ) . Passing another order of size ∆ ι makes it move again to approximately x(x( x, ∆ ι ) , ∆ ι ) = x( x, ι ) , etc. Passing tothe limit ∆ ι → but keeping the total trade size equal to δ provides asymptoticallya price move equal to ∆x( x, δ ) .This specific curve will play a central role in our analysis, see Section 1.3. We first consider the situation in which the number of shares the trader would liketo hold is given by a continuous Itô process Y of the form Y = Y + Z · b s ds + Z · a s dW s , (1.3)where ( a, b ) ∈ A := ∪ k A k , A k := { ( a, b ) ∈ L λ : | ( a, b ) | ≤ k dt × d P − a . e . } for k > . In order to derive our continuous time trading dynamics, we consider the cor-responding discrete time rebalancing policy set on a time grid t ni := iT /n, i = 0 , . . . , n, n ≥ , and then pass to the limit n → ∞ .If the trader only changes the composition of his portfolio at the discrete times t ni , then he holds Y t ni stocks on each time interval [ t ni , t ni +1 ) . Otherwise stated, thenumber of shares actually held at t ≤ T is Y nt := n − X i =0 Y t ni { t ni ≤ t 12 ( δ nt ni ) f ( X nt ni − ) , (1.7)or equivalently V n = V + n X i =1 [ t ni − ,T ] Y t ni − ( X n ·∧ t ni − − X nt ni − )+ n X i =1 [ t ni ,T ] (cid:20) 12 ( δ nt ni ) f ( X nt ni − ) + Y t ni − δ nt ni f ( X nt ni − ) (cid:21) , (1.8)in which V ∈ R . Let us comment this formula. The first term on the right-hand side corresponds to the evolution of the portfolio value strictly betweentwo trades ; it is given by the number of shares held multiplied by the priceincrement. When a trade of size δ nt ni occurs at time t ni , the costs of buying thestocks is − ( δ nt ni ) f ( X nt ni − ) + δ nt ni X nt ni − but it provides δ nt ni more stocks, on top of the Y nt ni − = Y t ni − units that are already in the portfolio. After the price’s move gener-ated by the trade, the stocks are evaluated at X nt ni . The increment in value du to theprice’s move and the additional position is therefore δ nt ni X nt ni + Y nt ni − ( X nt ni − X nt ni − ) .Since X nt ni − X nt ni − = δ nt ni f ( X nt ni − ) , we obtain (1.8), a compact version of which isgiven in (1.7).Our continuous time trading dynamics are obtained by passing to the limit n → ∞ , i.e. by considering faster and faster rebalancing strategies. Proposition 1.1. Let Z := ( X, Y, V ) where Y is defined as in (1.3) for some ( a, b ) ∈ A , and ( X, V ) solves X = X + Z · σ ( X s ) dW s + Z · f ( X s ) dY s + Z · ( µ ( X s ) + a s ( σf ′ )( X s )) ds (1.9) nd V = V + Z · Y s dX s + 12 Z · a s f ( X s ) ds. (1.10) Let Z n := ( X n , Y n , V n ) be defined as in (1.5) - (1.4) - (1.7) . Then, there exists aconstant C > such that sup [0 ,T ] E (cid:2) | Z n − Z | (cid:3) ≤ Cn − for all n ≥ . Proof. This follows standard arguments and we only provide the main ideas. Inall this proof, we denote by C a generic positive constant which does not dependon n nor i ≤ n , and may change from line to line. We shall use repeatedly (H1)and the fact that a and b are bounded by some constant k , in the dt × d P -a.e. sense.a. The convergence of the process Y n is obvious: sup [0 ,T ] E (cid:2) | Y n − Y | (cid:3) ≤ Cn − . (1.11)For later use, set ∆ X n := X − X n and also observe that the estimate sup [ t ni − ,t ni ) E (cid:2) | ∆ X n | (cid:3) ≤ E h | ∆ X nt ni − | i (1 + Cn − ) + Cn − , (1.12)is standard. We now set ˜ X nt := X nt + A nt + B nt , t ni − ≤ t ≤ t ni , where A nt := Z tt ni − f ( X ns ) dY s + Z tt ni − a s ( σf ′ )( X ns ) dsB nt := Z tt ni − ( Y s − Y t ni − )( µf ′ + 12 σ f ′′ )( X ns ) ds + Z tt ni − ( Y s − Y t ni − )( σf ′ )( X ns ) dW s . Since A nt ni + B nt ni = δ nt ni f ( X nt ni − ) , we have ˜ X nt ni = X nt ni . Set ∆ ˜ X n := X − ˜ X n , β := bf + aσf ′ and β := af , so that d | ∆ ˜ X nt | = 2∆ ˜ X nt [( µ + β t )( X t ) − ( µ + β t )( X nt )] dt + [( σ + β t )( X t ) − ( σ + β t )( X nt ) − ( Y t − Y t ni − )( σf ′ )( X nt )] dt − 2∆ ˜ X nt ( Y t − Y t ni − )( µf ′ + 12 σ f ′′ )( X nt ) dt + 2∆ ˜ X nt [( σ + β t )( X t ) − ( σ + β t )( X nt )] dW t − 2∆ ˜ X nt ( Y t − Y t ni − )( σf ′ )( X nt ) dW t . n view of (1.11)-(1.12), this implies E h | ∆ ˜ X nt | i ≤ E h | ∆ X nt ni − | i + C E "Z tt ni − ( | ∆ ˜ X ns | + | X s − X ns | + | Y s − Y t ni − | ) ds ≤ E h | ∆ X nt ni − | i (1 + Cn − ) + C E "Z tt ni − | ∆ ˜ X ns | ds + n − , and therefore sup [ t ni − ,t ni ] E h | ∆ ˜ X n | i ≤ E h | ∆ X nt ni − | i (1 + Cn − ) + Cn − , (1.13)by Gronwall’s Lemma. Since ˜ X nt ni = X nt ni , this shows that E h | ∆ X nt ni | i ≤ Cn − for all i ≤ n. Plugging this inequality in (1.12), we then deduce sup [ t ni − ,t ni ] E (cid:2) | ∆ X n | (cid:3) ≤ Cn − for all i ≤ n. (1.14)b. We now consider the difference V − V n . It follows from (1.8) that V nt ni = V nt ni − + Z t ni t ni − Y t ni − µ ( X ns ) ds + Z t ni t ni − Y t ni − σ ( X ns ) dW s + Z t ni t ni − (cid:18) a s f ( X ns ) + Y t ni − a s ( f ′ σ )( X ns ) (cid:19) ds + Z t ni t ni − Y t ni − f ( X ns ) dY s + Z t ni t ni − α ns ds + Z t ni t ni − α ns dW s where, by (1.11), α n and α n are adapted processes satisfying sup [ t ni − ,t ni ) E [ | α n | + | α n | ] ≤ Cn − . In view of (1.11)-(1.14), this leads to V nt ni = V nt ni − + V t ni − V t ni − + Z t ni t ni − γ ns ds + Z t ni t ni − γ ns dW s (1.15)where γ n and γ n are adapted processes satisfying sup [ t ni − ,t ni ) E [ | γ n | + | γ n | ] ≤ Cn − . (1.16) et ˜ V nt := V nt ni − + V t − V t ni − + Z tt ni − γ ns ds + Z tt ni − γ ns dW s , t ni − ≤ t ≤ t ni . Then, by applying Itô’s Lemma to | ˜ V nt − V t | , using (1.16) and Gronwall’s Lemma,we obtain sup [ t ni − ,t ni ] E h | ˜ V n − V | i ≤ E h | V nt ni − − V t ni − | i (1 + Cn − ) + Cn − , so that, by the identity ˜ V nt ni = V nt ni and an induction, E h | V nt ni − V t ni | i ≤ Cn − , i ≤ n. We conclude by observing that E (cid:2) | V nt − V t | (cid:3) ≤ C E h | V nt