Optimal reflection of diffusions and barrier options pricing under constraints

We introduce a new class of control problems in which the gain depends on the solution of a stochastic di(cid:27)erential equation re(cid:29)ected at the boundary of a bounded domain, along directions which are controlled by a bounded variation process. We provide a PDE characterization of the associated value function. This study is motivated by applications in mathematical (cid:28)nance where such equations are related to the pricing of barrier options under portfolio constraints.


Introduction
This paper is motivated by a previous work [1] where a new class of parabolic PDE with Neumann and Dirichlet conditions is introduced. Namely, [1] discusses the problem of super-hedging a barrier option under portfolio constraints and shows that, when there is no rebate, the super-hedging price is a viscosity solution of an equation of the form where O is an open domain of R d , E is some compact subset of R which depends on the constraints imposed on the portfolio, Lϕ = ∂ ∂t ϕ + 1 2 Tr σσ D 2 ϕ is the generator of the diffusion which models the evolution of the risky assets under the risk neutral probability measure, H e ϕ = δ(·, e)ϕ − γ(·, e), Dϕ , γ(x, e) is an (oblique) inward direction when x ∈ ∂O andĝ is a suitable function associated to the payoff function g of the option which, in a suitable sens, satisfies min e∈E H eĝ ≥ 0 andĝ ≥ g onŌ. See [1] for details and Section 4 below for an example. When the solution ϕ of the above equation is positive, it reduces to min e∈E H e ϕ = 0 on [0, T ) × ∂O, and, in particular cases, see [9] and [10], the constraint H e ϕ ≥ 0 at the parabolic boundary of  and W is a standard Brownian motion, recall that γ(x, e 0 ) is an inward direction for x ∈ ∂O, see e.g. [4]. Thus, the pricing of the barrier option is, at least formally, related to the expectation of a functional depending on the solution of a stochastic differential equation which is reflected at the boundary of O along the direction γ(x, e 0 ). This phenomenon was already observed in [9] in a particular setting and can be easily explained whenĝ ≥ 0 andĝ is non-decreasing on O, see Remark 4.4 below. By analogy, (1.2) should be associated to the control problem sup ∈E E e − T t δ(X (s), (s))dL (s)ĝ (X (T )) (1. 5) where (X , L ) is the solution on [t, T ] of X (s) = x + s t σ(X (r))dW (r) + s t γ(X (r), (r))dL (r) X (s) ∈Ō and L (s) = 2 and E is a suitable set of adapted processes with values in E. The difference with (1.3) is that the direction of reflection is now controlled by the process ∈ E.
This naturally leads to the introduction of a new class of control problems of the form (1.5), which, to the best of our knowledge, have not been studied so far.
In this paper, we first show that (1.6) admits a strong solution in the case where O is bounded, |γ| = 1 and (O, γ) satisfies a uniform exterior sphere condition: There is a huge literature on reflected SDEs and we refer to [5] for an overview of mains results. In the case where (X, ) is the solution of a SDE with Lipschitz coefficients, the existence of a strong solution under the exterior sphere condition (1.7) is easily deduced from [4]. Indeed, it suffices to consider the extended system (X, ) reflected at the boundary of O ×Ẽ for some open ballẼ = B(0,r) which contains the compact set E along a smooth directionγ such thatγ = (γ, 0) on O × E andγ = (γ, −e/r)/ √ 2 on O × ∂Ẽ. This system satisfies the exterior sphere condition of [4]. Since takes values in E, the reflection does not operate on this component and we therefore obtain existence of a solution to (1.6). However, this formulation is quite restrictive and we are interested by a more general class of controls. We therefore come back to the initial deterministic Skorokhod problem and follow the steps of [4] which are inspired by [7]. The existence to the Skorokhod problem with directions of reflection controlled by a continuous function with bounded variations is deduced from [4] by using the above arguments which consists in considering an extended system. Since the problem is deterministic and the reflection does not operate on , we can add jumps to this component without any difficulty. We then use suitable estimates on a family of test functions introduced in [3] to prove the existence of a solution to (1.6) in our general setting. Moreover, by considering SDEs with random coefficients, we are able to incorporate an other control on the direction which takes the form of an Itô process, see Section 2. We then introduce a control problem which generalizes (1.5) and prove that its value function is a viscosity solution of an equation of the form (1.2), for which we provide a comparison result. In the case where γ(x, e) does not depend on e, it essentially follows from the results of [3]. In our general setting, we need to introduce an additional condition which is satisfied whenever (1.2) admits a nonnegative subsolution and ρ is independent of x. These results are presented in Section 3. In the last section, we discuss the link between (1.5) and the pricing of barrier options under portfolio constraints. In a particular setting, we prove that (1.5) coincides with the super-hedging price of the option, when (1.2) admits a sufficiently smooth solution. This generalizes previous results of [9]. When E is reduced to a singleton, this leads to a natural Monte-Carlo approach for its estimation.
) the set of continuous maps ϕ from E 1 ×· · ·×E I into E that admit continuous (resp. bounded) derivatives up to order k i in their i-th component x i . We omit k i when it is equal to 0 and only write C k 1 (E 1 × · · · × E I , E) when k 1 = k 2 = . . . = k I . We omit E when E = R, and, in this case, we denote by D x i ϕ and D 2 x i ϕ the (partial) Jacobian and Hessian matrix with respect to x i . We simply write Dϕ and D 2 ϕ for D x 2 ϕ and D 2 x 2 ϕ if I = 2. For T > 0, we define BV([0, T ], E) as the set of cadlag maps from [0, T ] into E with a bounded total variation and a finite number of discontinuities. For We write E c for R m \E, ∂E andĒ denote the boundary and the closure of E, R m is the open ball centered on x with radius r, ·, · is the natural scalar product on R m . We denote by M m the set of square matrices of dimension m and we extend the definition of |·| to M m by identifying M m to R m×m . For x ∈ R m , diag [x] is the diagonal matrix of M m whose i-th diagonal element is x i , Tr [M ] is the trace of M ∈ M m . All inequalities between random variables have to be taken in the a.s. sens.

