A numerical scheme for the quantile hedging problem
Cyril Bénézet, Jean-François Chassagneux, Christoph Reisinger
AA numerical scheme for the quantile hedging problem
Cyril B´en´ezet ∗ , Jean-Fran¸cois Chassagneux ∗ , Christoph Reisinger † March 5, 2019
Abstract
We consider the numerical approximation of the quantile hedging price in anon-linear market. In a Markovian framework, we propose a numerical methodbased on a Piecewise Constant Policy Timestepping (PCPT) scheme coupled witha monotone finite difference approximation. We prove the convergence of our al-gorithm combining BSDE arguments with the Barles & Jakobsen and Barles &Souganidis approaches for non-linear equations. In a numerical section, we illus-trate the efficiency of our scheme by considering a financial example in a marketwith imperfections.
Key words:
Quantile hedging, BSDEs, monotone approximation schemes
In this work, we study the numerical approximation of the quantile hedging price of aEuropean contingent claim in a market with possibly some imperfections. The quantilehedging problem is a specific case of a broader class of approximate hedging problems.It consists in finding the minimal initial endowment of a portfolio that will allow thehedging a European claim with a given probability p of success, the case p “ ∗ UFR de Math´ematiques & LPSM, Universit´e Paris Diderot, Bˆatiment Sophie Germain, 8 placeAur´elie Nemours, 75013 Paris, France ( [email protected], [email protected] ) † Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United King-dom ( [email protected] ) a r X i v : . [ q -f i n . C P ] F e b e now present in more detail the quantile hedging problem and the new numericalmethod we introduce and study in this paper.On a complete probability space p Ω , F , P q , we consider a d -dimensional Brownianmotion p W t q t Pr ,T s and denote by p F t q t Pr ,T s its natural filtration. We suppose that allthe randomness comes from the Brownian motion and assume that F “ F T .Let µ : R d Ñ R d , σ : R d Ñ M d p R q , where M d p R q is the set of d ˆ d matrices with realentries, f : r , T s ˆ R d ˆ R ˆ R d Ñ R be Lipschitz continuous functions, with Lipschitzconstant L .For p t, x, y q P r , T s ˆ R d ˆ R and ν P H , which denotes the set of predictable square-integrable processes, we consider the solution p X t,x , Y t,x,y,ν q to the following stochasticdifferential equations: X s “ x ` ż st µ p X u q d u ` ż st σ p X u q d W u ,Y s “ y ´ ż st f p u, X u , Y u , ν u q d u ` ż st ν u d W u , s P r t, T s . In the financial applications we are considering, X will typically represent the log-priceof risky assets, the control process ν is the amount invested in the risky assets, and thefunction f is non-linear to allow to take into account some market imperfections in themodel. A typical financial example, which will be investigated in the numerical section,is the following: Example 1.1.
The underlying diffusion X is a one-dimensional Brownian motion withconstant drift µ P R and volatility σ ą . There is a constant borrowing rate R and alending rate r with R ě r . In this situation, the function f is given by: f p t, x, y, z q “ ´ ry ´ σ ´ µz ` p R ´ r qp y ´ σ ´ z q ´ . The quantile hedging problem corresponds to the following stochastic control problem:for p t, x, p q P r , T s ˆ R d ˆ r , s find v p t, x, p q : “ inf ! y ě D ν P H , P ´ Y t,x,y,νT ě g p X t,xT q ¯ ě p ) . (1.1)The main objective of this paper is to design a numerical procedure to approximate thefunction v by discretizing an associated non-linear PDE first derived in [8]. A key pointin the derivation of this PDE is to observe that the above problem can be reformulated asa classical stochastic target problem by introducing a new control process representingthe conditional probability of success. To this end, for α P H , we denote P t,p,αs : “ p ` ż st α s d W s , t ď s ď T , and by A t,p the set of α such that P t,p,α ¨ P r , s . The problem (1.1) can be rewritten as v p t, x, p q : “ inf ! y ě Dp ν, α q P p H q , Y t,x,y,νT ě g p X t,xT q t P t,p,αT ą u ) v p t, x, p q “ inf α P A t,p Y αt (1.2)where p Y α , Z α q is the solution to Y αs “ g p X t,xT q t P t,p,αT ą u ` ż Ts f p s, X s , Y αs , Z αs q d s ´ ż Ts Z αs d W s , t ď s ď T .
The article [7] justifies the previous representation and proves a dynamic programmingprinciple for the control problem in a general setting. In the Markovian setting, thiswould lead naturally to the following PDE for v in r , T q ˆ R d ˆ p , q :“ ´ B t ϕ ` sup a P R d F a p t, x, ϕ, Dϕ, D ϕ q “ p t, x, y q P r , T s ˆ R d ˆ R ` , q : “ ˆ q x q p ˙ P R d ` and A : “ ˆ A xx A xp A xp J A pp ˙ P S d ` , A xx P S d , denoting Ξ : “ p t, x, y, q, A q , we define F a p Ξ q : “ ´ f p t, x, y, z p x, q, a qq ´ L p x, q, A, a q , (1.4)with z p x, q, a q : “ q x σ p x q ` q p a , (1.5) L p x, q, A, a q : “ µ p x q J q x `
12 Tr “ σ p x q σ p x q J A xx ‰ ` | a | A pp ` a J σ p x q J A xp . (1.6)The PDE formulation in (1.3) is not entirely correct as the supremum part may degen-erate and it would require using semi-limit relaxation to be mathematically rigourous.We refer to [8], where it has been obtain in a more general context. We shall use analternative PDE formulation to this “natural” one (1.3), which we give at the start ofSection 2.Moreover, the value function v continuously satisfies the following boundary conditionsin the p -variable: v p t, x, q “ v p t, x, q “ V p t, x q on r , T s ˆ p , d , (1.7)where V is the super-replication price of the contingent claim with payoff g p¨q .It is also known that v has a discontinuity as t Ñ T . By definition, the terminalcondition is R d ˆ r , s Q p x, p q ÞÑ g p x q p ą P R ` , v p T ´ , x, p q “ pg p x q on R d ˆ r , s , (1.8)and we shall work with this terminal condition at t “ T from now on.To design the numerical scheme to approximate v , we use the following strategy:1. Bound and discretise the set where the controls α take their values.2. Consider an associated Piecewise Constant Policy Timestepping (PCPT) schemefor the control processes .3. Use a monotone finite difference scheme to approximate in time and space thePCPT solution resulting from 1. & 2.The approximation of controlled diffusion processes by ones where policies are piecewiseconstant in time was first analysed by [23]; in [24], this procedure is used in conjunc-tion with Markov chain approximations to diffusion processes to construct fully discreteapproximation schemes to the associated Bellman equations and to derive their conver-gence order. An improvement to the order of convergence from [23] was shown recentlyin [22] using a refinement of Krylov’s original, probabilistic techniques.Using purely viscosity solution arguments for PDEs, error bounds for such approxima-tions are derived in [3], which are weaker than those in [23] for the control approxi-mation scheme, but improve the bounds in [24] for the fully discrete scheme. In [27],using a switching system approximation introduced in [3], convergence is proven for ageneralised scheme where linear PDEs are solved piecewise in time on different meshes,and the control optimisation is carried out at the end of time intervals using possi-bly non-monotone, higher order interpolations. An extension of the analysis in [27] tojump-processes and non-linear expectations is given in [17].Our first contribution is to prove that the approximations built in step 1. and 2. aboveare convergent for the quantile hedging problem, which has substantial new difficultiescompared to the settings considered in the aforementioned works. For this we rely heav-ily on the comparison theorem for the formulation in (2.1) and we take advantage of themonotonicity property of the approximating sequences. The main new difficulties comefrom the non-linear form of the PDE including unbounded controls, and in particularthe boundaries in the p -variable. To deal with the latter especially, we rely on somefine estimates for BSDEs to prove the consistency of the scheme including the strongboundary conditions (see Lemma 2.2 and Lemma 2.3).Our second contribution is to design the monotone scheme in step 3. and to prove itsconvergence. The main difficulties come here from the non-linearity of the new termfrom the driver of the BSDE in the gradient combined with the degeneracy of thediffusion operator given in (1.6), and again the boundedness for the domain in p . Inparticular, a careful analysis of the consistency of the boundary condition is needed (seeProposition 3.4). 4o the best of our knowledge, this is the first numerical method for the quantile hedgingproblem in this non-linear market specification. In the linear market setting, usinga dual approach, [6] combines the solution of a linear PDE with Fenchel-Legendretransforms to tackle the problem of Bermudan quantile hedging. Their approach cannotbe directly adapted here due to the presence of the non-linearity. The dual approachin the non-linear setting would impose some convexity assumption on the f parameterand would require to solve fully non-linear PDEs. Note that here f is only requiredto be Lipschitz continuous in p y, z q . We believe that an interesting alternative to ourmethod would be to extend the work of [5] to the non-linear market setting we considerhere.The rest of the paper is organised as follows. In Section 2, we derive the controlapproximation and PCPT scheme associated with items 1 . & 2 . above and prove theirconvergence. In Section 3, we present a monotone finite difference approximation whichis shown to convergs to the semi-discrete PCPT scheme. In Section 4, we present nu-merical results for a specific application and analyse the observed convergence. Finally,the appendix contains some of the longer, more technical proofs and collects usefulbackground results used in the paper. Notations diag p x q is the diagonal matrix of size d , whose diagonal is given by x .Let us denote by S the sphere in R d ` of radius 1 and by D the set of vectors η P S such that their first component η “
0. For a vector η P S z D , we denote η : “ η p η , . . . , η d ` q J P R d . By extension, we denote, for Z Ă S z D , Z : “ t η P R d | η P Z u . We denote by BC : “ L pr , T s , C p R d ˆ r , sqq , namely the space of functions u thatare essentially bounded in time and continuous with respect to their space variable.The convergence in C pr , T s ˆ R d q considered here is the local uniform convergence. In this section, we design a Piecewise Constant Policy Timestepping (PCPT) schemewhich is convergent to the value function v defined in 1.2.Following [5], it has been shown in [10], that the function v is equivalently a viscositysolution of the following PDE (see Theorems 3.1 and 3.2 in [10]): H p t, x, ϕ, B t ϕ, Dϕ, D ϕ q “ p , T q ˆ R d ˆ p , q , where H is a continuous operator H p Θ q “ sup η P S H η p Θ q , (2.2)where for p t, x, y, b q P r , T s ˆ R d ˆ R ` ˆ R , q : “ ˆ q x q p ˙ P R d ` and A : “ ˆ A xx A xp A xp J A pp ˙ P d ` , and Θ : “ p t, x, y, b, q, A q , we define H η p Θ q “ p η q ´ ´ b ´ f p t, x, y, z p x, q, η qq ´ L p x, q, A, η q ¯ , for η P S z D . (2.3)Recall also the definition of L and z in (1.5) and (1.6).This representation and its properties are key in the proof of convergence. Looselyspeaking, it is obtained by “compactifying” the set t u ˆ R d to the unit sphere S . Acomparison theorem is shown in Theorem 3.2 in [10].As partially stated in the introduction, we will work under the following assumption: p H q (i) The functions b , σ are L -Lipschitz continuous and g is bounded and L g -Lipschitzcontinuous.(ii) The function f is measurable and for all t P r , T s , f p t, ¨ , ¨ , ¨q is L -Lipschitz contin-uous. For all p t, x, z q P r , T sˆ R d ˆ R d , the function y ÞÑ f p t, x, y, z q is decreasing.Moreover, f p t, x, , q “ . (2.4)Under the above Lipschitz continuity assumption, the mapping S z D Q η ÞÑ H η p Θ q P R extends continuously to S by setting, for all η P D , H η p Θ q “ ´ A pp , see Remark 3.1 in [10]. Remark 2.1. (i) In p H q (ii), the monotonicity assumption is not a restriction, as ina Lipschitz framework, the classical transformation ˜ v p t, x, p q : “ e λt v p t, x, p q for λ largeenough allows to reach this setting; see Remark 3.3 in [10] for details.(ii) The condition (2.4) is a reasonable financial modelling assumption: It says thatstarting out in the market with zero initial wealth and making no investments will leadto a zero value of the wealth process.(iii) Since f is decreasing and g is bounded, it is easy to see that | V | ď | g | , where V is the super-replication price. In order to introduce a discrete-time scheme which approximates the solution v of(2.1)–(1.8), we start by discretizing the set of controls S .Let p R n q n ě be an increasing sequence of closed subsets of S z D such that ď n ě R n “ S . (2.5)6or n ě
1, let v n : r , T s ˆ R d ˆ r , s Ñ R be the unique continuous viscosity solutionof the following PDE: H n p t, x, ϕ, B t ϕ, Dϕ, D ϕ q “ , (2.6)satisfying the boundary conditions (1.7)-(1.8), see Corollary 6.1. Above, the operator H n is naturally given by H n p Θ q : “ sup η P R n H η p Θ q . (2.7) Proposition 2.1.
The functions v n converge to v in C pr , T s ˆ R d q . Proof.