ni − − V t ni − | + | V nt ni − − V nt | + | V t ni − − V t | i ≤ C (cid:16) E h | V nt ni − − V t ni − | i + n − (cid:17) , for t ni − ≤ t < t ni . (cid:3) Remark 1.2. If the impact function δf ( x ) was replaced by a more general C b oneof the form F ( x, δ ) , with F ( x, 0) = ∂ δδ F ( x, 0) = 0 , the computations made in theabove proof would only lead to terms of the from ∂ δ F ( X, dY and aσ ( X ) ∂ xδ F ( X, in place of f ( X ) dY and a ( σf ′ )( X ) in the dynamics (1.9) . Similarly, the term a f ( X ) would be replaced by a ∂ δ F ( X, in (1.10) . We now explain how we incorporate jumps in our dynamics. Let U k denote theset of random { , · · · , k } -valued measures ν supported by [ − k, k ] × [0 , T ] that areadapted in the sense that t ν ( A × [0 , t ]) is adapted for all Borel subset A of [ − k, k ] . We set U := ∪ k ≥ U k . Note that an element ν of U can be written in the form ν ( A, [0 , t ]) = k X j =1 { ( δ j ,τ j ) ∈ A × [0 ,t ] } (1.17) n which ≤ τ < · · · < τ k ≤ T are stopping times and each δ j is a real-valued F τ j -random variable.Then, given ( a, b, ν ) ∈ A × U , we define the trading signal as Y = Y − + Z · b s ds + Z · a s dW s + Z · Z δν ( dδ, ds ) , (1.18)where Y − ∈ R .In view of the previous sections, we assume that the dynamics of the stock priceand portfolio value processes are given by (1.9)-(1.10) when Y has no jump. Weincorporate jumps by assuming that the trader follows the natural idea of splittinga large order δ j into small pieces on a small time interval. This is a current practicewhich aims at avoiding having a too large impact, and paying a too high liquiditycost. Given the asymptotic already derived in the previous section, we can reduceto the case where this is done continuously at a constant rate δ j /ε on [ τ j , τ j + ε ] , forsome ε > . We denote by ( X − , V − ) the initial price and portfolio values. Then,the number of stocks in the portfolio associated to a strategy ( a, b, ν ) ∈ A k × U k isgiven by Y ε = Y + k X j =1 [ τ j ,T ] (cid:2) − δ j + ε − δ j ( · ∧ ( τ j + ε ) − τ j ) (cid:3) , (1.19)and the corresponding stock price and portfolio value dynamics are X ε = X − + Z · σ ( X εs ) dW s + Z · f ( X εs ) dY εs + Z · ( µ ( X εs ) + a s ( σf ′ )( X εs )) ds (1.20) V ε = V − + Z · Y εs dX εs + 12 Z · a s f ( X εs ) ds. (1.21)When passing to the limit ε → , we obtain the convergence of Z ε := ( X ε , Y ε , V ε ) to Z = ( X, Y, V ) with ( X, V ) defined in (1.22)-(1.23) below. In the following, weonly state the convergence of the terminal values, see the proof for a more completedescription. It uses the curve x defined in (1.2) above, recall also (1.1). Proposition 1.2. Given ( a, b, ν ) ∈ A × U , let Z = ( X, Y, V ) be defined by (1.18) nd X = X − + Z · σ ( X s ) dW s + Z · f ( X s ) dY cs + Z · ( µ ( X s ) + a s ( σf ′ )( X s )) ds + Z · Z ∆x( X s − , δ ) ν ( dδ, ds ) (1.22) V = V − + Z · Y s dX cs + 12 Z · a s f ( X s ) ds + Z · Z ( Y s − ∆x( X s − , δ ) + I ( X s − , δ )) ν ( dδ, ds ) (1.23) where I ( x, z ) := Z z sf (x( x, s )) ds, for x, z ∈ R . (1.24) Set Z ε := ( X ε , V ε , Y ε ) . Then, there exists a constant C > such that E (cid:2) | Z εT + ε − Z T | (cid:3) ≤ C ( ε + P [sup t ≤ T ν ( R , [ t, t + ε ]) ≥ ) , for all ε ∈ (0 , . Moreover, lim ε → P [sup t ≤ T ν ( R , [ t, t + ε ]) ≥ 2] = 0 . Proof. In all this proof, we denote by C a generic positive constant which doesnot depend on ε , and may change from line to line. Here again, we shall userepeatedly (H1) and the fact that a and b are bounded by some constant k , in the dt × d P -a.e. sense.Let ν be of the form (1.17) for some k ≥ and note that the last claim simplyfollows from the fact that { τ j +1 − τ j ≥ ε } ↑ Ω up to a P -null set for all j ≤ k .Step 1. We first consider the case where τ j +1 ≥ τ j + ε for all j ≥ . Again, theestimate on | Z εT + ε − Z T | follows from simple observations and standard estimates,and we only highlight the main ideas. We will indeed prove that for ≤ j ≤ k + 1 E " sup [ τ j − + ε,τ j ) | Z − Z ε | + sup ≤ s ≤ ε E [ | Z τ j + s − Z ετ j + ε | ≤ Cε, (1.25)where we use the convention τ = 0 and τ k +1 = T . The result is trivial for ( Y ε , Y ) since they are equal on each intervalle [ τ j − + ε, τ j ) and ( a, b ) is bounded.a. We first prove a stronger result for ( X ε , X ) . Fix p ∈ { , } . Let x ε be thesolution of the ordinary differential equation x εt = X τ j − + Z t δ j ε f (x εs ) ds. et ∆ X ε := X ε − x ε ·− τ j . Itô’s Lemma leads to d (∆ X εt ) p = p (∆ X εt ) p − α ,εt dt + p ( p − X εt ) p − ( α ,εt ) dt + p (∆ X εt ) p − α ,εt dW t + p δ j ε (∆ X εt ) p − ( f ( X εt ) − f (x εt − τ j )) dt on [ τ j , τ j + ε ] , in which α ,ε and α ,ε are bounded processes. The inequality x p − ≤ x p − + x p , the Lipschitz continuity of f and Gronwall’s Lemma then imply sup ≤ t ≤ ε E h | X ετ j + t − x εt | p i ≤ C E (cid:20) | X ετ j − − X τ j − | p + Z ε | X ετ j + s − x εs | p − ds (cid:21) . We now use a simple change of variables to obtain x εε = x( X τ j − , δ j ) = X τ j , in which x is defined in (1.2), while sup ≤ t ≤ ε E (cid:2) | X τ j + t − X τ j | p (cid:3) ≤ Cε p . Since X and X ε have the same dynamics on [ τ j + ε, τ j +1 ) , this shows that E " sup [ τ j + ε,τ j +1 ) | X t − X εt | p ≤ C E h | X τ j + ε − X ετ j + ε | p i ≤ C E h | x εε − X ετ j + ε | p + | X τ j + ε − X τ j | p i ≤ C E (cid:20) | X ετ j − − X τ j − | p + Z ε | X ετ j + s − x εs | p − ds + ε p (cid:21) . For p = 2 , this provides E " sup [ τ j − + ε,τ j ) | X − X ε | p + sup ≤ s ≤ ε E [ | X τ j + s − X ετ j + ε | p ≤ Cε p , by induction over j , and the case p = 4 then follows from the above. For later use,note that the estimate sup ≤ t ≤ ε E h | X ετ j + t − x εt | i ≤ Cε (1.26)is a by-product of our analysis. . The estimate on V − V ε is proved similarly. We introduce v εt := V τ j − + Z t δ j ε sf (x εs ) ds + Y τ j − Z t δ j ε f (x εs ) ds = V τ j − + Z t Y εs δ j ε f (x εs ) ds, and obtain a first estimate by using (1.26): E h | V ετ j + t − v εt | i ≤ C E " | V ετ j − − V τ j − | + ε + (cid:18)Z ε ε − Y ετ j + s δ j | X ετ j + s − x εs | ds (cid:19) ≤ C E h | V ετ j − − V τ j − | + ε i , for ≤ t ≤ ε . Then, we observe that v εε = V τ j − + I ( X τ j − , δ j ) + Y τ j − ∆x( X τ j − , δ j ) = V τ j , while sup ≤ t ≤ ε E (cid:2) | V τ j + t − V τ j | (cid:3) ≤ Cε. By using the estimate on X − X ε obtained in a., we then show that E " sup [ τ j + ε,τ j +1 ) | V t − V εt | ≤ C E h | V τ j + ε − V ετ j + ε | + ε i , and conclude by using an induction over j .Step 2. We now consider the general case. We define τ εj +1 = ( ε + τ εj ) ∨ τ j +1 , δ εj +1 = Z ( τ εj ,τ εj +1 ] δν ( dδ, dt ) , j ≥ , where ( τ ε , δ ε ) = ( τ , δ ) . On E ε := { min j ≤ k − ( τ j +1 − τ j ) ≥ ε } , ( τ εj , δ εj ) j ≥ =( τ j , δ j ) j ≥ . Hence, it follows from Step 1. that E (cid:2) | Z εT + ε − Z T | (cid:3) ≤ Cε + C E h | ˜ Z εT + ε | + | Z T | i P [ E cε ] , in which ˜ Z ε stands for the dynamics associated to ( τ εj , δ εj ) j ≥ . It now follows fromstandard estimates that ( ˜ Z εT + ε ) <ε ≤ and Z T are bounded in L . (cid:3) We conclude this section with a proposition collecting some important proper-ties of the functions x and I which appear in Proposition 1.1. They will be usedin the subsequent section. roposition 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 immediate consequence of the Lipschitz continuity of the function f , which ensures uniqueness of the ODE defining x in (1.2). More generally, it hasthe flow property, which we shall use in the following arguments. The assertion(ii) is an immediate consequence of the definition of x : x(x( x, ι ) , y − ι ) = x( x, y ) for ι > and ∂ y x( x, 0) = f ( x ) , so that differentiating at ι = 0 provides (ii). Theidentity in (iii) follows from direct computations. As for (iv), it suffices to writethat I (x( x, ι ) , y − ι ) = R yι ( t − ι ) f (x( x, t )) dt for ι > , and again to differentiate at ι = 0 . (cid:3) Remark 1.3. It follows from Proposition 1.3 that our model allows round tripsat (exactly) zero cost. Namely, if x is the current stock price, v the wealth, and y the number of shares in the portfolio, then performing an immediate jump ofsize δ makes ( x, y, v ) jump to (x( x, δ ) , y + δ, v + y ∆x( x, δ ) + I ( x, δ )) . Passingimmediately the opposite order, we come back to the position (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 Proposition 1.3(i)-(iii). This is a desirable property if one wants to have achance to hedge options perfectly, or more generally to obtain a non-degeneratedsuper-hedging price. We now turn to the super-hedging problem. From now on, we define the admissiblestrategies as the Itô processes of the form Y = y + Z · b s ds + Z · a s dW s + Z · Z δν ( dδ, ds ) (2.1)in which y ∈ R , ( a, b, ν ) ∈ A × U and Y is essentially bounded. If | Y | ≤ k and ( a, b, ν ) ∈ A k × U k , then we say that ( a, b, ν ) ∈ Γ k , k ≥ , and we let Γ := ∪ k ≥ Γ k . We will comment in Remark 2.1 below the reason why we restrict to boundedcontrols. iven ( t, z ) ∈ D := [0 , T ] × R × R × R , we define Z t,z,γ := ( X t,z,γ , Y t,z,γ , V t,z,γ ) as the solution of (1.22)-(2.1)-(1.23) on [ t, T ] associated to γ ∈ Γ and with initialcondition Z t,z,γt − = z . A European contingent claim is defined by its payoff function, a measurable map x ∈ R ( g , g )( x ) ∈ R . The first component is the cash-settlement part, i.e. theamount of cash paid at maturity, while g is the delivery part, i.e. the number ofunits of stocks to be delivered.An admissible strategy γ ∈ Γ allows to super-hedge the claim associated to thepayoff g , starting from the initial conditions z at time t if Z t,z,γT ∈ G where G := { ( x, y, v ) ∈ R × R × R : v − yx ≥ g ( x ) and y = g ( x ) } . (2.2)Recall that V stands for the frictionless liquidation value of the portfolio, it is thesum of the cash component and the value Y X of the stocks held without takingthe liquidation impact into account.We set G k ( t, z ) := { γ ∈ Γ k : Z t,z,γT ∈ G } , G ( t, z ) := ∪ k ≥ G k ( t, z ) , and define the super-hedging price as w ( t, x ) := inf k ≥ w k ( t, x ) where w k ( t, x ) := inf { v : G k ( t, x, , v ) = ∅} . For later use, let us make precise what are the T -values of these functions. Proposition 2.1. Define G k ( x ) := inf { y x( x, y ) + g (x( x, y )) − I ( x, y ) : | y | ≤ k s.t. y = g (x( x, y )) } , x ∈ R , and G := inf k ≥ G k . Then, w k ( T, · ) = G k and w ( T, · ) = G. (2.3) roof. Set z = ( x, , v ) and fix γ = ( a, b, ν ) ∈ Γ . By (1.22)-(1.23), we have Z T,z,γT = (x( x, y ) , y, v + I ( x, y )) with y := Z δν ( dδ, { T } ) . In view of (2.2), Z T,z,γT ∈ G is then equivalent to v + I ( x, y ) − y x( x, y ) ≥ g (x( x, y )) and y = g (x( x, y )) . By definition of w (resp. w k ), we have to compute the minimal v for which thisholds for some y ∈ R (resp. | y | ≤ k ). (cid:3) Remark 2.1. Let us conclude this section with a comment on our choice of theset of bounded controls Γ .a. First, this ensures that the dynamics of X, Y and V are well-defined. Thiscould obviously be relaxed by imposing L λ bounds. However, note that the boundshould anyway be uniform. This is crucial to ensure that the dynamic programmingprinciple stated in Section 2.2 is valid, as it uses measurable selection arguments: ω ϑ [ ω ] ∈ L λ does not imply E h k ϑ [ · ] k L λ i < ∞ . See Remark 2.2 below for arelated discussion.b. In the proof of Theorem 2.1, we will need to perform a change of measureassociated to a martingale of the form dM = − M χ a dW in which χ a may explodeat a speed a if a is not bounded. See Step 1. of the proof of Theorem 2.1. In orderto ensure that this local martingale is well-defined, and is actually a martingale,one should impose very strong integrability