SDEs with controlled reflecting directions
The aim of this section is to construct a stochastic differential equation wich is reflected at the boundary of some bounded open set O ⊂ R d , d ≥ 1, along a direction which is controlled by an adapted cadlag process with bounded variations and a.s. a finite number of jumps taking values in a compact subset E of R , ≥ 1. We follow the arguments of [4] and start with the resolution of the associated (deterministic) Skorokhod problem. 4

The stochastic Skorokhod problem with controlled reflecting direction
We now consider some probability space (Ω, F, P) supporting a d-dimensional standard Brownian motion W . We denote by F = (F t ) t≤T the natural filtration induced by W , satisfying the usual conditions, and assume that F = F T . Given two uniformly Lipschitz functions µ and σ from R d into R d and M d respectively, it is shown in [4] that, under the condition (2.1), there exists a unique couple (X, L) of Fadapted continuous processes such that L is real valued, has bounded variations and (2.5) The aim of this section is to extend this result to the case where µ and σ are random, and γ is controlled by some bounded variation process with a.s. a finite number of jumps taking values in the compact set E.
In the following, given two subsets E 1 and E 2 of R m 1 and R m 2 , m 1 , m 2 ≥ 1, we denote by L F (E 1 , E 2 ) the set of measurable maps such that t → f t (·, x) is progressively measurable for each x ∈ E 1 , and for some K > 0 independent of (t, ω) ∈ [0, T ] × Ω. In the sequel, we shall only write f t (x) for f t (ω, x). We denote by BV F (E 2 ) the set of E 2 -valued nondecreasing cadlag adapted processes with bounded variations and P−a.s. a finite number of jumps. For ease of notations, we write E for BV F (E) and we denote by E b the set of elements of E whose total variation on [0, T ] is essentially bounded.
In the rest of this section, we fix (µ, σ) ∈ L F (R d , R d × M d ) and assume that the conditions (2.2) and (2.3) hold. Our first result extends Theorem 5.1 in [4].