1. For n ă n , we observe that v n is a super-solution of (2.6) as R n Ă R n .Using the comparison result of Proposition 6.1, we obtain that v n ě v n . Similarly, usingthe comparison principle ([10], Theorem 3.2), we obtain that v n ě v , for all n ě p t, x, p q P r , T s ˆ p , d ˆ r , s , let: v p t, x, p q “ lim j Ñ8 sup " v n p s, y, q q : n ě j and }p s, y, q q ´ p t, x, p q} ď j * , (2.8) v p t, x, p q “ lim j Ñ8 inf " v n p s, y, q q : n ě j and }p s, y, q q ´ p t, x, p q} ď j * . (2.9)From the above discussion, recalling that v and v are continuous, we have v ě v ě v ě v , which shows that v and v satisfy the boundary conditions (1.7)-(1.8).In order to prove the theorem, it is enough to show that v is a viscosity subsolutionof (2.1) and v is a viscosity supersolution (which follows similarly and is thereforeomitted). The comparison principle ([10], Theorem 3.2) then implies that v “ v “ v ,and it follows from [12], Remark 6.4 that the convergence v n Ñ v as n Ñ 8 is uniformon every compact set. Using Theorem 6.2 in [1], we obtain that v is a subsolution to H p t, x, ϕ, B t ϕ, Dϕ, D ϕ q “ p , T q ˆ p , d ˆ p , q , where H p Θ q “ lim j Ñ8 inf " H n p Θ q : n ě j and } Θ ´ Θ } ď j * . In the next step, we prove that H “ H , which concludes the proof of the proposition.2. Let us denote by P n : S Ñ ˜ S n the closest neighbour projection on the closed set ˜ S n .From (2.5), we have that lim n Ñ8 P n p η q “ η , for all η P S . We also have that H p Θ q “ H η ˚ p Θ q η ˚ P argmax η P S H η p Θ q as S is compact. Let us now introduce η n : “ P n p η ˚ q and by continuity of H , we have H η n p Θ q Ñ H p Θ q . We also observe that H η n p Θ q ď H n p Θ q ď H p Θ q . This proves the convergence H n p Θ q Ò H p Θ q , for all Θ. As H is continuous, we concludeby using Dini’s Theorem that the convergence is uniform on compact subsets, leadingto H “ H . l From now on, we fix n ě R n the associated discrete set of control. For p t, x, y q Pr , T s ˆ R d ˆ R ` , q P R d ` and A P S d ` , denoting Ξ : “ p t, x, y, q, A q , we define F n p Ξ q “ sup a P R n F a p Ξ q with F a p Ξ q : “ ´ f p t, x, y, z p x, q, a qq ´ L p x, q, A, a q . Following the proof of Corollary 6.1 in the appendix, we easily observe that v n is alsothe unique viscosity solution to ´B t ϕ ` F n p t, x, ϕ, Dϕ, D ϕ q “ r , T q ˆ R d ˆ p , q (2.10)with the same boundary conditions (1.7)–(1.8). The above PDE is written in a moreclassical way and we will mainly consider this form in the sequel. Let us observe inparticular that K : “ R n is a discrete subset of R d , such that (2.10) appears as a naturaldiscretisation of (1.3) and will be simpler to study.To approximate v n , we consider an adaptation of the PCPT scheme in [24, 3], andespecially [17], to our setting, as described below.For κ P N ˚ , we consider grids of the time interval r , T s : π “ t “ : t ă ¨ ¨ ¨ ă t k ă ¨ ¨ ¨ ă t κ : “ T u , and denote | π | : “ max ď k ď κ p t k ` ´ t k q .For 0 ď t ă s ď T , a P K and a continuous φ : R d ˆ r , s Ñ R , we denote by S a p s, t, φ q : r t, s s ˆ R d ˆ r , s Ñ R the unique solution of ´B t ϕ ` F a p t, x, ϕ, Dϕ, D ϕ q “ r t, s q ˆ R d ˆ p , q , (2.11) ϕ p s, x, p q “ φ p x, p q on R d ˆ r , s , (2.12) ϕ p r, x, q “ B p t, s, φ qp r, x q , ϕ p r, x, q “ B p t, s, φ qp r, x q on r t, s q ˆ R d . (2.13)The function B p p t, s, φ q for p P t , u is solution to ´B t ϕ ` F p r, x, ϕ, Dϕ, D ϕ q “ r t, s q ˆ R d , (2.14)8ith terminal condition B p p t, s, φ qp r, x qp s, x q “ φ p x, p q .The solution to the PCPT scheme associated with the grid π is then the function v n,π : r , T s ˆ R d ˆ r , s such that S p π, t, x, p, v n,π p t, x, p q , v n,π q “ , (2.15)where for a grid π , p t, x, p, y q P r , T s ˆ R d ˆ r , s ˆ R ` and a function u P BC , S p π, t, x, p, y, u q “ " y ´ min a P K S a p t ` π , t ´ π , u p t ` π , ¨qq p t, x, p q if t ă T,y ´ ˆ g p x q p otherwise , (2.16)with t ` π : “ inf t r P π | r ą t u and t ´ π : “ sup t r P π | r ď t u . (2.17)We will drop the subscript π for brevity whenever we consider a fixed mesh.Let us observe that the function v n,π can be alternatively described by the followingbackward algorithm:1. Initialisation: set v n,π p T, x, p q : “ g p x q p , x P R d ˆ r , s .2. Backward step: For k “ κ ´ , . . . ,
0, compute w k,a : “ S a p t k , t k ` , v n,π p t k ` , ¨qq and set v n,π p¨q : “ inf a P K w k,a . (2.18) Remark 2.2.
In our setting, we can easily identify the boundary values (of the scheme):(i) At p “ , the terminal condition is φ p T, x q “ (recall that v p T, x, p q “ g p x q t p ą u ),and this propagates through the backward iteration, so that v n,π p t, x, q “ for all p t, x q P r , T s ˆ R d .(ii) At p “ , the terminal condition is φ p T, x q “ g p x q and the boundary conditionis thus given by v n,π p t, x, q “ V p t, x q for all p t, x q P r , T s ˆ R d , where V is thesuper-replication price. The main result of this section is the following.
Theorem 2.1.
The function v n,π converges to v n in C as | π | Ñ . Proof.
1. We first check the consistency with the boundary condition. Let ˆ a P K and ˆ w be the (continuous) solution of ´B t ϕ ` F ˆ a p t, x, ϕ, Dϕ, D ϕ q “ r , T q ˆ R d ˆ p , q (2.19)with boundary condition v p t, x, p q “ pV p t, x q on r , T s ˆ R d ˆ t , u Ť t T u ˆ R d ˆ r , s .By backward induction on π , one gets that v n,π ď ˆ w . (2.20)9ndeed, we have v n,π p T, ¨q “ ˆ w p T, ¨q . Now if the inequality is true at time t k , k ě
1, wehave, using the comparison result for (2.19), recalling Proposition 6.1, that w k, ˆ a p t, ¨q ď ˆ w p t, ¨q for t P r t k ´ , t k s , and thus a fortiori ˆ w p t, ¨q ě v n,π p t, ¨q , for t P r t k ´ , t k s .We also obtain that v n,π p¨q ě v n p¨q (2.21)by backward induction. Indeed, we have v n,π p T, ¨q “ v n p T, ¨q . Assume that the inequal-ity is true at time t k , k ě
1. We observe that w k,a is a supersolution of (2.6), namelythe PDE satisfied by v n . By the comparison result, this implies that w k,a p t, ¨q ě v n p t, ¨q ,for t P r t k ´ , t k s . Taking the infimum over a P K yields then (2.21).Since v n ď w ď w ď ˆ w , (2.22)where w p t, x, p q “ lim sup p t ,x ,p , | π |qÑp t,x,p, q v n,π p t , x , p q and w “ lim inf p t ,x ,p , | π |qÑp t,x,p, q v n,π p t , x , p q , we obtain that w and w satisfy the boundary conditions (1.7)–(1.8).2. We prove below that the scheme is monotone , stable and consistent , see Proposition2.2, Proposition 2.3 and Proposition 2.4 respectively. Combining this with step 1. andTheorem 2.1 in [4] then ensures the convergence in C of v n,π to v n as | π | Ñ l Remark 2.3.
We prove the following properties by a combination of viscosity solutionarguments and, mostly, BSDE arguments, where they appear more natural. It shouldbe possible to derive these results purely using PDE arguments using similar main stepsas in [3] .
Proposition 2.2 (Monotonicity) . Let u ě v for u, v P BC , p t, x, p q P r , T s ˆ R d ˆ r , s , y P R . We have: S p π, t, x, p, y, u q ď S p π, t, x, p, y, v q . (2.23) Proof.
Let t ă T, x P R d , p P r , s . By definition of v n,π , recalling (2.18), it issufficient to prove that, for any a P K , we have: S a p t ` , t ´ , u p t ` , ¨qqp t, x, p q ě S a p t ` , t ´ , v p t ` , ¨qqp t, x, p q . (2.24)with t ` , t ´ defined in (2.17) . But this follows directly from the comparison result givenin Proposition 6.1. l We now study the stability of the scheme. We first show that the solution of the scheme v n,π is increasing in its third variable. This is not only an interesting property in itsown right which the piecewise constant policy solution inherits from the solution to theoriginal problem (1.1), but it also allows us to obtain easily a uniform bound for v n,π ,namely the boundary condition at p “
1. 10 emma 2.1.
The scheme (2.16) has the property, for all t P r , T s and x P R d : v n,π p t, x, q q ď v n,π p t, x, p q if ď q ď p ď . (2.25) Proof.
We are going to prove the assertion by induction on k P t , . . . , κ u .For t “ T “ t κ and every x P R d , we have p x, p q ÞÑ v n,π p T, x, p q : “ g p x q p , which is anincreasing function of p .Let 1 ď k ă κ ´
1. Assume now that v n,π p t, x, ¨q is an increasing function for all t ě t k ` and x P R d . We show that v n,π p t, x, ¨q is also increasing for t P r t k , t k ` q and x P R d .Let 0 ď q ď p ď
1. By the definition of v n,π in (2.18), it is sufficient to show that foreach a P K , we have, for p t, x q P r , T s ˆ R d , w k,a p t, x, q q ď w k,a p t, x, p q . From Lemma 6.1(i) in the appendix, these two quantities admit a probabilistic repre-sentation with two different random terminal times τ q “ inf t s ě t : P t,q,as P t , uu ^ t k ` , (2.26) τ p “ inf t s ě t : P t,p,as P t , uu ^ t k ` . (2.27)However, using Lemma 6.1(ii), we can write probabilistic representations with BSDEswith terminal time t k ` : we have that S a p t k ` , t k , w π p t k ` , ¨qqp t, x, p q “ ˜ Y t,x,p,at , where˜ Y t,x,p,at is the first component of the solution of the following BSDE: Y s “ v n,π p t k ` , X t,xt k ` , ˜ P t,p,at k ` q ` ż t k ` s f p u, X t,xu , Y u , Z u q d u ´ ż t k ` s Z u d W u , (2.28)where ˜ P t,p,a is the process defined by:˜ P t,p,as “ p ` ż st a t u ď τ p u d W u , (2.29)and a similar representation holds for S a p t k ` , t k , w π p t k ` , ¨qqp t, x, q q .It remains to show that v n,π p t k ` , X t,xt k ` , ˜ P t,p,at k ` q ě v n,π p t k ` , X t,xt k ` , ˜ P t,q,at k ` q . (2.30)If this is true, the classical comparison theorem for BSDEs (see e.g. Theorem 2.2 in[18]), concludes the proof.First, we observe that P t,p,aτ p ě P t,q,aτ p . On t τ p “ T u , (2.30) holds straightforwardly bythe induction hypothesis. On t τ p ă T u , if P t,p,aτ p “ P t,p,aT “ P t,q,aT ď
1; if P t,p,aτ p “ a fortiori P t,q,aτ p “ P t,p,aT “ P t,q,aT “
0, which concludes the proof. l Proposition 2.3 (Stability) . The solution to scheme (2.16) is bounded. roof. For any π and any p t, x, p q P r , T s ˆ R d ˆ r , s , we have v n,π p t, x, p q ď v n,π p t, x, q “ V p t, x q . l To prove the consistency of the scheme, we will need the two following lemmata.
Lemma 2.2.
For ď τ ď t ď θ ď T , ξ P R , and φ P C pr , T s ˆ R d ˆ r , sq , thefollowing holds | S a p τ, θ, φ p θ, ¨q ` ξ qp t, ¨q ´ S a p τ, θ, φ p θ, ¨qqp t, ¨q ´ ξ | ď C | θ ´ t || ξ | . Proof.
We denote w “ S a p τ, θ, φ p¨qq and ˜ w “ S a p τ, θ, φ p¨q ` ξ q . Using Lemma 6.1, wehave that, for p t, x, p q P r τ, θ s ˆ R d ˆ r , s , w p t, x, p q “ Y t and ˜ w p t, x, p q “ ˆ Y t where p Y, Z q and p ˆ Y , ˆ Z q are solutions to, respectively, Y r “ φ p X t,xθ , ˜ P t,p,aθ q ` ż Tr f p s, X t,xs , Y s , Z s q d s ´ ż θr Z s d W s , t ď r ď θ , ˆ Y r “ φ p X t,xθ , ˜ P t,p,aθ q ` ξ ` ż Tr f p s, X t,xs , ˆ Y s , ˆ Z s q d s ´ ż θr ˆ Z s d W s , t ď r ď θ . Denoting Γ : “ Y ` ξ and f ξ p t, x, y, z q “ f p t, x, y ´ ξ, z q , one observes then that p Γ , Z q is the solution toΓ r “ φ p X t,xθ , ˜ P t,p,aθ q ` ξ ` ż Tr f ξ p s, X t,xs , Γ s , Z s q d s ´ ż θr Z s d W s , t ď r ď θ . Let ∆ : “ Γ ´ ˆ Y , δZ “ Z ´ ˆ Z and δf s “ f ξ p s, X t,xs , Γ s , Z s q ´ f p s, X t,xs , Γ s , Z s q , for s P r t, θ s . We then get∆ r : “ ż θr ` f p s, X t,xs , Γ s , Z s q ´ f p s, X t,xs , Y s , Z s q ` δf s ˘ d s ´ ż θr δZ s d W s . Classical energy estimates for BSDEs [18, 11] lead to E « sup r Pr t,θ s | ∆ r | ff ď C E „ż θt | ∆ s δf s | d s . (2.31)Next, we compute ż θt | ∆ s δf s | d s ď C sup s Pr t,θ s | ∆ s | ` C ˆż θt | δf s | d s ˙ . Combining the previous inequality with (2.31), we obtain E « sup r Pr t,θ s | ∆ r | ff ď C E «ˆż θt | δf s | ˙ ff . f , we get from the definition of f ξ , | δf s | ď Lξ , which eventually leads to E « sup r Pr t,θ s | ∆ r | ff ď C | θ ´ t | ξ (2.32)and concludes the proof. l Lemma 2.3.