conditions on a .In order to simplify the presentation, we therefore stick to bounded controls.Many other choices are possible. Note however that, in the case f ≡ , a large classof options leads to hedging strategies in our set Γ , up to a slight payoff smoothingto avoid the explosion of the delta or the gamma at maturity. This implies that,although the perfect hedging strategy may not belong to Γ , at least it is a limit ofelements of Γ and the super-hedging prices coincide. Our control problem is a stochastic target problem as studied in [18]. The aimof this section is to show that it satisfies a version of their geometric dynamicprogramming principle.However, the value function w is not amenable to dynamic programming perse. The reason is that it assumes a zero initial stock holding at time t , whilethe position Y θ will (in general) not be zero at a later time θ . It is therefore apriori not possible to compare the later wealth process V θ with the correspondingsuper-hedging price w ( θ, X θ ) . till, a version of the geometric dynamic programming principle can be obtainedif we introduce the process ˆ X t,z,γ := x( X t,z,γ , − Y t,z,γ ) (2.4)which represents the value of the stock immediately after liquidating the stockposition.We refer to Remark 2.2 below for the reason why part (ii) of the followingdynamic programming principle is stated in terms of ( w k ) k ≥ instead of w . Proposition 2.2 (GDP) . Fix ( t, x, v ) ∈ [0 , T ] × R × R . (i) If v > w ( t, x ) then there exists γ ∈ Γ and y ∈ R such that V t,z,γθ ≥ w ( θ, ˆ X t,z,γθ ) + I ( ˆ X t,z,γθ , Y t,z,γθ ) , for all stopping time θ ≥ t , where z := (x( x, y ) , y, v + I ( x, y )) . (ii) Fix k ≥ . If v < w k +2 ( t, x ) then we can not find γ ∈ Γ k , y ∈ [ − k, k ] and astopping time θ ≥ t such that V t,z,γθ > w k ( θ, ˆ X t,z,γθ ) + I ( ˆ X t,z,γθ , Y t,z,γθ ) with z := (x( x, y ) , y, v + I ( x, y )) . Proof. Step 1. In order to transform our stochastic target problem into a timeconsistent one, we introduce the auxiliary value function corresponding to an initialholding y in stocks: ˆ w ( t, x, y ) := inf k ≥ ˆ w k ( t, x, y ) where ˆ w k ( t, x, y ) := inf { v : G k ( t, x, y, v ) = ∅} . Note that w k +1 ( t, x ) ≤ inf { v : ∃ y ∈ [ − k, k ] s.t. G k ( t, x( x, y ) , y, v + I ( x, y )) = ∅} .This follows from (1.22)-(1.23). Since x(x( x, − y ) , y ) = x , see Proposition 1.3, thisimplies that ˆ w k ( t, x, y ) ≥ w k +1 ( t, 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 Proposi-tion 1.3, we have ˆ w k +1 ( t, x, y ) ≤ w k ( t, x( x, − y )) + I (x( x, − y ) , y ) . (2.6)Step 2. a. Assume that v > w ( t, x ) . The definition of w implies that we can find y ∈ R and γ ∈ G ( t, z ) where z := (x( x, y ) , y, v + I ( x, y )) . By the arguments of 18, Step 1 proof of Theorem 3.1], V t,z,γθ ≥ ˆ w ( θ, X t,z,γθ , Y t,z,γθ ) , for all stopping time θ ≥ t . Then, (2.5) applied for k → ∞ provides (i).b. Assume now that we can find γ ∈ Γ k , y ∈ [ − k, k ] and a stopping time θ ≥ t such that V t,z,γθ > ( w k + I )( θ, ˆ X t,z,γθ , Y t,z,γθ ) , where z := (x( x, y ) , y, v + I ( x, y )) . By(2.4)-(2.6), V t,z,γθ > ˆ w k +1 ( θ, X t,z,γθ , Y t,z,γθ ) , and it follows from [18, Step 2 proof ofTheorem 3.1] and Corollary A.1 that v + I ( x, y ) ≥ ˆ w k +1 ( t, x( x, y ) , y ) . We con-clude that (ii) holds by appealing to (2.5) and the identities x(x( x, y ) , − y ) = x and I (x(x( x, y ) , − y ) , y ) = I ( x, y ) , see Proposition 1.3. (cid:3) We conclude this section with purely technical considerations that justify theform of the above dynamic programming principle. They are of no use for the laterdevelopments but may help to clarify our approach. Remark 2.2. Part (ii) of Proposition 2.2 can not be stated in terms of w . Thereason is that measurable selection technics can not be used with the set Γ . Indeed,if ω γ [ ω ] ∈ Γ , then the corresponding bounds depend on ω and are not uniform: ameasurable family of controls { γ [ ω ] , ω ∈ Ω } does not permit to construct an elementin Γ . Part (i) of Proposition 2.2 only requires to use a conditioning argument, whichcan be done within Γ . Remark 2.3. A version of the geometric dynamic programming principle alsoholds for ( ˆ w k ) k ≥ , this is a by-product of the above proof. It is therefore temptingto try to derive a pde for the function ˆ w . However, the fact that the control b appears linearly in the dynamics of ( X, Y, V ) makes this problem highly singular,and “standard approaches” do not seem to work. We shall see in Lemma 2.1 that thissingularity disappears in the parameterization x( X, − Y ) used in Proposition 2.2.Moreover, hedging implies a control on the diffusion part of the dynamics whichtranslates into a strong relation between Y and the space gradient D ˆ w ( · , X, Y ) .This would lead to a pde set on a curve on the coordinates ( t, x, y ) depending on D ˆ w (the solution of the pde). In order to understand what is the partial differential equation that w should solve,let us state the following key lemma. Although the control b appears linearly inthe dynamics of ( X, Y, V ) , the following shows that the singularity this may createdoes indeed not appear when applying Itô’s Lemma to V − ( ϕ + I )( · , ˆ X , Y ) , recall(2.4), it is absorbed by the functions x and I (compare with Remark 2.3). Theproof of this Lemma is postponed to Section 2.5. emma 2.1. Fix ( t, x, y, v ) ∈ D , z := ( x, y, v ) , γ = ( a, b, ν ) ∈ Γ . Then, ˆ X t,z,γ = x( x, − y )+ Z · t [ˆ µ ( ˆ X t,z,γs , Y t,z,γs ) + ( ∂ x x µ − ∂ x x a s f f ′ )( X t,z,γs , − Y t,z,γs )] ds + Z · t ˆ σ ( ˆ X t,z,γs , Y t,z,γs ) dW s . Given ϕ ∈ C ∞ b , set E t,z,γ := V t,z,γ − ( ϕ + I )( · , ˆ X t,z,γ , Y t,z,γ ) . Then, E t,z,γ − E t,z,γt = Z · t [ Y t,z,γs − ˇ Y t,z,γs ]( µ − f ′ f a s / X t,z,γs ) ds + Z · t [ Y t,z,γs − ˇ Y t,z,γs ] σ ( X t,z,γs ) dW s + Z · t ˆ F ϕ ( s, ˆ X t,z,γs , Y t,z,γs ) ds in which ˇ Y t,z,γ := Y t,z,γ + ˆ X t,z,γ − X t,z,γ f ( X t,z,γ ) + ∂ x ϕ ( · , ˆ X t,z,γ ) f ( ˆ X t,z,γ ) f ( X t,z,γ ) and ˆ F ϕ := − ∂ t ϕ − ˆ µ∂ x [ ϕ + I ] − 12 ˆ σ ∂ xx [ ϕ + I ] , where for ( x ′ , y ′ ) ∈ R × R ˆ µ ( x ′ , y ′ ) := 12 [ ∂ xx x σ ](x( x ′ , y ′ ) , − y ′ ) and ˆ σ ( x ′ , y ′ ) := ( σ∂ x x)(x( x ′ , y ′ ) , − y ′ ) . Let us now appeal to Proposition 2.2 and apply Lemma 2.1 to ϕ = w , assumingthat w is smooth and that Proposition 2.2(i) is valid even if we start from v = w ( t, x ) , i.e. assuming that the inf in the definition of w is a min . With the notationsof the above lemma, Proposition 2.2(i) formally applied to θ = t + leads to ≤ d E t,z,γt = ( y − ˆ y ) (cid:8) [ µ − f f ′ a t / x, y ))] dt + σ (x( x, y )) dW t (cid:9) + ˆ F w ( t, ˆ x, y ) dt in which ˆ y = y + ˆ x − x( x, y ) f (x( x, y )) + ∂ x w ( t, ˆ x ) f (ˆ x ) f (x( x, y )) and ˆ x = x(x( x, y ) , − y ) = x. emaining at a formal level, this inequality cannot hold unless y = ˆ y , because σ = 0 , and ˆ F w ( t, x, ˆ y ) = ˆ F w ( t, ˆ x, y ) ≥ . This means that w should be a super-solution of F ϕ ( t, x ) := ˆ F ϕ ( t, x, ˆ y [ ϕ ]( t, x )) = 0 (2.7)where, for a smooth function ϕ , ˆ y [ ϕ ]( t, x ) := x − ( x, x + f ( x ) ∂ x ϕ ( t, x )) and x − denotes the inverse of x( x, · ) .From (ii) of Proposition 2.2, we can actually (formally) deduce that the aboveinequality should be an equality, and therefore that w should solve (2.7).In order to give a sense to the above, we assume that (cid:26) x( x, · ) is invertible for all x ∈ R ( x, z ) ∈ R × R x − ( x, z ) is C . (H2) In view of (2.3), we therefore expect w to be a solution of F ϕ [0 ,T [ + ( ϕ − G ) { T } = 0 on [0 , T ] × R . (2.8)Since w may not be smooth and (ii) of Proposition 2.2 is stated in terms of w k instead of w , we need to consider the notion of viscosity solutions and the relaxedsemi-limits of ( w k ) k ≥ . We therefore define w ∗ ( t, x ) := lim inf ( t ′ ,x ′ ,k ) → ( t,x, ∞ ) w k ( t ′ , x ′ ) and w ∗ ( t, x ) := lim sup ( t ′ ,x ′ ,k ) → ( t,x, ∞ ) w k ( t ′ , x ′ ) , in which the limits are taken over t ′ < T , as usual. Note that w ∗ actually coincideswith the lower-semicontinuous enveloppe of w , this comes from the fact that w =inf k ≥ w k = lim k →∞ ↓ w k , by construction.We are now in position to state the main result of this section. In the following,we assume that (cid:26) G is continuous and G k ↓ G uniformly on compact sets. w ∗ and w ∗ are finite on [0 , T ] × R . (H3) The first part of (H3) will be used to obtain the boundary condition. The secondpart is natural since otherwise our problem would be ill-posed. heorem 2.1 (Pricing equation) . The functions w ∗ and w ∗ are respectively aviscosity super- and a subsolution of (2.8) . If they are bounded and inf f > ,then w = w ∗ = w ∗ and w is the unique bounded viscosity solution of (2.8) . Ifin addition G is bounded and C with G, G ′ , G ′′ Hölder continuous, then w ∈ C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) . The proof is reported in Section 2.5. Let us now discuss the verification coun-terpart. Remark 2.4 (Verification) . Assume that ϕ is a smooth solution of (2.8) and thatwe can find ( a, b ) ∈ A such that the following system holds on [ t, T ) : X = x + ∆x( x, ˆ y [ ϕ ]( t, x )) + Z · t σ ( X s ) dW s + Z · f ( X s ) dY cs + Z · ( µ ( X s ) + a s ( σf ′ )( X s )) ds + ∆x( X T − , − Y T − ) { T } Y = ˆ y [ ϕ ]( t, x ) + Z · t b s ds + Z · t a s dW s − Y T − { T } = x − ( ˆ X, ˆ X + ( f ∂ x ϕ )( · , ˆ X )) − Y T − { T } ˆ X := x( X, − Y ) V = ϕ ( t, x ) + I ( x, ˆ y [ ϕ ]( t, x )) + Z · t Y s dX cs + 12 Z · a s f ( X s ) ds + ( Y T − ∆x( X T − , − Y T − ) + I ( X T − , − Y T − )) { T } . a. Note that ˆ X t = x( X t , − Y t ) = x(x( x, ˆ y [ ϕ ]( t, x )) , − ˆ y [ ϕ ]( t, x )) = x , recall Propo-sition 1.3(i), so that Y t = ˆ y [ ϕ ]( t, x ) = x − ( ˆ X t , ˆ X t + ( f ∂ x ϕ )( t, ˆ X t )) . We thereforeneed to find ( a, b ) such that X = x( ˆ X, Y ) = ˆ X + ( f ∂ x ϕ )( · , ˆ X ) . This amounts tosolving: σ ( X ) + f ( X ) a = ˆ σ ( ˆ X, Y ) ∂ x ψ ( · , ˆ X ) f ( X ) b + ( µ + aσf ′ )( X ) = (ˆ µ ( ˆ X, Y ) + ( ∂ x x µ − ∂ x x a s f f ′ )( X, − Y )) ∂ x ψ ( · , X )+ 12 ˆ σ ( ˆ X, Y ) ∂ xx ψ ( · , ˆ X ) where ψ ( t, x ) := x + ( f ∂ x ϕ )( t, x ) . Since f > , this system has a solution. Underadditional smoothness and boundedness assumption, ( a, b ) ∈ A .b. Let ˇ Y be as in Lemma 2.1 for the above dynamics. Since X = x( ˆ X, Y ) =ˆ X + ( f ∂ x ϕ )( · , ˆ X ) on [ t, T ) by construction, we have ˇ Y = Y on [ t, T ) . Then, itfollows from Lemma 2.1 and (2.7) - (2.8) that V T − = ϕ ( T, ˆ X T − ) + I ( ˆ X T − , Y T − ) = G ( ˆ X T − ) + I ( ˆ X T − , Y T − ) . ince X T = ˆ X T − and Y T − ∆x( X T − , − Y T − ) + I ( X T − , − Y T − ) + I ( ˆ X T − , Y T − ) = 0 ,see Proposition 1.3, this implies that V T = G ( X T ) . Hence, the hedging strategyconsists in taking an initial position is stocks equal to Y t = ˆ y [ ϕ ]( t, x ) and thento use the control ( a, b ) up to T . A final immediate trade is performed at T . Inparticular, the number of stocks Y is continuous on ( t, T ) . In this section, we consider the simple case of a constant impact function f : f ( x ) = λ > for all x ∈ R . This is certainly a too simple model, but this allows us tohighlight the structure of our result as the pde simplifies in this case. Indeed, for x( x, y ) = x + yλ and I ( x, y ) = 12 y λ, we have ˆ µ ( x, y ) = 0 , ˆ σ ( x, y ) := σ ( x + yλ ) , ˆ y [ ϕ ] := ∂ x ϕ. The pricing equation is given by a local volatility model in which the volatilitydepends on the hedging price itself, and therefore on the claim ( g , g ) to be hedged: − ∂ t ϕ ( t, x ) − σ ( x + ∂ x ϕλ ) ∂ xx ϕ ( t, x ) . As for the process Y in the verification argument of Remark 2.4, it is given by Y = ∂ x ϕ ( · , ˆ X ) = ∂ x ϕ ( · , X − λY ) . This