Let X be an other continuous semimartingales with values inŌ and assume that
In order to prove Lemma 2.1, we shall appeal to the following technical result.
Proof. For e ∈ E, we can define the family (f ε (·, e)) ε>0 associated to γ(·, e) as in [3] and [4]. The bound on |D e f ε (x, y, e)| follows from the construction in the proof of Theorem 4.1 in [3], see in particular page 1136. The existence of h is deduced from [3] and [4] by increasing the dimension of the reflection problem as in 1. of the proof of Corollary 2.1. 2
Proof of Lemma 2.1. First observe that we can always assume that |Y − Y | ≤ δ where δ is defined as in Remark 2.1, for a given θ ∈ (0, 1). Indeed, we can always replace (X, X , Y, Y , L, L ) by (X, X , Y, Y , L, L )/η with η ≥ 1 such that B(0, ηδ/2) ⊃ O and change (µ, σ, γ) accordingly so that the equations in Lemma 2.1 holds for these new processes. From now on, we therefore assume that |Y − Y | ≤ δ.
Recall the definitions of h and f ε for θ defined as above. We fix ε, λ > 0 and define the smooth functionf ε onŌ ×Ō × E bỹ , (t))) t≤T and following the arguments of the proof of Theorem 5.1 in [4], we obtain that where C λ , C are two positive constant such that the second one does not depend on λ, and where c stands for the continuous part. Using the bounds on f ε and D e f ε of Lemma 2.2, we observe that A t + B t ≤ 0 forK large enough with respect to K and λ. Since where C is a positive constant. The required result is then obtained by sending ε → 0 and using Doob's inequality and Gronwall's Lemma. 2 We can now provide the main result of this section, which ensures the strong existence and uniqueness of a SDE with random coefficients and controlled reflecting directions.
Then, there exists a unique continuous adapted process (X, L) such that L ∈ BV F (R + ) and Proof. Observe that Lemma 4.7 in [4] can be easily extended to our setting by appealing to the arguments already used in the proof of Corollary 2.1. The existence when | |(T ) is uniformly bounded then follows from Corollary 2.1, Lemma 2.1 and the same arguments as in [4], see the discussion after their Corollary 5.2, or as in the proof of Proposition 4.1 in [7]. In the case where | |(T ) is not uniformly bounded, we use a localization argument. For each n ≥ 1, we define τ n := inf{s ≥ t : | | |(s) | ≥ n} and let (X n , L n ) be the unique solution of (2.7) associated to n (·) := (· ∧ τ n ). We then define (X, L) by with the convention τ 0 = 0. It solves (2.7) associated to . The same argument provides uniqueness. 2 takes values in a compact set F of R . Then, it follows from Theorem 2.2 that existence and uniqueness holds for for some r ∈ (0, 1). This is easily checked by arguing as in the proof of Corollary 2.1. This allows us to introduce a new control on the direction of reflection which corresponds to an Itô process.

Optimal control
As in the previous section, we consider a bounded open set O ⊂ R d and γ ∈ C 2 (R d+ , R d ) such that |γ| = 1 and (2.3) holds.

Definitions and assumptions
We fix a compact subset A of R and denote by A the set of predictable processes with values in A.
Let µ and σ be two continuous maps on R d × A × E with values in R d and M d respectively. We assume that both are Lipschitz with respect to their first variable uniformly in the two other ones, so that (µ α, , σ α, ) defined by The aim of this section is to provide a PDE characterization for the control problem t,x (r) , and ρ, g, f are continuous real valued maps onŌ × E,Ō andŌ × A × E respectively. In order to ensure that J is well defined, we assume that ρ ≥ 0. We also assume that, as a function onŌ × E, ρ is C 1 with Lipschitz first derivative in its first variable, uniformly in the second one, and Lipschitz in its second variable, uniformly in the first one.