Let ď τ ă θ ď T and φ P C pr , T s ˆ R d ˆ r , sq . For p t, x, p q Pr τ, θ q ˆ R d ˆ p , q , φ p t, x, p q ´ S a p τ, θ, φ p θ, ¨qqp t, x, p q ´ p θ ´ t q G a φ p t, x, p q “ o p θ ´ t q . where G a φ p t, x, p q : “ ´B t φ p t, x, p q ` F a p t, x, p, φ, Dφ, D φ q . Proof.
We first observe that S a p τ, θ, φ p¨qqp t, x, p q “ Y t , where p Y a , Z a q is solution to Y r “ Φ θ ` ż θr f p s, X t,xs , Y s , Z s q d s ´ ż θr Z s d W s with, for t ď s ď θ , Φ s “ φ p s, X t,xs , P t,p,αs q and α : “ a r ,τ s . By a direct application of Ito’s formula, we observe thatΦ r “ Φ θ ´ ż θr tB t φ ` L α φ up s, X t,xs , P t,p,αs q d s ´ ż θr Z s d W s , t ď r ď θ , where Z s : “ z p X t,xs , Dφ p s, X t,xs , P t,p,as q , α s q , t ď s ď θ .For ease of exposition, we also introduce an “intermediary” process p ˆ Y , ˆ Z q as the solu-tion to ˆ Y r “ Φ θ ` ż θr f p s, X t,xs , Φ s , Z s q d s ´ ż θr ˆ Z s d W s , t ď r ď θ . Now, we computeˆ Y t ´ Φ t ` p θ ´ t q G a φ p t, x, p q“ E „ż θt ` tB t φ p s, X t,xs , P t,p,as q ´ B t φ p t, x, p qu ` t F a φ p s, X t,xs , P t,p,as q ´ F a φ p t, x, p qu ˘ d s . Using the smoothness of φ , the Lipschitz property of f and the following control E “ | X t,xs ´ x | ` | P t,p,αs ´ p | ‰ ď C a | θ ´ t | , (2.33)13e obtain | ˆ Y t ´ Φ t ` p θ ´ t q G a φ p t, x, p q| ď C a,φ p θ ´ t q . (2.34)We also have ˆ Y r ´ Φ r “ ż θr G a φ p s, X t,xs , P t,p,αs q d s ´ ż θr p ˆ Z s ´ Z s q d W s Applying classical energy estimates for BSDEs, we obtain E « sup r Pr t,θ s | ˆ Y r ´ Φ r | ` ż θt | ˆ Z s ´ Z s | d s ff ď C E «ˆż θt | G a φ p s, X t,xs , P t,p,αs | d s ˙ ff ď C a,φ p θ ´ t q , (2.35)where for the last inequality we used the smoothness of φ and the linear growth of f and σ .We also observe thatˆ Y r ´ Y r “ ż θr t δf s ` f p s, X t,xs , ˆ Y s , ˆ Z s q ´ f p s, X t,xs , Y s , Z s qu d s ´ ż θr t ˆ Z s ´ Z s u d W s , where δf s : “ f p s, X t,xs , Φ s , Z s q ´ f p s, X s , ˆ Y s , ˆ Z s q , for t ď s ď θ . Once again, fromclassical energy estimates [18, 11], we obtain | ˆ Y t ´ Y t | ď C E «ˆż θt δf s d s ˙ ff . Using the Cauchy-Schwarz inequality and the Lipschitz property of f , | ˆ Y t ´ Y t | ď C p θ ´ t q E « sup r Pr t,θ s | ˆ Y r ´ Φ r | ` ż θt | ˆ Z s ´ Z s | d s ff . This last inequality, combined with (2.35), leads to | ˆ Y t ´ Y t | ď C p θ ´ t q . The proof is concluded by combining the above inequality with (2.34). l Finally, we can prove the following consistency property.
Proposition 2.4 (Consistency) . Let φ P C pr , T s ˆ R d ˆ r , sq . For p t, x, p q P r , T q ˆ R d ˆ p , q , ˇˇˇˇ t ` π ´ t S ` π, t, x, p, φ p t, x, p q ` ξ, φ p¨q ` ξ ˘ ` B t φ ´ F n p t, x, p, φ, Dφ, D φ q ˇˇˇˇ Ñ as p| π | , ξ q Ñ . roof. We first observe that by Lemma 2.2, it is sufficient to prove ˇˇˇˇ t ` π ´ t S ` π, t, x, p, φ p t, x, p q , φ p¨q ˘ ` B t φ ´ F n p t, x, p, φ, Dφ, D φ q ˇˇˇˇ ÝÑ | π |Ó ˇˇˇˇ t ` π ´ t S ` π, t, x, p, φ p t, x, p q , φ p¨q ˘ ` B t φ ´ F n p t, x, p, φ, Dφ, D φ q ˇˇˇˇ “ ˇˇˇˇ t ` π ´ t t φ p t, x, p q ´ min a P K S a ` t ´ π , t ` π , φ p t ` π , ¨q ˘ p t, x, p qu ´ max a P K G a p t, x, p q φ ˇˇˇˇ ď max a P K ˇˇˇˇ t ` π ´ t t φ p t, x, p q ´ S a ` t ´ π , t ` π , φ p t ` π , ¨q ˘ p t, x, p qu ´ G a φ ˇˇˇˇ . The proof is then concluded by applying Lemma 2.3. l To conclude this section, let us observe that we obtain the following result, combiningProposition 2.1 and Theorem 2.1 .
Corollary 2.1.
In the setting of this section, assuming (H), the following holds lim n Ñ8 lim | π |Ó v n,π “ v . Remark 2.4.
An important question, from numerical perspective, is to understand howto fix the parameters n and π in relation to each other. The theoretical difficulty here isto obtain a precise rate of convergence for the approximations given in Proposition 2.1and Theorem 2.1, along the lines of the continuous dependence estimates with respectto control discretisation in [21, 17], and estimates of the approximation by piecewiseconstant controls as in [23, 22]. To answer this question in our general setting is achallenging task, extending also to error estimates for the full discretisation in the nextsection, which is left for further research. The goal of this section is to introduce a fully implementable scheme and to prove itsconvergence. The scheme is obtained by adding a finite difference approximation tothe PCPT procedure described in Section 2.2. Then in Section 4, we present numericaltests that demonstrate the practical feasibility of our numerical method. From now on,we will assume that the log-price process X is a one-dimensional Brownian motion withdrift, for p t, x q P r , T s ˆ R : X t,xs “ x ` µ p s ´ t q ` σ p W s ´ W t q , s P r t, T s , (3.1)with µ P R and σ ą X .We take advantage of the specific dynamics to design a simple to implement numericalscheme, which also simplifies the notation.We shall moreover work under the following hypothesis. Assumption 3.1.
The coefficient µ is non-negative. Remark 3.1.
This assumption is introduced without loss of generality in order to alle-viate the notation in the scheme definition. We detail in Remark 3.2(ii) how to modifythe schemefor non-positive drift µ . The convergence properties are the same. We now fix n ě R n the associated discrete set of controls (see Section 2.1). Wedenote K “ R n assuming that 0 R K and recall that v n is the solution to (2.10). Weconsider the grid π “ t “ : t ă ¨ ¨ ¨ ă t k ă ¨ ¨ ¨ ă t κ : “ T u on r , T s and approximate v n by a PCPT scheme, extending Section 2.2.The main point here is that we introduce a finite difference approximation for thesolution S a p¨q , a P K to (2.11)–(2.13). This approximation, denoted by S aδ p¨q for aparameter δ ą
0, will be specified in Section 3.1 below. For δ ą a P K , eachapproximation S aδ p¨q is defined on a spatial grid G aδ : “ δ Z ˆ Γ aδ Ă R ˆ r , s . (3.2)where Γ aδ is a uniform grid of r , s , with N aδ ` { N aδ . Atypical element of G aδ is denoted p x k , p l q : “ p kδ, l { N aδ q , and an element of (cid:96) p G aδ q is u k,l : “ u p x k , p l q , for all k P Z and 0 ď l ď N aδ . For 0 ď t ă s ď T , and ϕ : δ Z ˆr , s Ñ R a bounded function, we have that S aδ p s, t, ϕ q P (cid:96) p G aδ q .In order to define our approximation of v n , it is not enough to replace S a p¨q in theminimisation (2.16), or similarly (2.18), by S aδ p¨q , as the approximations are not definedon the same grid for the p -variable. (The flexibility of different grids will be importantlater on.) We thus have to consider a supplementary step which consists in a linearinterpolation in the p -variable. Namely any mapping u P (cid:96) p G aδ q is extended into I aδ p u q : δ Z ˆr , s Ñ R by linear interpolation in the second variable: if u P (cid:96) p G aδ q , k P Z and p P r p l , p l ` q with 0 ď l ă N aδ , I aδ p x k , p q “ p l ` ´ pp l ` ´ p l u k,l ` p ´ p l p l ` ´ p l u k,l ` , and obviously I aδ p x k , q “ u k,N aδ .The solution to the numerical scheme associated with π, δ is then v n,π,δ : π ˆ δ Z ˆr , s Ñ R satisfying p S p π, δ, t, x, p, v n,π,δ p t, x, p q , v n,π,δ q “ , (3.3)where, for any 0 ď t P π, x P δ Z , p P r , s , y P R ` and any bounded function u : π ˆ δ Z ˆ r , s Ñ R : p S p π, δ, t, x, p, y, u q “ " y ´ min a P K I aδ p S aδ p t ` π , t k , u p t ` π , ¨qqq p t k , x, p q if k ă κ,y ´ g p x q p otherwise , (3.4)16here t ` π “ inf t s P π : s ě t u .Alternatively, the approximation v n,π,δ is defined by the following backward induction:1. Initialisation: set v n,π,δ p T, x, p q : “ g p x q p , x P R d ˆ r , s .2. Backward step: For k “ κ ´ , . . . ,
0, compute w k,aδ : “ S aδ p t k , t k ` , v n,π,δ p t k ` , ¨qq and set, for p x, p q P δ Z ˆ r , s , v n,π,δ p t k , x, p q : “ inf a P K I aδ p w k,aδ qp t k , x, p q . (3.5)Before stating the main convergence result of this section, see Theorem 3.1 below, wegive the precise definition of S aδ p¨q using finite difference operators. Let 0 ď t ă s ď T, δ ą , ϕ : δ Z ˆ r , s Ñ R . We set h : “ s ´ t ą a P K , we will describe the grid G aδ “ δ Z ˆ Γ aδ Ă δ Z ˆ r , s and the finitedifference scheme used to define S aδ .First, we observe that for the model specification of this section, (2.10) can be rewrittenas sup a P K ˆ ´ D a ϕ ´ µ ∇ a ϕ ´ σ a ϕ ´ f p t, x, ϕ, σ ∇ a ϕ q ˙ “ , (3.6)with: ∇ a ϕ : “ B y ϕ ` aσ B p ϕ, (3.7)∆ a ϕ : “ B yy ϕ ` aσ B yp ϕ ` a σ B pp ϕ, (3.8) D a ϕ : “ B t ϕ ´ aσ µ B p ϕ. (3.9)Exploiting the degeneracy of the operators ∇ a and ∆ a in the direction p a, ´ σ q , weconstruct Γ aδ so that the solution to (3.6) is approximated by the solution of an implicitfinite difference scheme with only one-directional derivatives.To take into account the boundaries p “ , p “
1, we set N aδ : “ min " j ě j | a | σ δ ě * “ R σ | a | δ V (3.10)and a p a, δ q : “ sgn p a q σδN aδ , (3.11)where a ‰
0. We have N aδ “ σ { δ | a p a, δ q| . We finally set:Γ aδ : “ " , | a p a, δ q| σ δ, . . . , N aδ | a p a, δ q| σ δ “ * “ " jN aδ : j “ , . . . , N aδ * . (3.12)17e now define the finite difference scheme. To use the degeneracy of the operators ∇ a p a,δ q and ∆ a p a,δ q in the direction p a p a, δ q , ´ σ q , we define the following finite differenceoperators, for v “ p v k,l q k P Z , ď l ď N aδ “ p v p x k , p l qq k P Z , ď l ď N aδ P (cid:96) p G aδ q and w “ p w k q k P Z P (cid:96) p k Z q : ∇ aδ v k,l : “ δ ` v k ` ,l ` sgn p a q ´ v k ´ ,l ´ sgn p a q ˘ , ∇ δ w k : “ δ p w k ` ´ w k ´ q , ∇ a ` ,δ v k,l : “ δ ` v k ` ,l ` sgn p a q ´ v k,l ˘ , ∇ ` ,δ w k : “ δ p w k ` ´ w k q , ∆ aδ v k,l : “ δ ` v k ` ,l ` sgn p a q ` v k ´ ,l ´ sgn p a q ´ v k,l ˘ , ∆ δ w k : “ δ p w k ` ` w k ´ ´ w k q . Let θ ą p t, x, y, q, q ` , A q P r , T s ˆ R . F p t, x, y, q, A q : “ ´ µq ´ σ A ´ f p t, x, y, σq q , and (3.13) p F p t, x, y, q, q ` , A q : “ ´ µq ` ´ ˆ σ ` θ δ h ˙ A ´ f p t, x, y, σq q . (3.14)Now, S aδ p s, t, ϕ q P (cid:96) p G aδ q is defined as the unique solution to (see Proposition 3.1 belowfor the well-posedness of this definition) S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ q “ , (3.15) v k, “ v k , v k,N aδ “ v k , (3.16)where, for k P Z , ă l ă N aδ , p v, v ` , v ´ q P R , and any bounded function u : δ Z ˆr , s Ñ R : S p k, l, v, q, q ` , A, u q “ v ´ u p x k , p a p p l qq ` h p F p t, kδ, v, q, q ` , A q , (3.17)with, for p P r , s , p a p p q : “ p ´ µ a p a, δ q σ h, (3.18)and where p v k q k P Z (resp. p v k q k P Z ) is the solution to S b p k, v k , ∇ δ v k , ∇ ` ,δ v k , ∆ δ v k , p ϕ k q k P Z q “ , (3.19) p resp. S b p k, v k , ∇ δ v k , ∇ ` ,δ v k , ∆ δ v k , p ϕ k q k P Z q “ q (3.20)with ϕ k “ ϕ p kδ, q (resp. ϕ k “ ϕ p kδ, N aδ q ) and, for k P Z , p v, v ` , v ´ q P R , u P (cid:96) p Z q : S b p k, v, q, q ` , A, u q “ v ´ u k ` h p F p t, kδ, v, q, q ` , A q . (3.21) Remark 3.2. (i) Here, as stated before, we have assumed µ ě . If the oppositeis true, one has to consider ∇ a ´ p δ q v k,l : “ δ ` v k,l ´ v k ´ ,l ´ sgn p a q ˘ (resp. ∇ ´ p δ q w k : “ δ p w k ´ w k ´ q ) instead of ∇ a ` ,δ v k,l (resp. ∇ ` ,δ w k ), in the definition of S aδ p s, t, ϕ q (resp. v k , v k ), to obtain a monotone scheme.(ii) For the nonlinearity f , we used the Lax-Friedrichs scheme [13, 17], adding the term θ p v ` ` v ´ ´ v q term in the definition of p F to enforce monotonicity.