shows that the hedging strategy (if it is well-defined) consists in following theusual ∆ -hedging strategy but for a ∆ = ∂ x ϕ computed at the value of the stock ˆ X which would be obtained if the position in stocks was liquidated.Note that we obtain the usual heat equation when σ is constant. This isexpected, showing the limitation of the fixed impact model. To explain this, let usconsider the simpler case g = 0 and use the notations of Remark 2.4. We also set µ = 0 for ease of notations. Since σ is constant, the strategy Y does not affect thecoefficients in the dynamics of X , it just produces a shift λdY each time we buy orsell. Since Y T = 0 , and Y t − = 0 , the total impact is null: X T = X t − + σ ( W T − W t ) . s for the wealth process, we have V T = ϕ ( t, x ) + 12 Y t λ + Z Tt Y s dX cs + 12 Z Tt a s λds − Y T − λ + 12 Y T − λ = ϕ ( t, x ) + Z Tt Y s σdW s + 12 λ ( Y t − Y T − ) + Z Tt λY s dY cs + 12 Z Tt a s λds = ϕ ( t, x ) + Z Tt Y s σdW s . Otherwise stated, the liquidation costs are cancelled: when buying, the trader paysa costs but moves the price up, when selling back, he pays a cost again but sellat a higher price. If there is no effect on the underlying dynamics of X and f isconstant, this perfectly cancels.However, the hedging strategy is still affected: Y = ∂ x ϕ ( · , X − λY ) . We first provide the proof of our key result. Proof of Lemma 2.1. To alleviate the notations, we omit the super-scripts.a. We first observe from Proposition 1.3(i) that x( X, − Y ) has continuous paths,while Proposition 1.3(ii) implies that f ∂ x x − ∂ y x = 0 (and therefore f ′ ∂ x x+ f ∂ xx x − ∂ xy x = 0 ). Using Itô’s Lemma, this leads to d x( X s , − Y s ) = ( µ − a s f f ′ )( X s ) ∂ x x( X s , − Y s ) ds + σ ( X s ) ∂ x x( X s , − Y s ) dW s + 12 (cid:2) σ ∂ xx x − a s f ∂ xy x + a s ∂ yy x (cid:3) ( X s , − Y s ) ds. We now use the identity f ∂ xy x − ∂ yy x = 0 , which also follows from Proposition1.3(ii), to simplify the above expression into d x( X s , − Y s ) = [ ∂ x x( µ − a s f f ′ ) + 12 ∂ xx x σ ]( X s , − Y s ) ds + ( σ∂ x x)( X s , − Y s ) dW s . b. Similarly, it follows from Proposition 1.3(iii) that V − I ( ˆ X , Y ) has continu-ous paths, and so does E by a. Before to apply Itô’s lemma to derive the dy-namics of E , let us observe that ∂ y I (x( x, − y ) , y ) = yf (x(x( x, − y ) , y )) = yf ( x ) nd that ∂ yy I (x( x, − y ) , y ) = y ( f f ′ )( x ) + f ( x ) . Also note that ˆ σ (x( x, − y ) , y ) = σ ( x ) ∂ x x( x, − y ) . Then, using the dynamics of ˆ X derived above, we obtain d E s =( Y s − ˇ Y s ) σ ( X s ) dW s + ( Y s − ˇ Y s )[ µ − a s ( f f ′ )]( X s ) ds + ˆ F ϕ ( s, ˆ X s , Y s ) ds + a s σ ( X s )[ Y s f ′ ( X s ) − ∂ x x( X s , − Y s ) ∂ xy I ( ˆ X s , Y s )] ds, where ˇ Y := ∂ x ( ϕ + I )( · , ˆ X , Y ) ∂ x x( X, − Y ) . By Proposition 1.3(ii)(iv), f ( x ) ∂ xy I ( 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 Proposition 1.3(ii), it follows that ∂ x x( X, − Y ) ∂ xy I (x( X, − Y ) , Y ) = Y f ′ ( X ) , which implies d E s = ( Y s − ˇ Y s ) σ ( X s ) dW s + ( Y s − ˇ Y s )[ µ − a s ( f f ′ )]( X s ) ds + ˆ F ϕ ( s, ˆ X s , Y s ) ds. We now deduce from Proposition 1.3 that ∂ x I ( ˆ X, Y ) = − ∆x( ˆ X, Y ) + Y f (x( ˆ X, Y )) f ( ˆ X ) = ˆ X − X + Y f ( X ) f ( ˆ X ) ∂ x x( X, − Y ) = f ( ˆ X ) /f ( X ) , so that ˇ Y = ∂ x ϕ ( · , ˆ X ) f ( ˆ X ) f ( X ) + ˆ X − Xf ( X ) + Y. (cid:3) We now prove the super- and subsolution properties of Theorem 2.1. Supersolution property. We first prove the supersolution property. It followsfrom similar arguments as in [5]. Let ϕ be a C ∞ b function, and ( t o , x o ) ∈ [0 , T ] × R be a strict (local) minimum point of w ∗ − ϕ such that ( w ∗ − ϕ )( t o , x o ) = 0 .a. We first assume that t o < T and F ϕ ( t o , x o ) < , and work towards acontradiction. In view of (2.7), ˆ F ϕ ( t, x, y ) < if ( t, x ) ∈ B and | y − ˆ y [ ϕ ]( t, x ) | ≤ ε, or some open ball B ⊂ [0 , T [ × R which contains ( t o , x o ) , and some ε > . Since x − is continuous, this implies that ˆ F ϕ ( t, x, y ) < if ( t, x ) ∈ B and | x + ∂ x ϕ ( t, x ) f ( x ) − x( x, y ) | ≤ εf (x( x, y )) , (2.9)after possibly changing B and ε . Let ( t n , x n ) n be a sequence in B that convergesto ( t o , x o ) and such that w ( t n , x n ) → w ∗ ( t o , x o ) (recall that w ∗ coïncides with thelower-semicontinuous enveloppe of w ). Set v n := w ( t n , x n ) + n − . It follows fromProposition 2.2(i) that we can find ( a n , b n , ν n ) = γ n ∈ Γ and y n ∈ R such that V t n ,z n ,γ n θ n ≥ w ( θ n , ˆ X t n ,z n ,γ n θ n ) + I ( ˆ X t n ,z n ,γ n θ , Y t n ,z n ,γ n θ n ) , (2.10)where z n := (x( x n , y n ) , y n , v n + I ( x n , y n )) and θ n is the first exit time after t n of ( · , ˆ X t n ,z n ,γ n ) from B (note that ˆ X t n ,z n ,γ n t n = x(x( x n , y n ) , − y n ) = x n ). In thefollowing, we use the simplified notations X n , ˆ X n , V n and Y n for the correspondingquantities indexed by ( t n , z n , γ n ) . Since ( t o , x o ) reaches a strict minimum w ∗ − ϕ ,this implies V nθ n ≥ ϕ ( θ n , ˆ X nθ n ) + I ( ˆ X nθ , Y nθ n ) + ι, (2.11)for some ι > . Let ˇ Y n be as in Lemma 2.1 and observe that ˇ Y n − Y n = ˆ X n + ∂ x ϕ ( · , ˆ X n ) f ( ˆ X n ) − x( ˆ X n , Y n ) f (x( ˆ X n , Y n )) . (2.12)Set χ n := ( µ − f ′ f ( a ns ) / X n ) σ ( X n ) + ˆ F ϕ ( · , ˆ X n , Y n )( Y n − ˇ Y n ) σ ( X n ) | Y n − ˇ Y n |≥ ε and consider the measure P n defined by d P n d P = M nθ n where M n = 1 − Z ·∧ θ n t n M ns χ ns dW s . Then, it follows from (2.11), Lemma 2.1, (2.9) and (2.12) that ι ≤ E P n [ V nθ n − ( ϕ + I )( θ n , ˆ X nθ n , Y nθ n )] ≤ v n + I ( x n , y n ) − ( ϕ + I )( t n , x(x( x n , y n ) , − y n ) , y n )= v n − ϕ ( t n , x n ) . The right-hand side goes to , which is the required contradiction.b. We now explain how to modify the above proof for the case t o = T . Afterpossibly replacing ( t, x ) ϕ ( t, x ) by ( t, x ) ϕ ( t, x ) − √ T − t , we can assume that t ϕ ( t, x ) → ∞ as t → T , uniformly in x on each compact set. Then (2.9) still holdsfor B of the form [ T − η, T ) × B ( x o ) in which B ( x o ) is an open ball around x o and η > small. Assume that ϕ ( T, x o ) < G ( x o ) . Then, after possibly changing B ( x o ) ,we have ϕ ( T, · ) ≤ G − ι on B ( x o ) , for some ι > . Then, with the notations ofa., we deduce from (2.3)-(2.10) that V nθ n ≥ ϕ ( θ n , ˆ X nθ n ) + I ( ˆ X nθ , Y nθ n ) + ι ∧ ι , in which ι := min { ( w ∗ − ϕ )( t, x ) : ( t, x ) ∈ [ t o − η, T ) × ∂B ( x o ) } > and θ n is nowthe minimum between T and the first time after t n at which ˆ X n exists B ( x o ) . Thecontradiction is then deduced from the same arguments as above. (cid:3) Subsolution property. We now turn to the subsolution property. Again theproof is close to [5], except that we have to account for the specific form of thedynamic programming principle stated in Proposition 2.2(ii). Let ϕ be a C ∞ b function, and ( t o , x o ) ∈ [0 , T ] × R be a strict (local) maximum point of w ∗ − ϕ such that ( w ∗ − ϕ )( t o , x o ) = 0 . By [2, Lemma 4.2], we can find a sequence ( k n , t n , x n ) n ≥ such that k n → ∞ , ( t n , x n ) is a local maximum point of w ∗ k n − ϕ and ( t n , x n , w k n ( t n , x n )) → ( t o , x o , w ∗ ( t o , x o )) .a. As above, we first assume that t o < T . Set ϕ n ( t, x ) := ϕ ( t, x ) + | t − t n | + | x − x n | and assume that F ϕ ( t o , x o ) > . Then, F ϕ n > on a open neighborhood B of ( t o , x o ) which contains ( t n , x n ) , for all n large enough. Since we are going tolocalize the dynamics, we can modify ϕ n , σ, µ and f in such a way that they areidentically equal to outside a compact A ⊃ B . It then follows from Remark 2.4a. that, after possibly changing n ≥ , we can find ( b n , a n ) ∈ A k n such that thefollowing admits a strong solution: X n = x n + ∆x( x n , ˆ y [ ϕ n ]( t n , x n )) + Z · t n σ ( X ns ) dW s + Z · t n f ( X ns ) dY n,cs + Z · t n ( µ ( X s ) + a ns ( σf ′ )( X ns )) dsY n = ˆ y [ ϕ n ]( t n , x n ) + Z · t n b ns ds + Z · t n a ns dW s = x − ( ˆ X n , ˆ X n + ( f ∂ x ϕ n )( · , ˆ X n ))ˆ X n := x( X n , − Y n ) V n = v n + I ( x n , ˆ y [ ϕ n ]( t n , x n )) + Z · t n Y ns dX n,cs + 12 Z · t n ( a ns ) f ( X ns ) ds. In the above, we have set v n := w k n ( t n , x n ) − n − . Observe that the constructionof Y n ensures that it coincides with the corresponding process ˇ Y n of Lemma 2.1.Also note that ˆ X nt n = x(x( x n , y n ) , − y n ) = x n , and let θ n be the first time after n at which ( · , ˆ X n ) exists B . By applying Itô’s Lemma, using Lemma 2.1 and thefact that F ϕ n ≥ on B , we obtain V nθ n ≥ ( ϕ n + I )( θ n , ˆ X nθ n , Y nθ n ) + v n − ϕ n ( t n , x n ) . Let ε := min {| t − t o | + | x − x o | , ( t, x ) ∈ ∂B } . For n large enough, the aboveimplies V nθ n ≥ ( w k n − + I )( θ n , ˆ X nθ n , Y nθ n ) + ε + ι n , where ι n := ( ϕ n − w k n − )( t n − , x n − ) + v n − ϕ n ( t n , x n ) converges to . Hence, wecan find n such that V nθ n > ( w k n − + I )( θ n , ˆ X nθ n , Y nθ n ) . Now observe that we can change the subsequence ( k n ) n ≥ in such a way that k n ≥ k n − + 2 . Then, v n = w k n ( t n , x n ) − n − < w k n − +2 ( t n , x n ) , which leads toa contradiction to Proposition 2.2(ii).b. It remains to consider the case t o = T . As in Step 1., we only explain howto modify the argument used above. Let ( v n , k n , t n , x n ) be as in a. We now set ϕ n ( t, x ) := ϕ ( t, x ) + √ T − t + | x − x n | . Since ∂ t ϕ n ( t, x ) → −∞ as t → T , we canfind n large enough so that F ϕ n ≥ on [ t n , T ) × B ( x o ) in which B ( x o ) is an openball around x o . Assume that ϕ ( T, x o ) > G ( x o ) + η for some η > . Then, afterpossibly changing B ( x o ) , we can assume that ϕ n ( T, · ) ≥ G + η on B ( x o ) . We nowuse the same construction as in a. but with θ n defined as the minimum between T and the first time where ˆ X n exists B ( x o ) . We obtain V nθ n ≥ ( ϕ n + I )( θ n , ˆ X nθ n , Y nθ n ) + v n − ϕ n ( t n , x n ) . Let ε := min {| x − x o | , x ∈ ∂B ( x o ) } . For n large enough, the above implies V nθ n ≥ w k n − ( θ n , ˆ X nθ n ) θ n Let v be a supersolution (resp. subsolution) of (2.8) . Fix ρ > .Then, ˜ v defined by ˜ v ( t, x ) = e ρt v ( t, Φ( x )) , is a supersolution (resp. subsolution) of ρϕ − ∂ t ϕ − (cid:20) B (Φ , e − ρt ∂ x ϕ ) /f (Φ) − A (Φ , e − ρt ∂ x ϕ ) f ′ (Φ) /f (Φ) (cid:21) ∂ x ϕ − A (Φ , e − ρt ∂ x ϕ ) ∂ xx ϕ/f (Φ) − e ρt L (Φ , e − ρt ∂ x ϕ ) (2.16) with the terminal condition ϕ ( T, · ) = e ρT G (Φ) . (2.17)To prove that comparison holds for (2.8), it suffices to prove that it holds for(2.16)-(2.17). For the latter, this is a consequence of the following result. It israther standard but we provide the complete proof by lack of a precise reference. Theorem 2.2. Let O be an open subset of R , u (resp. v ) be a upper-semicontinuoussubsolution (resp. lower-semicontinuous supersolution) on [0 , T ) × O of: ρϕ − ∂ t ϕ − ¯ B ( · , e − ρt ∂ x ϕ ) ∂ x ϕ − 12 ¯ A ( · , e − ρt ∂ x ϕ ) ∂ xx ϕ − e ρt ¯ L ( · , e − ρt ∂ x ϕ ) = 0 (2.18) where ρ > is constant, ¯ A, ¯ B and ¯ L : ( t, x, p ) ∈ [0 , T ] × O × R → R are Lipschitzcontinuous functions. Suppose that u and v are bounded and satisfy u ≤ v on theparabolic boundary of [0 , T ) × O , then u ≤ v on the closure of [0 , T ] × O . roof. Suppose to the contrary that sup [0 ,T ] ×O ( u − v ) > , and define, for n > , Θ n := sup ( t,x,y ) ∈ [0 ,T ) ×O (cid:18) u ( t, x ) − v ( t, y ) − n | x − y | − n | x | (cid:19) . Then, there exists ι > , such that Θ n ≥ ι for n large enough. Since u and v are bounded and u ≤ v on the parabolic boundary of the domain, we can find ( t n , x n , y n ) ∈ [0 , T ) × O which achieves the above supremum.As usual, we apply Ishii’s Lemma combined with the sub- and