18e now assume that the following conditions on the parameters are satisfied: δ ď , (3.22) hL δ ď θ ă , (3.23) µh ď δ ď M h, (3.24)for a constant M ą
0. Under these conditions, we prove that S aδ p s, t, ϕ q is uniquelydefined, and that it can be obtained by Picard iteration. Remark 3.3.
Since µh ď δ , we have | µ a p a,δ q σ h | ď | a p a,δ q| σ δ , which ensures that from (3.18) , p a p p l q P r , s for all ă l ă N aδ . Proposition 3.1.
For every bounded function ϕ : δ Z ˆ r , s Ñ R , there exists a uniquesolution to (3.15) – (3.16) . Proof.
First, v P (cid:96) p δ Z q (resp. v P (cid:96) p δ Z q ) is uniquely defined by (3.19) (resp. (3.20)),see Proposition 6.2.We consider the following map: (cid:96) p G aδ q Ñ (cid:96) p G aδ q ,v ÞÑ ψ p v q , where ψ p v q is defined by, for k P Z and l P t , . . . , N a ´ u : ψ p v q k,l “ ` hδ µ ` σ hδ ` θ p ϕ p kδ, p a p p l qq ` (3.25) hδ µv k ` ,l ` sgn p a q ` σ hδ p v k ` ,l ` sgn p a q ` v k ´ ,l ´ sgn p a q q` hf ´ t ´ , kδ, v k,l , σ δ p v k ` ,l ` sgn p a q ´ v k ´ ,l ´ sgn p a q q ¯ ` θ p v k ` ,l ` sgn p a q ` v k ´ ,l ´ sgn p a q q ¯ ,ψ p v q k, “ v k , ψ p v q k,N a “ v k . (3.26)Notice that v is a solution to (3.15)-(3.16) if and only if v is a fixed point of ψ . It isnow enough to show that ψ maps (cid:96) p G aδ q into (cid:96) p G aδ q and is contracting.If v P (cid:96) p G aδ q , by boundedness of ϕ, v and v , it is clear that ψ p v q is bounded.If v , v P (cid:96) p G aδ q , we have, for all k P Z and 1 ď l ď N a ´ | ψ p v q k,l ´ ψ p v q k,l | ď hδ µ ` σ hδ ` θ ` hL ` hLδ ` hδ µ ` σ hδ ` θ | v ´ v | . (3.27)Since δ ď hL ` hLδ ď hLδ ď θ , thus: | ψ p v q ´ ψ p v q| ď θ ` hδ µ ` σ hδ ` θ ` hδ µ ` σ hδ ` θ | v ´ v | . (3.28)Since 4 θ ă x ÞÑ θ ` x ` x is increasing on r , with limit 1 when x Ñ `8 , this proves that ψ is a contracting mapping. l Proposition 3.2.
Let ϕ , ϕ : δ Z ˆr , s Ñ R two bounded functions satisfying ϕ ď ϕ on δ Z ˆ r , s .1. (Monotonicity) For all k P Z , ď l ď N a , p v, q, q ` , A q P R , we have: S p k, l, v, q, q ` , A, ϕ q ď S p k, l, v, q, q ` , A, ϕ q . (3.29)
2. (Comparison theorem) Let p v , v q P (cid:96) p G aδ q satisfy, for all k P Z and ď l ď N aδ ´ : S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ qď S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ q (3.30) v k, ď v k, , (3.31) v k,N aδ ď v k,N aδ . (3.32) Then v ď v .3. We have S aδ p s, t, ϕ q k,l ď S aδ p s, t, ϕ q k,l for all k P Z and ď l ď N aδ . Proof.
Let ϕ , ϕ as stated in the proposition.1. We have, for k P Z and 0 ă l ă N aδ : S p k, l, v, q, q ` , A, ϕ q ´ S p k, l, v, q, q ` , A, ϕ q“ p ϕ ´ ϕ q p x k , p a p p l qq ď .
2. We assume here that a ą
0. For k P Z , let M k “ max ď l ď N aδ p v k ` l,l ´ v k ` l,l q ă 8 (if a ă
0, we have to consider max ď l ď N aδ p v k ´ l,l ´ v k ´ l,l )). We want to provethat M k ď k . Assume to the contrary that there exists k P Z such that M k ą
0. Then there exists 0 ď l ď N aδ such that v k ` l,l ´ v k ` l,l “ M k ą . (3.33)First, we have v k, ď v k, and v k ` N aδ ,N aδ ď v k ` N aδ ,N aδ . Thus 0 ă l ă N aδ .Moreover, using (3.30), re-arranging the terms, using the fact that f is non-increasing with respect to its third variable and Lipschitz-continuous, by (3.33), p ` θ q M k ď hL δ ˇˇ v k ` l ` ,l ` ´ v k ` l ` ,l ` ˇˇ ´ θ p v k ` l ` ,l ` ´ v k ` l ` ,l ` q` hL δ ˇˇ v k ` l ´ ,l ´ ´ v k ` l ´ ,l ´ ˇˇ ´ θ p v k ` l ´ ,l ´ ´ v k ` l ´ ,l ´ q . (3.34)For j P t l ´ , l ` u , we observe that hL δ | v k ` j,j ´ v k ` j,j | ´ θ p v k ` j,j ´ v k ` j,j q ď ˆ hL δ ` θ ˙ M k . (3.35)20ndeed, if v k ` j,j ě v k ` j,j then hL δ | v k ` j,j ´ v k ` j,j | ´ θ p v k ` j,j ´ v k ` j,j q “ ˆ hL δ ´ θ ˙ p v k ` j,j ´ v k ` j,j q ď , since hL δ ď θ . Otherwise, if v k ` j,j ă v k ` j,j hL δ | v k ` j,j ´ v k ` j,j | ´ θ p v k ` j,j ´ v k ` j,j q “ ˆ hL δ ` θ ˙ p v k ` j,j ´ v k ` j,j qď ˆ hL δ ` θ ˙ M k . Inserting (3.35) into (3.34), we get p ` θ q M k ď ˆ hL δ ` θ ˙ M k . (3.36)Thus, ˆ ´ hLδ ˙ M k ď , (3.37)which is a contradiction to M k ą hLδ ď θ ă .3. Let v i “ S aδ p s, t, ϕ i q for i “ ,
2. Since ϕ ď ϕ and ϕ ď ϕ , we get by Proposition6.2 that v k, ď v k, and v k,N aδ ď v k,N aδ for all k P Z .By monotonicity, we get, for all k P Z and 0 ă l ă N aδ , S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ qď S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ q Moreover, S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ q“ S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ q “ S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ qď S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , ϕ q and the proof is concluded applying the previous point. l We last give a refinement of the comparison theorem, which will be useful in the sequel.21 roposition 3.3.
Let u : δ Z ˆr , s Ñ R be a bounded function, and let v , v P (cid:96) p G aδ q .Assume that, for all k P Z and ă l ă N aδ , we have S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , u qď ď S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , u q . Then: v k,l ´ v k,l ď e ´ a p a,δ q σ C p h,δ q l p N aδ ´ l q ´ |p v ¨ , ´ v ¨ , q ` | ` |p v ¨ ,N aδ ´ v ¨ ,N aδ q ` | ¯ , (3.38) where C p h, δ q : “ ln ˆ ` hδ µ ` σ hδ ` θ ` hL δhδ µ ` σ hδ ` θ ` hL δ ˙ δ . (3.39) Moreover, C p h, δ q ě ` p µ ` L q M ` θM ˘ h ` σ h ´ M σ . (3.40) Remark 3.4. (i) To prove the consistency of the scheme, we define in Lemma 3.2smooth functions w ˘ so that p w ˘ p x k , p l qq P l p G aδ q satisfy S ě or S ď , but wecannot use the comparison theorem as the values at the boundary cannot be controlled.The previous proposition will be used in Lemma 3.3 to show that the difference between w ˘ and the linear interpolant of a solution of S “ is small.(ii) The coefficient exp ´ ´ a p a,δ q σ C p h, δ q l p N aδ ´ l ¯ that appears in the first equation ofthe previous proposition shows that the dependance on the boundary values decays ex-ponentially with the distance to the boundary. This was to be expected and was alreadyobserved in similar situations, see for example Lemma 3.2 in [3] for Hamilton-Jacobi-Bellman equations. We now can state the main result of this section.
Theorem 3.1.
The function v n,π,δ converges to v n uniformly on compact sets, as | π | , δ Ñ satisfying conditions (3.22) – (3.24) for all h “ t i ` ´ t i , where π “ t “ t ă t ă ¨ ¨ ¨ ă t κ “ T u . We prove below that the scheme is monotone (see Proposition 3.5), stable (seeProposition 3.6), consistent with (2.10) in r , T q ˆ R ˆ p , q (see Proposition 3.7)and with the boundary conditions (see Proposition 3.4). The theorem then follows byidentical arguments to [4]. 22 .2 Proof of Theorem 3.1 We first show that the numerical scheme is consistent with the boundary conditions.For any discretisation parameters π, δ , we define V π,δ : π ˆ δ Z Ñ R as the solution tothe following system: S b p k, v jk , ∇ δ v jk , ∇ ` ,δ v jk , ∆ δ v jk , v j ` k q “ , k P Z , ď j ă κ (3.41) v κk “ g p x k q , k P Z , (3.42)where v jk : “ v p t j , x k q for 0 ď j ď κ and k P Z . We set p U π,δ q jk : “ ∇ δ p V π,δ q jk “ δ pp V π,δ q jk ` ´p V π,δ q jk ´ q . We recall from Proposition 6.3 that V π,δ and U π,δ are bounded,uniformly in π, δ , and, by [4], that V π,δ converges to V uniformly on compact sets as | π | Ñ δ Ñ Proposition 3.4.
There exists constants K , K , K ą such that, for all discretisa-tion parameters π, δ with | π | small enough, we have, for p t j , x k , p q P π ˆ δ Z ˆ r , s : pV π,δ p t j , x k q ´ K p T ´ t j q ď v n,π,δ p t j , x k , p q ď pV π,δ p t j , x k q ` K p T ´ t j q ,pV π,δ p t j , x k q ´ p ´ e ´ K p qp ´ e ´ K p ´ p q q ď v n,π,δ p t j , x k , p qď pV π,δ p t j , x k q ` p ´ e ´ K p qp ´ e ´ K p ´ p q q . Proof.
We only prove, by backward induction, the lower bounds, while the proof ofthe upper bounds is similar. We need to introduce first some notation. For 0 ď j ď κ , k P δ Z and 0 ď l ď N aδ , we set V jk : “ V π,δ p t j , x k q and U jk : “ U π,δ p t j , x k q . For (cid:15) P t , u ,we define: (cid:15) w p t j , x k , p q : “ pV jk ´ (cid:15) c p t j , p q , (3.43)with (cid:15) c p t j , p q : “ (cid:15)K p T ´ t j q ` p ´ (cid:15) qp ´ e ´ K p qp ´ e ´ K p ´ p q q , (3.44)and (cid:15) w jk,l “ (cid:15) w p t j , x k , p l q , (cid:15) c jl “ (cid:15) c p t j , p l q , p l P Γ aδ . The proof now procedes in two steps.1. First, we have (cid:15) w p T, x k , p q ď pV π,δ p T, x k q “ pg p x k q “ v n,π,δ p T, x k , p q on δ Z ˆ r , s .Suppose that, for 0 ď j ă κ , on δ Z ˆ r , s , we have (cid:15) w p t j ` , x k , p q ď v n,π,δ p t j ` , x k , p q . We want to prove on δ Z ˆ r , s (cid:15) w p t j , x k , p q ď v n,π,δ p t j , x k , p q . Since (cid:15) w is convex in p , (cid:15) w p t j , x k , ¨q ď I aδ p (cid:15) w jk, ¨ q on r , s . By definition, we have v n,π,δ p t j , x k , p q “ min a P K I aδ p S aδ p t j ` , t j , v n,π,δ p t j ` , ¨qqqp x k , p q , we are thus going to prove (cid:15) w jk,l ď S aδ p t j ` , t j , v n,π,δ p t j ` , ¨qqp t j , x k , p l q (3.45)23or all a P K and all k P Z , ď l ď N aδ .For a P K , by induction hypothesis, (cid:15) w p t j ` , ¨q ď v n,π,δ p t j ` , ¨q , so if we are able to get S b p k, (cid:15) w k , ∇ δ(cid:15) w k , ∇ ` ,δ(cid:15) w k , ∆ δ(cid:15) w k , (cid:15) w j ` k, q ď , k P Z , (3.46) S b p k, (cid:15) w k , ∇ δ(cid:15) w k , ∇ ` ,δ(cid:15) w k , ∆ δ(cid:15) w k , (cid:15) w j ` k,N aδ q ď , k P Z , (3.47) S p k, l, (cid:15) w k,l , ∇ aδ (cid:15) w k,l , ∇ a ` ,δ(cid:15) w k,l , ∆ aδ (cid:15) w k,l , (cid:15) w p t j ` , ¨qq ď , k P Z , ă l ă N aδ , (3.48)where (cid:15) w jk “ (cid:15) w p t j , x k , q , (cid:15) w jk “ (cid:15) w p t j , x k , q , we obtain that (3.45) holds true by thecomparison result in Proposition 3.2, which concludes the proof. We now proceed withthe proof of (3.46), (3.47) and (3.48).2.a Now, observe that (cid:15) w jk “ ´ (cid:15)K p T ´ t j q , for k P Z . We have, since f p t j , x k , , q “ f is non-increasing in its third variable, S b p k, (cid:15) w k , ∇ δ(cid:15) w k , ∇ ` ,δ(cid:15) w k , ∆ δ(cid:15) w k , (cid:15) w j ` k, q “ ´ (cid:15)Kh ´ hf p t j , x k , ´ (cid:15)K p T ´ t j q , q ď . (cid:15) w jk “ V jk ´ (cid:15)K p T ´ t j q , for k P Z . Since f p t j , x k , V jk ´ (cid:15)K p T ´ t j q , U jk q ě f p t j , x k , V jk , U jk q , and by definition of V π,δ : S b p k, (cid:15) w k , ∇ δ(cid:15) w k , ∇ ` ,δ(cid:15) w k , ∆ δ(cid:15) w k , (cid:15) w j ` k,N aδ q “ ´ (cid:15)Kh ` S b p k, V jk , ∇ δ V jk , ∇ ` ,δ V jk , ∆ δ V jk qď ´ (cid:15)Kh ď . k P Z , 0 ă l ă N aδ . We have, by definition (3.17) of S : S p k, l, (cid:15) w jk,l , ∇ aδ (cid:15) w jk,l , ∇ a ` ,δ(cid:15) w jk,l , ∆ aδ (cid:15) w jk,l , (cid:15) w p t j ` , ¨qq“ (cid:15) w jk,l ´ (cid:15) w p t j ` , x k , p a p p l qq` h p F p t, kδ, (cid:15) w jk,l , ∇ aδ (cid:15) w jk,l , ∇ a ` ,δ(cid:15) w jk,l , ∆ aδ (cid:15) w jk,l qď ´ (cid:15) c jk,l ` µ a p a, δ q σ hV j ` k ` (cid:15) c p t j ` , x k , p a p p l qq´ p l h p F p t, x k , V jk , U jk , ∇ ` ,δ V jk , ∆ δ V jk q` h p F p t, x k , p l V jk , ∇ aδ (cid:15) w jk,l , ∇ a ` ,δ(cid:15) w jk,l , ∆ aδ (cid:15) w jk,l q , where we have used (3.41) and f p t, x k , (cid:15) w jk,l , σ ∇ aδ (cid:15) w jk,l q ě f p t, x k , p l V jk , σ ∇ aδ (cid:15) w jk,l q .By adding ˘ p l hf p t j , x k , p l V jk , σ ∇ aδ (cid:15) w jk,l q , using the Lipschitz continuity of f and ∇ aδ (cid:15) w jk,l “ p l U jk ` a p a, δ q σ p V jk ` ` V jk ´ q ` δ ´ (cid:15) c jl ´ sgn p a q ´ (cid:15) c jl ` sgn p a q ¯ ,
24e get, by definition (3.14) of p F , S p k, l, (cid:15) w k,l , ∇ aδ (cid:15) w jk,l , ∇ a ` ,δ(cid:15) w jk,l , ∆ aδ (cid:15) w jk,l , (cid:15) w j p t j ` , ¨qqď h a p a, δ q σ µ p V j ` k ´ V jk ` q ´ hσ a p a, δ q U jk ´ θ a p a, δ q σ δ U jk ` hLp l p ´ p l qp V jk ` | U jk |q ` hL | a p a, δ q| σ p V jk ` ` V jk ´ q´ ˆ (cid:15) c jl ´ (cid:15) c p t j ` , p a p p l qq ´ µh ∇ a ` ,δ(cid:15) c jl ´ ˆ σ h ` θδ ˙ ∆ aδ (cid:15) c jl ˙ ` hL | ∇ aδ (cid:15) c jl | . Since | a p a, δ q| ď max t| a | , a P K u ď n and V and U are bounded uniformly in h, δ (seeProposition 6.3 in the appendix), there exists a constant K n,θ,M,L ą h a p a, δ q σ µ p V j ` k ´ V jk ` q ´ hσ a p a, δ q U jk ´ θ a p a, δ q σ δ U jk ` hLp l p ´ p l qp V jk ` | U jk |q ` hL | a p a, δ q| σ p V jk ` ` V jk ´ q ď hK n,θ,M,L . When (cid:15) “
1, the terms of the last three lines all vanish except the first one, and c jl ´ c p t j ` , p l ´ µ a p a,δ q σ h q “ K h . Thus we get: S p k, l, (cid:15) w k,l , ∇ aδ (cid:15) w k,l , ∇ a ` ,δ(cid:15) w k,l , ∆ aδ (cid:15) w k,l , (cid:15) w p t j ` , ¨qq ď h p´ K ` K n,θ,M,L q . Hence, chosing K large enough gives the result.We now deal with the case (cid:15) “
0. By Taylor expansions of (cid:15) c around p t j , p l q , we get: S p k, l, (cid:15) w k,l , ∇ aδ (cid:15) w jk,l , ∇ a ` ,δ(cid:15) w jk,l , ∆ aδ (cid:15) w jk,l , (cid:15) w j p t j ` , ¨qqď hK n,θ,M,L ` hL | a p a, δ q| σ |B p (cid:15) c p t j , p l q| ` h B t (cid:15) c p t j , p l q ` h a p a, δ q B pp (cid:15) c p t j , p l q ` hε p h ; K q , with lim h Ñ ε p h ; K q “
0. By definition of (cid:15) c , we get, for h ą h P r , h s : S p k, l, (cid:15) w k,l , ∇ aδ (cid:15) w jk,l , ∇ a ` ,δ(cid:15) w jk,l , ∆ aδ (cid:15) w jk,l , (cid:15) w j p t j ` , ¨qqď h „ K n,θ,M,L ` K L | a p a, δ q| σ e ´ K p l ` K L a p a, δ q σ e ´ K p ´ p l q ´ K a p a, δ q e ´ K p l ´ K a p a, δ q e ´ K p ´ p l q ` | ε p h ; K q| ď h „ max h Pr ,h s | ε p h ; K q| ` K n,θ,M,L ` K | a p a, δ q|p e ´ K p l ` e ´ K p ´ p l q q ˆ Lσ ´ | a p a, δ q| K ˙ . To conclude, one can choose K large enough so that K n,θ,M,L ` K | a p a, δ q|p e ´ K p l ` e ´ K p ´ p l q qp Lσ ´ | a p a,δ q K q ď ´ η ă
0, and then consider h ą | ε p h ; K q| ď η for h P r , h s . l roposition 3.5 (Monotonicity) . Let π be a grid of r , T s and δ ą satisfying (3.22) – (3.24) . Let y P R , ď k ď κ, j P Z and p P r , s , and let U , V : π ˆ δ Z ˆ r , s Ñ R betwo bounded functions such that U ď V . Then: p S p π, δ, k, j, p, y, U q ě p S p π, δ, k, j, p, y, V q . (3.49) Proof.
The result is clear for k “ κ . If k ă κ , it is sufficient to show that: I aδ p S aδ p t k ` , t k , U p t k ` , ¨qqq ď I aδ p S aδ p t k ` , t k , V p t k ` , ¨qqq , for all a P K , recalling (3.4). This is a consequence of the comparison result in Propo-sition 3.2 and the monotonicity of the linear interpolator. l We now prove the stability of the scheme. Here, in contrast to Lemma 2.1, we arenot able to prove that the solution of the scheme is increasing in p . However, due tothe boundedness of the terminal condition, we obtain uniform bounds for v n,π,δ . Proposition 3.6 (Stability) . For all π and δ ą , there exists a unique solution v n,π,δ to (3.4) , which satisfies: ď v n,π,δ ď | g | on π ˆ δ Z ˆ r , s . (3.50) Proof.
We prove the proposition by backward induction.First, since v n,π,δ is a solution to (3.4), v n,π,δ p T, x, p q “ pg p x q on δ Z ˆ r , s , and wehave 0 ď v n,π,δ p T, x, p q ď | g | for all p x, p q P δ Z ˆ r , s .Let 0 ď j ď κ ´ v n,π,δ p t k , ¨q is uniquely determined for k ą j , andthat 0 ď v n,π,δ p t j ` , ¨q ď | g | . Since v n,π,δ is a solution to (3.4), we have v n,π,δ p t j , x, p q “ min a P K I aδ p S aδ p t j ` , t j , v n,π,δ p t j ` , ¨qqq , and for each a P K , S aδ p t j ` , t j , v n,π,δ p t j ` , ¨qq is uniquely determined by Proposition 3.1,so v n,π,δ p t j , ¨q is uniquely determined. Next, we show that, for all k P Z and 0 ď l ď N a :0 ď S aδ p t j ` , t j , v n,π,δ p t j ` , ¨qq ď | g | . Then it is easy to conclude that 0 ď v n,π,δ p t j , ¨q ď e LT | g | on R ˆ r , s , by propertiesof the linear interpolation and the minimisation.First, it is straightforward that ˇ u defined by ˇ u k,l “ k P Z and 0 ď l ď N a satisfies ˇ u “ S aδ p t j ` , t j , q .The comparison theorem gives 0 ď S aδ p t j ` , t j , v n,π,δ p t j ` , ¨qq , since 0 ď v n,π,δ p t j ` , ¨q .To obtain the upper bound, we notice that ˆ u defined by ˆ u k,l : “ | g | for all k P Z and0 ď l ď N a satisfies S p k, l, ˆ u k,l , ∇ aδ ˆ u k,l , ∇ a ` ,δ ˆ u k,l , ∆ aδ ˆ u k,l , ˆ u q “ ´ hf p t j , x k , ˆ u, q ě ´ hf p t j , x k , , q ě . Hence the comparison result in Proposition 3.2 yields S aδ p t j ` , t j , v n,π,δ p t j ` , ¨qq ď | g | . l
26e now prove the consistency. The proof requires several lemmata. First, we showthat the perturbation induced by the change of controls vanishes as δ Ñ Lemma 3.1.
For all a P K , a and a p a, δ q have the same sign, and: ď | a | ´ | a p a, δ q| ď n σ δ. (3.51) Moreover, there exists c ą such that for all a P K and δ ą , | a p a, δ q| ě c ą . Proof.
By definition of N aδ , p N aδ ´ q | a | σ δ ă “ N aδ | a p a, δ q| σ δ ď N aδ | a | σ δ, thus | a | ´ | a | N aδ ă | a p a, δ q| ď | a | . Also, we observe | a | N aδ “ | a | r σ | a | δ s ď | a | σ | a | δ “ a δσ ď n σ δ, which concludes the proof of (3.51).By (3.10), we have: N aδ ď σ | a | δ ` ď σa m δ ` ď σcδ where a m “ min t| a | : a P K u and c ą a, δ .Now, by (3.11), we get: | a p a, δ q| “ σδN aδ ě cδσδσ “ c. l Last, we give explicit supersolutions and subsolutions satisfying appropriate condi-tions. Let 0 ď t ă s ď T , δ ą a P K be fixed. For (cid:15) ą
0, we set f (cid:15) p t, x, y, ν q : “ p f p t, ¨ , ¨ , ¨q ˚ ρ (cid:15) qp t, y, ν q : “ ż R ˆ R ˆ R f p t, x ´ u, y ´ z, ν ´ η q ρ (cid:15) p u, z, η q d u d z d η, where ˚ is the convolution operator and, for (cid:15) ą ρ (cid:15) p x q : “ (cid:15) ´ ρ p x { (cid:15) q with ρ : R Ñ R is a mollifier, i.e. a smooth function supported on r´ , s satisfying ş R ρ “
1. We set F (cid:15) p t, x, y, q, A q “ ˆ σ ´ µ ˙ q ´ σ A ´ f (cid:15) p t, x, y, σq q . emark 3.5. Since f is L-Lipschitz continuous with respect to its three last variables,we have | f (cid:15) ´ f | ď L(cid:15) . The lengthy proof of the following lemma by insertion is given in the appendix.
Lemma 3.2.
Let ď t ă s ď T, ϕ P C b p R ˆ R , R q , a P K . We set h : “ s ´ t . Let (cid:15) ą such that (cid:15) Ñ and δ(cid:15) Ñ as h Ñ , observing (3.24) .Then there exist bounded functions S a, ˘ δ p s, t, ϕ q : δ Z ˆ r , s Ñ R of the form S a, ˘ δ,(cid:15) p s, t, ϕ qp x, p q “ ϕ p x, p a p p qq (3.52) ´ hF (cid:15) p t, x, ϕ p x, p a p p qq , ∇ a p a,δ q ϕ p x, p a p p qq , ∆ a p a,δ q ϕ p x, p a p p qqq˘ C ϕ,n p h, (cid:15) q , where p a is defined in (3.18) , and where C ϕ,n p h, (cid:15) q ą satisfies C ϕ,n p h,(cid:15) q h Ñ as h Ñ ,such that w ˘ : “ p S a, ˘ δ,(cid:15) p s, t, ϕ qp x k , p l qq k P Z , ď l ď N aδ P (cid:96) p G aδ qq satisfy S p k, l, w ` k,l , ∇ aδ w ` k,l , ∇ a ` ,δ w ` k,l , ∆ aδ w ` k,l q ě , (3.53) S p k, l, w ´ k,l , ∇ aδ w ´ k,l , ∇ a ` ,δ w ´ k,l , ∆ aδ w ´ k,l q ď , (3.54) for all k P Z and ă l ă N aδ .Furthermore, for all x P δ Z , S a, ˘ δ,(cid:15) p s, t, ϕ qp x, ¨q P C pr , s , R q , and |B pp S a, ˘ δ,(cid:15) p s, t, ϕ q| ď C ϕ p h q (cid:15) for some constant C ϕ p h q ą independent of (cid:15) . Lemma 3.3.
Let ď t ă s ď T, δ ą , a P K, ϕ P C b p R ˆ R q be fixed. Let h “ s ´ t , k P Z , x k P δ Z , p P p , q , and assume that h is sufficiently small so that p P r p , p N aδ ´ s ,observing (3.24) . Let (cid:15) ą such that (cid:15) Ñ and δ(cid:15) Ñ as h Ñ . Then we have: S a, ´ δ,(cid:15) p s, t, ϕ qp x k , p q ´ I aδ p S aδ p s, t, ϕ qqp x k , p q ď C ϕ,n p h, (cid:15) q , (3.55) I aδ p S aδ p s, t, ϕ qqp x k , p q ´ S a, ` δ,(cid:15) p s, t, ϕ qp x k , p q ď C ϕ,n p h, (cid:15) q , (3.56) where C ϕ,n p h, (cid:15) q ą satisfies C ϕ,n p h,(cid:15) q h Ñ , as h Ñ and where the functions S a, ˘ δ,(cid:15) p s, t, ϕ q are introduced in Lemma 3.2. Proof.
We prove the first identity, the second one is similar.Set w : “ S aδ p s, t, ϕ q and w ´ : “ S a, ´ δ,(cid:15) p s, t, ϕ q . By definition of w and by (3.54), one canapply Proposition 3.3. For all k P Z and 0 ă l ă N aδ : w ´ k,l ´ w k,l ď Be ´ a p a,δ q σ C p h,δ q l p N aδ ´ l q ď Be ´ a p a,δ q σ C p h,δ qp N aδ ´ q , (3.57)with B “ |p w ´¨ , ´ w ¨ , q ` | ` |p w ´¨ ,N aδ ´ w ¨ ,N aδ q ` | and C p h, δ q is defined in (3.39). ByLemma 3.1, there exists a constant c ą | a p a, δ q| ě c . In addition, using283.40), we get: Bh e ´ C p h,δ q a p a,δ q σ p N aδ ´ q ď Bh e ´ c σ ˜ p p µ ` L q M ` θM q h ` σ h ´ M σ ¸ “ Be c M σ e ´ c σ p p µ ` L q M ` θM q h ` σ h h Ñ , as h Ñ p P r p , p N aδ ´ q and k P Z . By definition of I aδ , one has: I aδ p S aδ p s, t, ϕ qqp x k , p q “ λw k,l ` p ´ λ q w k,l ` , (3.58)where p P r p l , p l ` q with 0 ă l ă N aδ ´
1, and λ “ p l ` ´ pp l ` ´ p l . Thus: S a, ´ δ,(cid:15) p s, t, ϕ qp x k , p q ´ I aδ p w qp x k , p q“ S a, ´ δ,(cid:15) p s, t, ϕ qp x k , p q ´ I aδ p w ´ qp x k , p q ` I aδ p w ´ qp x k , p q ´ I aδ p w qp x k , p q (3.59) “ S a, ´ δ,(cid:15) p s, t, ϕ qp x k , p q ´ I aδ p w ´ qp x k , p q ` λ p w ´ k,l ´ w k,l q ` p ´ λ qp w ´ k,l ` ´ w k,l ` q . The two last terms are controlled using (3.57), and, by properties of linear interpolationof the function p ÞÑ S a, ´ δ,(cid:15) p t ` , t ´ , ϕ qp x k , p q P C pr , s , R q with |B pp S a, ´ δ,(cid:15) p t ` , t ´ , ϕ q| ď C ϕ p h q (cid:15) (recall the previous Lemma) the first term is of order δ (cid:15) “ o p h q since (3.24) is inforce and δ(cid:15) Ñ l Lemma 3.4.
For ď t ă s ď T such that L p s ´ t q ď , ξ ą , ϕ : δ Z ˆ r , s Ñ R abounded function, the following holds for all a P K : S aδ p s, t, ϕ q ` ξ ´ L p s ´ t q ξ ď S aδ p s, t, ϕ ` ξ q ď S aδ p s, t, ϕ q ` ξ, where L is the Lipschitz constant of f . Proof.
Let v “ S aδ p s, t, ϕ q , w “ v ` ξ ´ L p s ´ t q ξ . Since v satisfies (3.15), we have, for k P Z and 0 ă l ă N aδ , S p k, l, w k,l , ∇ aδ w k,l , ∇ a ` ,δ w k,l , ∆ aδ w k,l , ϕ ` ξ q “ ´ L p s ´ t q ξ ` p s ´ t q p f p t, x k , v k,l , ∇ aδ v k,l q ´ f p t, x k , w k,l , ∇ aδ v k,l qq . Since f is non-increasing in its third variable and Lipschitz continuous, we get: S p k, l, w k,l , ∇ aδ w k,l , ∇ a ` ,δ w k,l , ∆ aδ w k,l , ϕ ` ξ q ď . The same computation with l “ l “ N aδ and S b instead of S gives S b p k, l, w k,l , ∇ δ w k,l , ∇ ` ,δ w k,l , ∆ δ w k,l , ϕ ` ξ q ď , and the comparison theorem given in Proposition 6.2 gives w k,l ď S aδ p s, t, ϕ ` ξ q k,l for k P Z and l P t , N aδ u .The comparison result from Proposition 3.2 gives the first inequality of the lemma. Thesecond one is proved similarly. l roposition 3.7 (Consistency) . Let ϕ P C b pr , T s ˆ R ˆ R , R q , p t, x, p q P r , T q ˆ R ˆp , q . We have, with the notation in (3.6) : ˇˇˇˇ t j ` ´ t j p S p π, δ, t j , x k , q, ϕ p t j , x k , q q ` ξ, ϕ ` ξ q´ sup a P K r´ D a ϕ p t, x, p q ` F p t, x, ϕ p t, x, p q , ∇ a ϕ p t, x, p q , ∆ a ϕ p t, x, p qqs ˇˇˇˇ Ñ , as δ, | π | Ñ satisfying (3.22) – (3.24) , π ˆ δ Z ˆ r , s Q p t j , x k , q q Ñ p t, x, p q , ξ Ñ . Proof.
Let ϕ, j, k, p, l as in the statement of the Proposition.Without loss of generality, we can consider π, δ, t j , x k , q such that, for all a P K :0 ď p a p q q ď , Since ϕ is smooth and p t k , x j , p l q Ñ p t, x, p q , we have | sup a P K r´ D a ϕ p t, x, p q ` F p t, x, p, ϕ p t, x, p q , ∇ a ϕ p t, x, p q , ∆ a ϕ p t, x, p qqs´ sup a P K r´ D a ϕ p t j , x k , p l q ` F p t j , x k , p l , ϕ p t j , x k , p l q , ∇ a ϕ p t j , x k , p l q , ∆ a ϕ p t j , x k , p l qqqs | Ñ . Thanks to Lemma 3.4, it suffices to prove: ˇˇˇˇ t j ` ´ t j p S p π, δ, t j , x k , q, ϕ p t j , x k , q q , ϕ q´ sup a P K p´ D a ϕ p t j , x k , q q ` F p t j , x k , ϕ p t j , x k , q q , ∇ a ϕ p t j , x k , q q , ∆ a ϕ p t j , x k , q qqq ˇˇˇˇ Ñ , as | π | Ñ π ˆ δ Z ˆ p , q Q p t j , x k , q q Ñ p t, x, p q .Let (cid:15) ą (cid:15) Ñ δ(cid:15) Ñ | inf ´ sup | ď sup | ¨ ´ ¨ | , adding ˘ ˆ t j ` ´ t j ϕ p t j ` , x k , p a p q qq` F (cid:15) p t j , x k , ϕ p t j ` , x k , p a p q qq , ∇ a p a,δ q ϕ p t j ` , x k , p a p q qq , ∆ a p a,δ q ϕ p t j ` , x k , p a p q qqq ¯ and using Lemma 3.1, it is enough to show that, for all a P K , ˇˇˇˇ t j ` ´ t j p ϕ p t j ` , x k , p a p q qq ´ I aδ p S aδ p t j ` , t j , ϕ p t j ` , ¨qqqp x k , q qq (3.60) ´ F (cid:15) p t j , x k , ϕ p t j ` , x k , p a p q qq , ∇ a p a,δ q ϕ p t j ` , x k , p a p q qq , ∆ a p a,δ q ϕ p t j ` , x k , p a p q qqq ˇˇˇ Ñ . | ¨ | “ max p¨ , ´¨q and the two followinginequalities, obtained by Lemma 3.3, and by definition (3.53)-(3.54) of S a, ˘ δ,(cid:15) :1 t j ` ´ t j p ϕ p t j ` , x k , p a p q qq ´ I aδ p S aδ p t j ` , t j , ϕ p t j ` , ¨qqqp x k , q qq´ F (cid:15) p t j , x k , ϕ p t j ` , x k , p a p q qq , ∇ a p a,δ q ϕ p t j ` , x k , p a p q qq , ∆ a p a,δ q ϕ p t j ` , x k , p a p q qqqď t j ` ´ t j p ϕ p t j ` , x k , p a p q qq ´ S a, ´ δ,(cid:15) p t j ` , t j , ϕ p t j ` , ¨qqqp x k , q q ` o p t j ` ´ t j qq´ F (cid:15) p t j , x k , ϕ p t j ` , x k , p a p q qq , ∇ a p a,δ q ϕ p t j ` , x k , p a p q qq , ∆ a p a,δ q ϕ p t j ` , x k , p a p q qqq , and F (cid:15) p t j , x k , ϕ p t j ` , x k , p a p q qq , ∇ a p a,δ q ϕ p t j ` , x k , p a p q qq , ∆ a p a,δ q ϕ p t j ` , x k , p a p q qqq´ t j ` ´ t j p ϕ p t j ` , x k , p a p q qq ´ I aδ p S aδ p t j ` , t j , ϕ p t j ` , ¨qqqp x k , q qqď F (cid:15) p t j , x k , ϕ p t j ` , x k , p a p q qq , ∇ a p a,δ q ϕ p t j ` , x k , p a p q qq , ∆ a p a,δ q ϕ p t j ` , x k , p a p q qqq´ t j ` ´ t j p ϕ p t j ` , x k , p a p q qq ´ S a, ` δ,(cid:15) p t j ` , t j , ϕ p t j ` , ¨qqqp x k , q q ` o p t j ` ´ t j qq . l We now present a numerical application of the previous algorithm.
We keep the notation of the previous section: the process X is a Brownian motion withdrift. In this numerical example, the drift of the process Y is given by the followingfunctions: f p t, x, y, z q : “ ´ σ ´ µz, and f p t, x, y, z q : “ ´ σ ´ µz ` R p y ´ σ ´ z q ´ , where, for x P R , x ´ “ max p´ x, q denotes the negative part of x . The function f corresponds to pricing in a linear complete Black & Scholes market. It is well-knownthat there are explicit formulae for the quantile hedging price for a vanilla put or call,see [19].In both cases, we compute the quantile hedging price of a put option with strike K “ T “
1, i.e. g p x q “ max p K ´ exp p x q , q .The parameters of X are σ “ .
25 and µ “ . b “ .
05 in the dynamics of the associated geometric Brownian motion, where µ “ b ´ σ { f , we observe the convergence of v π,δ towards v n for a fixed discretecontrol set, and we estimate the rate of convergence with respect to δ . Second, weshow that the conditions (3.22) to (3.24) are not only theoretically important, but alsonumerically. Last, we use the fact that the analytical solution to the quantile hedgingproblem with driver f is known (see [19]) to assess the convergence (order) of the schememore precisely. We observe that a judicious choice of control discretisation, time andspace discretisation leads to convergence of v π,δ to v . However, the unboundedness ofthe optimal control as t Ñ T leads to expensive computations.The scheme obtained in the previous section deals with an infinite domain in the x variable. In practice, one needs to consider a bounded interval r B , B s and to addsome boundary conditions. Here, we choose B “ log p q and B “ log p q , andthe approximate Dirichlet boundary values for v p t, B i , p q are the limits lim v th p t, x, p q as x Ñ x Ñ `8 , where v th is the analytical solution obtained in [19] for thelinear driver f . Since the non-linearity in f is small for realistic parameters (wechoose R “ .