super-solutionproperties of u and v , and the Lipschitz continuity of ¯ A, ¯ B and ¯ L to obtain, withthe notation p n := n ( x n − y n ) , ρ ( u ( t n , x n ) − v ( t n , y n )) ≤ [ ¯ B ( x n , e − ρt n ( p n + 1 n x n )) − ¯ B ( y n , e − ρt n p n )] p n + 1 n x n ¯ B ( x n , e − ρt n ( p n + 1 n x n ))+ 3 n A ( x n , e − ρt n ( p n + 1 n x n )) − ¯ A ( y n , e − ρt n p n )] + 12 n ¯ A ( x n , e − ρt n ( p n + 1 n x n ))+ e ρt n (cid:18) ¯ L ( x n , e − ρt n ( p n + 1 n x n )) − ¯ L ( y n , e − ρt n p n ) (cid:19) ≤ C (cid:18) n ( x n − y n ) + | x n − y n | + 1 n x n + 1 n (cid:19) for some constant C which does not depend on n . In view of Lemma 2.3 below, andsince ρ > and u ( t n , x n ) − v ( t n , y n ) ≥ Θ n ≥ ι , the above leads to a contradictionfor n large enough. (cid:3) We conclude with the proof of the technical lemma that was used in our argu-ments above. Lemma 2.3. Let Ψ be a bounded upper-semicontinuous function on [0 , T ] × R ,and Ψ i , i = 1 , , be two non-negative lower-semicontinuous functions on R suchthat { Ψ = 0 } = { } . For n > , set Θ n := sup ( t,x,y ) ∈ [0 ,T ] × R (cid:18) Ψ( t, x, y ) − n Ψ ( x − y ) − n Ψ ( x ) (cid:19) nd assume that there exists (ˆ t n , ˆ x n , ˆ y n ) ∈ [0 , T ] × R such that: Θ n = Ψ(ˆ t n , ˆ x n , ˆ y n ) − n Ψ (ˆ x n − ˆ y n ) − n Ψ (ˆ x n ) . Then, after possibly passing to a subsequence, (i) lim n →∞ n Ψ (ˆ x n − ˆ y n ) = 0 and lim n →∞ n Ψ (ˆ x n ) = 0 . (ii) lim n →∞ Θ n = sup ( t,x ) ∈ [0 ,T ] ×O Ψ( t, x, x ) . Proof. For later use, set ¯ R := R ∪ {−∞} ∪ {∞} and note that we can ex-tend Ψ as a bounded upper-semicontinuous function on [0 , T ] × ¯ R . Set M :=sup ( t,x ) ∈ [0 ,T ] × R Ψ( t, x, x ) , and select a sequence ( t n , x n ) n ≥ such that lim n →∞ Ψ( t n , x n , x n ) = M and lim n →∞ n Ψ ( x n ) = 0 . Let C be a upper-bound for Ψ . Then, C − n Ψ (ˆ x n − ˆ y n ) − n Ψ (ˆ x n ) ≥ Ψ(ˆ t n , ˆ x n , ˆ y n ) − n Ψ (ˆ x n − ˆ y n ) − n Ψ (ˆ x n ) ≥ Ψ( t n , x n , x n ) − n Ψ ( x n ) ≥ M − ε n where ǫ n → . Since Ψ and Ψ are non-negative, letting n → ∞ in the aboveinequality leads to lim n →∞ Ψ (ˆ x n − ˆ y n ) = 0 which implies lim n →∞ (ˆ x n − ˆ y n ) = 0 by the assumption { Ψ = 0 } = { } .After possibly passing to a subsequence, we can then assume that lim n →∞ ˆ x n =lim n →∞ ˆ y n = ˆ x ∈ ¯ R and that lim n →∞ ˆ t n = ˆ t ∈ [0 , T ] . Since Ψ is upper semi-continuous, the above leads to M − lim inf n →∞ (cid:18) n Ψ (ˆ x n − ˆ y n ) + 1 n Ψ (ˆ x n ) (cid:19) ≥ Ψ(ˆ t, ˆ x, ˆ x ) − lim inf n →∞ (cid:18) n Ψ (ˆ x n − ˆ y n ) − n Ψ (ˆ x n ) (cid:19) ≥ lim sup n →∞ (cid:18) Ψ(ˆ t n , ˆ x n , ˆ y n ) − n Ψ (ˆ x n − ˆ y n ) − n Ψ (ˆ x n ) (cid:19) ≥ M, and our claim follows. (cid:3) emark 2.5. It follows from the above that, whenever they are bounded, e.g. if G is bounded, then w ∗ ≥ w ∗ . Since by construction w ∗ ≤ w ≤ w ∗ , the three functionsare equal to the unique bounded viscosity solution of (2.8) . We conclude here the proof of Theorem 2.1 by showing that existence of a smoothsolution holds when inf f > , G is bounded and C with G, G ′ , G ′′ Hölder continuous. (2.19)Note that the assumptions inf f > and (H1) imply that Φ − is C , recall (2.15).Hence, by the same arguments as in Section 2.5.3, existence of a C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) solution to (2.16)-(2.17) implies the existence of a C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) solution to (2.8). As for (2.16)-(2.17), this is a consequence of [11,Thm 14.24], under (H1) and (2.19).It remains to show that the solution can be taken bounded, then the comparisonresult of Section 2.5.3 will imply that w is this solution. Again, it suffices to workwith (2.16)-(2.17). Let ϕ be a C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) solution of (2.16)-(2.17). Let S t,x be defined by S t,xs = x + Z st µ S ( s, S t,xs ) ds + Z st σ S ( s, S t,xs ) dW s , s ≥ t, where µ S := B (Φ , e − ρt ∂ x ϕ ) /f (Φ) − A (Φ , e − ρt ∂ x ϕ ) f ′ (Φ) /f (Φ) σ S := A (Φ , e − ρt ∂ x ϕ ) /f (Φ) . Note that the coefficients of the sde may only be locally Lipschitz. However, theyare bounded (recall (H1) and (2.19)), which is enough to define a solution by astandard localization procedure. Since σ S is bounded, Itô’s Lemma implies that ϕ ( t, x ) e − ρt = E (cid:20) G (Φ( S t,xT )) + Z Tt L (Φ( X t,xs ) , e − ρs ∂ x ϕ ( s, X t,xs )) ds (cid:21) . Since G and L are bounded, by (H1) and (2.19), ϕ is bounded as well. (cid:3) Remark 2.6. We refer to [10] for conditions under which additional smoothnessof the solution can be proven. Appendix We report here the measurability property that was used in the course of Proposi-tion 2.2.In the following, A k is viewed as a closed subset of the Polish space L λ endowedwith the usual strong norm topology k · k L λ .We consider an element ν ∈ U k as a measurable map ω ∈ Ω ν ( ω ) ∈ M k where M k denotes the set of non-negative Borel measures on R × [0 , T ] with totalmass less than k , endowed with the topology of weak convergence. This topologyis generated by the norm k m k M := sup { Z R × [0 ,T ] ℓ ( δ, s ) m ( dδ, ds ) : ℓ ∈ Lip } , in which Lip denotes the class of -Lipschitz continuous functions bounded by ,see e.g. [4, Proposition 7.2.2 and Theorem 8.3.2]. Then, U k is a closed subset of thespace M k, of M k -valued random variables. M k, is made complete and separableby the norm k ν k M := E (cid:2) k ν k M (cid:3) . See e.g. [8, Chap. 5]. We endow the set of controls Γ k with the natural producttopology k γ k L λ × M := k ϑ k L λ + k ν k M , for γ = ( ϑ, ν ) . As a closed subset of the Polish space L λ × M k, , Γ k is a Borel space, for each k ≥ . See e.g. [3, Proposition 7.12].The following stability result is proved by using standard estimates. In thefollowing, we use the notation Z = ( X, Y, V ) . Proposition A.1. 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