05 in our tests), it is expected that the prices are close (see also [20]).Furthermore, we will consider values obtained for points p t, x, p q with x far from to theboundary. In this situation, the influence of our choice of boundary condition should besmall, as noticed for example in Proposition 3.3. This was studied more systematically,for example, in [2]. v n with the non-linear driver In this section, we consider the non-linear driver f defined above, where there is noknown analytical expression for the quantile heding price. We now fix a discrete controlset, and we compute the value function v π,δ for various discretisation parameters π, δ satisfying (3.22) to (3.24). We consider the following control set with 22 controls: ˆ r´ , s X Z ´ t u ˙ ď ˆ r´ , s X Z ´ t u ˙ (4.1) “ t´ , ´ . , . . . , . , u Y " ´ , ´ ` , . . . , ´ , * , and δ P t . , . , . u . For a fixed δ , we set h “ Cδ with C : “ min p , θL , | σ ´ µ | q , θ “ and L “ | µ | ` R , so that (3.22) to (3.24) are satisfied.We get the graphs shown in Figure 1 for the function p ÞÑ v π,δ p t, x, p q , where p t, x q “p , q , p , q .We observe, while not proved, that the numerical approximation always gives anupper bound for v n , which is itself greater than the quantile hedging price v . This is apractically useful feature of this numerical method.The scheme preserves a key feature of the exact solution, namely that the quantilehedging price is 0 exactly for p below a certain threshold, depending on t, x . Thisis a consequence of the diffusion stencil ∆ aδ respecting the degeneracy of the diffusionoperator ∆ a in (3.8), which acts only in direction p , p q , and by the specific constructionof the meshes. 32 v p v (0,30,p)v (0,30,p)v (0,30,p) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 v p v (0,37,p)v (0,37,p)v (0,37,p) Figure 1: The functions v π,δ p t, x, ¨q , t “ x P t , u .In Table 1, we show some discretisation parameters obtained by this constructionwith selected values of δ . Here, N x is the number of points for the x -variable, N c thenumber of controls, and N p the total number of points for the p variable (i.e., for allmeshes combined). Moreover, a max (resp. a min ) is the greatest (resp. smallest) controlobtained, using the modification of the control set (4.1) as described in Section 3.1.Also recall that different meshes are applied in each step of the PCPT algorithm fordifferent a , hence we also report N p p a max q (resp. N p p a min q ), the number of points forthe p variable for the control a max (resp. a min ). With our choice of parameters, we have h “ δ , so the number of time steps is always δ . δ N t N x N c N p a max N p p a max q a min N p p a min q time (in seconds)0 . . . . .
05 20 33 15 38 2 . . . .
01 100 153 22 255 2 .
77 10 0 .
33 76 1660 .
005 200 304 22 585 2 .
94 18 0 .
33 151 4608Table 1: Parameters for selected values of δ . Using the same discrete control set as above, we now fix h “ . v π,δ for δ chosen as above. The conditions (3.22) to (3.24) are then not satisfied anymore.First, while π is coarse, we observe that the computational time to get v π,δ p t j , ¨q from v π,δ p t j ` , ¨q is larger. In fact, since the conditions are not satisfied anymore, the resultsof Proposition 3.1 are not valid anymore. While convergence to a fixed point is stillobserved, many more Picard iterations are needed. For example, for δ “ .
005 and h “ .
1, we observe that 3000 Picard iterations are needed, while in the example where(3.22) to (3.24) were satisfied, 250 iterations sufficed to obtain convergence (with atolerance parameter of 10 ´ ). 33he second observation is that, while we observe convergence to some limit (at leastwith our choice of δ : it might start to diverge for smaller δ , as seen for the case δ fixed and varying h below), it is not the limit observed in the previous subsection. Weshow in Figure 2 the difference between the solution obtained with δ “ . , h “ . δ “ . , h “ Cδ . When the conditions are not met, we are dealing with a non-monotone scheme, and convergence to the unique viscosity solution of the PDE, whichequals the value function of the stochastic target problem, is not guaranteed.Conversely, when δ is fixed and we vary h , the situation is different. There is no issuewith the Picard iterations, as the conditions needed for Proposition 3.1 are still satisfied.The issue here is that the consistency hypothesis is not satisfied, and convergence isnot observed: when h is too close to 0, the value v π,δ gets bigger, as seen in Figure 3.Here, δ is fixed to 0 .
05 and h goes from 0 .
025 to 1 . ˆ ´ . v p v Cdelta,0.005 (0,37,p)v (0,37,p)
Figure 2: Comparison of v . , . p , , ¨q and v Cδ, . p , , ¨q . v p v (0,30,p)v (0,30,p)v (0,30,p) Figure 3: Comparison of v h, . p , , ¨q forsome h . We now consider the linear driver f . In that case, the quantile hedging price can befound explicitly (see [19]). For each p t, x, p q , the optimal control can also be computedexplicitly: α p t, x, p q “ a π p T ´ t q exp ˆ ´ N ´ p p q ˙ , where N is the cumulative distribution function of the standard normal distribution.In particular, if the uniform grid π “ t , h, . . . , κh “ T u is fixed, one obtains that theoptimal controls are contained in the interval r , ? πh s .On the other hand, if δ is fixed, one sees from (3.11) that the greatest control one canreach (with a non-trivial grid for the p variable) is σ δ .We set our parameters as follows: we first choose n ě
2, we pick δ such that σnδ ě ? πCδ ,34nd we set h “ Cδ . It is easy to see that δ is proportional to n ´ .We now pick the controls in t σmδ , m ě n u to obtain K n : “ t a i : “ σm i δ , i “ , . . . , M u as follows: let m “ n so that a “ a n max “ σnδ . If m , . . . , m i are constructed, we set m i ` “ inf t m ě m i , σm i δ ´ σmδ ě n u and a i ` “ σm i ` δ . If m i ` ă n ´ , then we set M “ i ` p t, x, p q “ p , , . q , has aconvergence rate of about 1 . n in the construction described previously.Last, in Table 2, we report the values of δ and a max obtained for different choices of n . v p v(0,30,p)v (0,30,p)v (0,30,p)v (0,30,p) Figure 4: v n p , , ¨q and v p , , ¨q for n “ , , . . . . n | v _n − v | ( , , . ) l l l l l ll Figure 5: Convergence rate estimationof v n p , , . q to v p , , . q and log-logplot. n δ a max N x N c N p .
04 1 .
91 37 5 335 0 .
01 3 .
18 97 12 2447 0 .
006 5 .
09 248 26 1138Table 2: Discretisation parameters for selected values of n . We have introduced semi-discrete and discrete schemes for the quantile hedging prob-lem, proven their convergence, and illustrated their behaviour in a numerical test.The scheme, based on piecewise constant policy time-stepping, has the attractivefeature that semi-linear PDEs for individual controls can be solved independently onadapted meshes. In the example of the Black-Scholes dynamics this had the effectthat in spite of the degeneracy of the diffusion operator it was possible to construct35n each mesh a local scheme, i.e. one where only neighbouring points are involved inthe discretisation. This does not contradict known results on the necessity of non-local stencils for monotone consistent schemes in this degenerate situation (see e.g.[26]), because of the superposition of different highly anisotropic meshes to arrive at ascheme which is consistent overall.A more accurate scheme could be constructed by exploiting higher order, limitedinterpolation in the p -variable, such as in [27]. It should be possible to deduce conver-gence from the results of this paper and the properties of the interpolator using thetechniques in [28]. Proof. [Proposition 3.3] For ease of notation, we set, e p l q : “ e ´ a p a,δ q σ C p h,δ q l p N a ´ l q ,e ‹ : “ min x Pr ,N aδ s e p x q “ e p N aδ { q “ e ´ a p a,δ q σ C p h,δ qp N aδ q “ e ´ C p h,δ q δ ,B : “ |p v ¨ , ´ v ¨ , q ` | ` |p v ¨ ,N aδ ´ v ¨ ,N aδ q ` | . By the comparison theorem, it is enough to show that w P (cid:96) p G aδ q defined by w k,l : “ v k,l ` Be p l q (6.1)satisfies w k, ě v k, , w k,N aδ ě v k,N aδ and S p k, l, w k,l , ∇ aδ w k,l , ∇ a ` ,δ w k,l , ∆ aδ w k,l , u q ě
0, forall k P Z and 0 ă l ă N aδ .The boundary conditions are easily checked: if k P Z and l P t , N a u , we have, since e p q “ e p N a q “ w k,l “ v k,l ` B ě v k,l ` p v k,l ´ v k,l q ` ě v k,l . For k P Z , 1 ď l ď N aδ ´
1, we prove S p k, l, w k,l , ∇ aδ w k,l , ∇ a ` ,δ w k,l , ∆ aδ w k,l , u q ě ˘ hf p t ´ , e kδ , v k,l , δ p w k ` ,l ` sgn p a q ´ w k ´ ,l ´ sgn p a q qq , since S p k, l, v k,l , ∇ aδ v k,l , ∇ a ` ,δ v k,l , ∆ aδ v k,l , u q ě f is non-increasing with respect toits third variable and Lipschitz continuous with respect to its fourth variable, we have: S p k, l, w k,l , ∇ aδ w k,l , ∇ a ` ,δ w k,l , ∆ aδ w k,l , u q (6.2) ě B „ˆ ` µ hδ ` σ hδ ` θ ˙ e p l q ´ ˆ µ hδ ` σ hδ ` θ ˙ e p l ` sgn p a qq´ ˆ σ hδ ` θ ˙ e p l ´ sgn p a qq ´ hL δ | e p l ` sgn p a qq ´ e p l ´ sgn p a qq| . (6.3)36e have | e p l ` sgn p a qq ´ e p l ´ sgn p a qq| ď ´ e ‹ , thus: S p k, l, w k,l , ∇ aδ w k,l , ∇ a ` ,δ w k,l , ∆ aδ w k,l , u q (6.4) ě B „ˆ ` µ hδ ` σ hδ ` θ ` hL δ ˙ e ‹ ´ µ hδ ´ σ hδ ´ θ ´ hL δ . It is thus enough to have ˆ ` µ hδ ` σ hδ ` θ ` hL δ ˙ e ‹ ´ µ hδ ´ σ hδ ´ θ ´ hL δ ě , (6.5)and one can easily check that this is the case with our choice of C p h, δ q .It remains to prove (3.40). Since ln p ` x q ą x ´ x for all x ą
0, we have, by (3.24): C p h, δ q ą δ ˜ µ hδ ` σ hδ ` θ ` hL δ ´
12 1 ` µ hδ ` σ hδ ` θ ` hL δ ˘ ¸ “ p µ ` L q δh ` σ h ` θδ ´ ` p µ ` L q h ` σ hδ ` δθ ˘ ě ` p µ ` L q M ` θM ˘ h ` σ h ´ ´ pp ` θ q µ ` L q h ` σ M ¯ ě ` p µ ` L q M ` θM ˘ h ` σ h ´ M σ . Proof. [Lemma 3.2] We show the result for S a, ` δ,(cid:15) , the proof is similar for S a, ´ δ,(cid:15) .For k P Z and 0 ď l ď N aδ , let z k,l : “ ϕ p x k , p a p p l qq ´ hF (cid:15) p t, x k , ϕ p x, p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq . (6.6)It is enough to show that, for all k P Z and 0 ă l ă N aδ , S p k, l, z k,l , ∇ aδ z k,l , ∇ a ` ,δ z k,l , ∆ aδ z k,l , ϕ q ě ´ C ϕ,n p h, (cid:15) q . Then, since f is non-increasing in its third variable, it is then easy to show that w ` “ z ` C ϕ,n p h, (cid:15) q satisfies (3.53).Let k P Z and 1 ď l ď N a ´
1. We have, by definition (3.17): S p k, l, z k,l , ∇ aδ z k,l , ∇ a ` ,δ z k,l , ∆ aδ z k,l , ϕ q“ h ´ p F p t, x k , z k,l , ∇ aδ z k,l , ∇ a ` ,δ z k,l , ∆ aδ z k,l q´ F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq ¯ , so it is enough to show p F p t, x k ,z k,l , ∇ aδ z k,l , ∇ a ` ,δ z k,l , ∆ aδ z k,l q´ F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq ě ´ C φ,n p h, (cid:15) q h .
37e split the sum into three terms: A “ p F p t, x k , z k,l , ∇ aδ z k,l , ∇ a ` ,δ z k,l , ∆ aδ z k,l q´ p F p t, x k , ϕ p x k , p a p p l qq , ∇ aδ ϕ p x k , p a p p l qq , ∇ a ` ,δ ϕ p x k , p a p p l qq , ∆ aδ ϕ p x k , p a p p l qqq ,B “ p F p t, x k , ϕ p x k , p a p p l qq , ∇ aδ ϕ p x k , p a p p l qq , ∇ a ` ,δ ϕ p x k , p a p p l qq , ∆ aδ ϕ p x k , p a p p l qqq´ F p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq ,C “ F p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq´ F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq . First, we have C “ f (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qqq ´ f p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qqqě ´| f (cid:15) ´ f | . Secondly, by (3.13)-(3.14), we have, B “ ´ θ δ h ∆ aδ ϕ p x k , p a p p l qq` µ p ∇ a p a,δ q ϕ p x k , p a p p l qq ´ ∇ a ` ,δ ϕ p x k , p a p p l qqq ` σ p ∆ a p a,δ q ϕ p x k , p a p p l qq ´ ∆ aδ ϕ p x k , p a p p l qqq` p f p t, x k , ϕ p x k , p a p p l qq , σ ∇ a p a,δ q ϕ p x k , p a p p l qqq ´ f p t, x k , ϕ p x k , p a p p l qq , σ ∇ aδ ϕ p x k , p a p p l qqqqě ´ θ δ h | ∆ aδ ϕ p x k , p a p p l qq|´ µ | ∇ a p a,δ q ϕ p x k , p a p p l qq ´ ∇ a ` ,δ ϕ p x k , p a p p l qqq ´ σ | ∆ a p a,δ q ϕ p x k , p a p p l qq ´ ∆ aδ ϕ p x k , p a p p l qq|´ σL | ∇ a p a,δ q ϕ p x k , p a p p l qq ´ ∇ aδ ϕ p x k , p a p p l qq| The first term goes to 0 since δ h Ñ h Ñ aδ ϕ p x k , p a p p l qq is bounded. Thelast three terms go to 0 by Taylor expansion and Lemma 3.1, since ϕ is smooth.Finally, by (3.13)-(3.14), using the linearity of the discrete differential operators and(6.6), and since f is Lipschitz-continuous, we have, A ě ´ hµ | ∇ a ` ,δ F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq|´ h p σ ` θ δ h q| ∆ aδ F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq|´ Lh | F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq|´ Lσh | ∇ aδ F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq| .
38e can show that each term goes to 0 as h Ñ
0. By example: h σ | ∆ aδ F (cid:15) p t, x k , ϕ p x k , p a p p l qq , ∇ a p a,δ q ϕ p x k , p a p p l qq , ∆ a p a,δ q ϕ p x k , p a p p l qqq|ě h σ µ | ∆ aδ ∇ a p a,δ q ϕ p x k , p a p p l qq|´ h σ | ∆ aδ ∆ a p a,δ q ϕ p x k , p a p p l qq|´ h σ | ∆ aδ f (cid:15) p t, x k , ϕ p x k , p a p p l qq , σ ∇ aδ ϕ p x k , p a p p l qqq| . The first two terms go to 0 with h since | ∆ aδ ∇ a p a,δ q ϕ p x k , p a p p l qq| and | ∆ aδ ∆ a p a,δ q ϕ p x k , p a p p l qq| are bounded, by smoothness of ϕ and by Lemma 3.1.We can control the derivatives of f (cid:15) : p x, p q ÞÑ f (cid:15) p t, x, ϕ p x, p q , σϕ p x, p qq with respect to (cid:15) : for any α “ p α , α q P N , we have | D α f (cid:15) | ď C ϕ,α (cid:15) α ` α , (6.7)for a constant C ϕ,α ą ´ h σ | ∆ aδ f (cid:15) p t, x k , ϕ p x k , p a p p l qq , σ ∇ aδ ϕ p x k , p a p p l qqq|ě ´ h σ |p ∆ aδ ´ ∆ a p a,δ q q f (cid:15) p t, x k , ϕ p x k , p a p p l qq , σ ∇ aδ ϕ p x k , p a p p l qqq|´ h σ | ∆ a p a,δ q f (cid:15) p t, x k , ϕ p x k , p a p p l qq , σ ∇ aδ ϕ p x k , p a p p l qqq|ě ´ C h σ δ (cid:15) ´ C h σ (cid:15) , where C , C ą
0, and this quantity goes to 0 by our choice of (cid:15) .Last, the smoothness of S a, ˘ δ,(cid:15) is straightforward by (3.52) and the control on its secondderivative with respect to p is obtained by (6.7). l For p t, x, y q P r , T s ˆ R d ˆ R ` , q : “ ˆ q x q p ˙ P R d ` and A : “ ˆ A xx A xp A xp J A pp ˙ P S d ` , A xx P S d , denoting Ξ : “ p t, x, y, q, A q , we define, recalling (1.4)-(1.5)-(1.6), F p Ξ q “ sup a P R F a p Ξ q with F a p Ξ q : “ ´ f p t, x, y, z p x, q, a qq ´ L p x, q, A, a q , where R Ă S z D with a finite number of elements.39 roposition 6.1. Let ď τ ă θ ď T and u (resp. u ) be a lower semi-continuoussuper-solution (resp. upper semi-continuous sub-solution) with polynomial growth, of ´B t ϕ ` F p t, x, ϕ, Dϕ, D ϕ q “ on r τ, θ q ˆ R d ˆ p , q (6.8) with u ě u on r τ, θ s ˆ R d ˆ t , u Ť t θ u ˆ R d ˆ r , s , then u ě u on r τ, θ s ˆ R d ˆ r , s . Corollary 6.1.
There exists a unique continuous solution w to (6.8) or equivalently sup η P R H η p t, x, ϕ, B t ϕ, Dϕ, D ϕ q “ on r τ, θ q ˆ R d ˆ p , q (6.9) satisfying w p¨q “ Ψ p¨q on r τ, θ s ˆ R d ˆ t , u Ť t θ u ˆ R d ˆ r , s , where Ψ P C Proof.
This is a direct application of the comparison principles. The equivalencebetween (6.8) and (6.9), comes from the fact that H η p Θ q and ´ b ´ F η p Ξ q have thesame sign. l Lemma 6.1. (i) Let a P R d and w a be the unique solution to ´B t ϕ ` F a p t, x, ϕ, Dϕ, D ϕ q “ on r τ, θ q ˆ R d ˆ p , q (6.10) satisfying w a p¨q “ Ψ p¨q on r τ, θ s ˆ R d ˆ t , u Ť t θ u ˆ R d ˆ r , s , where Ψ P C . Thenit admits the following probabilistic representation: w a p t, x, p q “ Y t , (6.11) where Y is the first component of the solution p Y, Z q to the following BSDE with randomterminal time Y ¨ “ Ψ p T , X t,x T , P t,p,a T q ` ż T ¨ f p s, X t,xs , Y s , Z s q d s ´ ż T ¨ Z s d W s , (6.12) with T : “ inf t s ě t : P t,p,as P t , uu ^ θ and P t,p,a ¨ “ p ` a p W ¨ ´ W t q X t,x ¨ “ x ` ş ¨ t µ p X t,xs q d s ` ş ¨ t σ p X t,xs q d W s . (ii) Assume moreover, that Ψ p T, ¨q “ φ p¨q and Ψ p¨ , q “ B p¨ , φ q , Ψ p¨ , q “ B p¨ , φ q ,with the notation of (2.14) . Then the solution p ˜ Y , ˜ Z q to Y ¨ “ φ p X t,xθ , ˜ P t,p,aθ q ` ż θ ¨ f p s, X t,xs , Y s , Z s q d s ´ ż θ ¨ Z s d W s , (6.13) where ˜ P t,p,a ¨ : “ P t,p,a ¨^ T , satisfies Y “ ˜ Y on r t, T s . roof. (i) The probabilistic representation is proved in [14]. Note that uniqueness tothe PDE comes from the previous lemma in the special case where R is reduced to oneelement.(ii) Let A : “ t T “ θ u , B : “ t T ă θ, P p,a T “ u , and C : “ t T ă θ, P p,a T “ u , so thatΩ “ A Y B Y C . For e P t , u , let p e Y t,x , e Z t,x q the solution to Y ¨ “ φ p X t,xθ , e q ` ż θ ¨ f p s, X t,xs , Y s , Z s q d s ´ ż θ ¨ Z s d W s . By (2.14), we have B e p τ, X t,xτ , φ q “ e Y τ for e P t , u .We introduce the following auxiliary processes, for s P r t, θ s ,ˇ Y s : “ Y s t ď s ď T ` Y s s ą T B ` Y s s ą T C , ˇ Z s : “ Z t t ď s ď T ` Z t s ą T B ` Z t s ą T C . First, by construction, we have Y “ ˇ Y on r t, T s . To prove the proposition it is thussufficient to show that ˜ Y “ ˇ Y on r t, θ s . To this effect, we show that p ˇ Y , ˇ Z q is solutionof (6.13).We have, for all s P r t, θ s ,ˇ Y s “ „ Ψ p T , X t,x T , P t,p,a T q ` ż T s f p u, X t,xu , Y u , Z u q d u ´ ż T s Z u d W u s ď T ` „ φ p X t,xθ , q ` ż θs f p u, X t,xu , Y u , Z u q d u ´ ż θs Z u d W u T ă s B ` „ φ p X t,xθ , q ` ż θs f p u, X t,xu , Y u , Z u q d u ´ ż θs Z u d W u T ă s C . By our hypotheses and by the definition of p Y ¨ , since ˜ P t,p,aθ “ B and ˜ P t,p,aθ “ C , we haveΨ p T , X t,x T , P t,p,a T q “ φ p X t,xθ , P t,p,aθ q A ` B p T , X t,x T , φ q B ` B p T , X t,x T , φ q C “ φ p X t,xθ , P t,p,aθ q A ` Y T B ` Y T C “ φ p X t,xθ , ˜ P t,p,aθ q` „ż θ T f p u, X t,xu , Y u , Z u q d u ´ ż θ T Z u d W u B ` „ż θ T f p u, X t,xu , Y u , Z u q d u ´ ż θ T Z u d W u C . A X t T ă s u “ H , we deduceˇ Y s “ φ p X t,xθ , ˜ P t,p,aθ q` „ż θ T f p u, X t,xu , Y u , Z u q d u s ď T B ` ż θ T f p u, X t,xu , Y u , Z u q d u s ď T C ` „ż T s f p u, X t,xu , Y u , Z u q d u s ď τ ` ż T s f p u, X t,xu , Y u , Z u q d u T ă s B ` ż T s f p u, X t,xu , Y u , Z u q d u T ă s C ´ „ż θ T Z u d W u s ď T B ` ż θ T Z u d W u s ď T C ´ „ż T s Z u d W u s ď τ ` ż T s Z u d W u T ă s B ` ż T s Z u d W u T ă s C . Now, since p Y u , Z u q “ p ˇ Y u , ˇ Z u q on p T , θ s X B and p Y u , Z u q “ p ˇ Y u , ˇ Z u q on p T , θ s X B ,and ş θ T f p u, X t,xu , ˇ Y u , ˇ Z u q d u A “
0, we get ż θ T f p u, X t,xu , Y u , Z u q d u s ď T B ` ż θ T f p u, X t,xu , Y u , Z u q d u s ď T C “ ż θ T f p u, X t,xu , ˇ Y u , ˇ Z u q d u s ď T . A similar analysis for the other terms shows that p ˇ Y , ˇ Z q is a solution to (6.13) andconcludes the proof. l S b We recall here the main results about the finite difference approximation defined by theoperator S b , see (3.21). Proposition 6.2 (Comparison theorem) . Let ď t ă s ď T, δ ą , h “ s ´ t such that (3.22) - (3.23) - (3.24) is satisfied.Let p u , u , v , v q P (cid:96) p δ Z q such that u k ď u k for all k P Z .1. For all k P Z , p v, ∇ , ∇ ` , ∆ q P R : S b p k, v, ∇ , ∇ ` , ∆ , u k q ď S b p k, v, ∇ , ∇ ` , ∆ , u k q .
2. Assume that, for all k P Z : S b p k, v k , ∇ δ v k , ∇ ` ,δ v k , ∆ δ v k , u k q ď , S b p k, v k , ∇ δ v k , ∇ ` ,δ v k , ∆ δ v k , u k q ě . Then v k ď v k for all k P Z . . Assume that, for all k P Z : S b p k, v k , ∇ δ v k , ∇ ` ,δ v k , ∆ δ v k , u k q “ , S b p k, v k , ∇ δ v k , ∇ ` ,δ v k , ∆ δ v k , u k q “ . Then v k ď v k for all k P Z . Proof.
The proof is similar to the proof of Proposition 3.2 and is ommited. l Proposition 6.3.
Let π, δ ą such that (3.22) - (3.23) - (3.24) is satisfied for all h “ t j ` ´ t j , j “ , . . . , κ ´ .Let V π,δ : π ˆ δ Z Ñ R the solution to: v κk “ g p x k q , k P Z ,S b p k, V jk , ∇ δ v jk , ∇ ` ,δ v jk , ∆ δ v jk , v j ` k q “ , k P Z , ď j ă κ. For all k P Z , let U π,δ q k : “ p V π,δ q k ` ´p V π,δ q k ´ δ . Then:1. p V π,δ , U π,δ q P (cid:96) p δ Z q and their bound is independant of π, δ .2. V π,δ converges uniformly on compacts sets to V , the super-replication price of thecontingent claim with payoff g . Proof.
We only show the first point, the second one is obtained by applying thearguments of [4], after proving monotonicity, stability and consistency following thesteps of Subsection 3.2.Since g is bounded, it is easy to show that V π,δ is also bounded independently of π, δ ,and the proof is similar to the proof of Proposition 3.6.Since g is Lipschitz-continuous, we get that U π,δ p T, ¨q is bounded. Using the Lipschitz-continuity of f , one deduces easily that U π,δ is a solution of u jk ´ u j ` k ´ h ˆ ´ µ ∇ ` ,δ u jk ´ p σ ` θ δ h q ∆ δ u jk ´ L ´ L | u jk | ´ L | ∇ δ u jk | ˙ ě , k P Z , ď j ă κ,u jk ´ u j ` k ´ h ˆ ´ µ ∇ ` ,δ u jk ´ p σ ` θ δ h q ∆ δ u jk ` L ` L | u jk | ` L | ∇ δ u jk | ˙ ď , k P Z , ď j ă κ,u κk “ g p x k ` q ´ g p x k ´ q δ P r´
L, L s , k P Z . Again, comparison theorems can be proved, and it is now enough to show that thereexists p u, u q P (cid:96) p π ˆ δ Z q which are bounded uniformly in π, δ such that u jk ´ u j ` k ´ h ˆ ´ µ ∇ ` ,δ u jk ´ p σ ` θ δ h q ∆ δ u jk ´ L ´ L | u jk | ´ L | ∇ δ u jk | ˙ ď , k P Z , ď j ă κ,u jk ´ u j ` k ´ h ˆ ´ µ ∇ ` ,δ u jk ´ p σ ` θ δ h q ∆ δ u jk ` L ` L | u jk | ` L | ∇ δ u jk | ˙ ě , k P Z , ď j ă κ,u κk ď ´ L, u κk ě L, k P Z . We deal with u only, we obtain similar results for u .One can easily show that u j : “ ´p L ` q ś κk “ j ` ´ h k L , where h k : “ t k ´ t k ´ , satisfiesthe requirements. Furthermore, one gets u j ě u ě ´ p L ` q T L , thus one gets that u is lower bounded by 1 ´ p L ` q T L . l cknowledgements This work was partially funded in the scope of the research project “Advanced tech-niques for non- linear pricing and risk management of derivatives” under the aegis ofthe Europlace Institute of Finance, with the support of AXA Research Fund.44 eferences [1] Yves Achdou, Guy Barles, Hitoshi Ishii, and Grigorii Lazarevich Litvinov.
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