Dynamic programming
We first provide some useful estimates on X α, t,x and J which will be used to derive the dynamic programming principle of Lemma 3.2 below.
Proof. We write (X, L, β) and (X , L , β ) for (X α, t,x , L α, t,x , β α, t,x ) and (X α, t ,x , L α, t ,x , β α, t ,x ). It follows from Lemma 2.1 that where C > 0 denotes a generic constant independent of (t, t , x, x ). Choosing some largeK > 0, applying Itô's Lemma to (e −K| |(t)f ε (X(t), y, (t))) t≤T , y ∈Ō andf ε defined as in (2.6), and using the same arguments as in Lemma 2.1 leads to This proves (3.2) and (3.3). We now prove (3.4). Recalling that γ ∈ C 2 (R d+ , R d ) and |γ| = 1, we deduce from Itô's Lemma applied to X, γ(x, ) − x, γ(x, ) and Cauchy-Schwartz inequality which in view of (3.6) and Cauchy-Schwartz inequality implies that This proves (3.4). We finally prove (3.5). We first assume that ρ ∈ C 2,1 (R d+ , R) and apply Itô's Lemma to X − X , γ(X, )ρ(X, ) on [t , T ]. Using the above estimates, we obtain where C depends on ρ only through the bounds on |ρ|, on the first and second derivatives in its first variable and on the first derivative in its second variable. In view of the previous estimates and Cauchy-Schwartz inequality, the result follows for ρ smooth enough. Since the estimate of (3.4) clearly does not depend on ρ, this result is easily extended to the general case by a standard approximation argument. We can now prove the following dynamic programming principle.
follows from the Markov feature of our model. We now prove the converse inequality.

Let ϕ be a continuous map on
Let (B n ) n≥1 be a partition of [0, T ] ×Ō and (t n , x n ) n≥1 be a sequence such that (t n , x n ) ∈ B n for each n ≥ 1. It follows from Lemma 3.1 that, for each n ≥ 1, we can find ξ n := (α n , n ) ∈ A × E b such that where ε > 0 is a fix parameter. Moreover, by continuity of ϕ and J(·, ξ) for ξ ∈ A × E b , see Lemma 3.1, we can choose (B n , t n , x n ) n≥1 in such a way that By arbitrariness of ε > 0, this shows that 3. By replacing ϕ by a sequence (ϕ k ) k≥1 of continuous functions satisfying we deduce from (3.10) and the dominated convergence Theorem that, for all ξ ∈ Using the lower semi-continuity of v and the same localization argument as in the proof of Theorem 2.2 shows that the above inequality actually holds for all ξ ∈ A×E.

PDE characterization for the optimal control problem
In this section, we show that v is a solution of where σ is the transposed matrix associated to σ.

Definitions
Since v may not be smooth, we need to consider the above equation in the viscosity sens. Moreover, the boundary conditions may not be satisfied in a strong sens and, as usual, we have to consider a relaxed version, see e.g. [2]. We therefore introduce the operator K + and K − defined as

Definition 3.1 We say that a lower-semicontinuous (resp. upper-semicontinuous) function w on [0, T ] ×Ō is a viscosity supersolution (resp. subsolution) of
We say that a locally bounded function w is a (discontinuous) viscosity solution of (3.11) if w * (resp. w * ) is a supersolution (resp. subsolution) of (3.11) where means that Dϕ/ϕ ∈ K, see e.g. [8]. In this case, the term H e ϕ ≥ 0 can be assimilated to a constraint on the gradient of the logarithm of the solution at the boundary of O. A similar constraint appears in [1], but in the whole domain.

Super and subsolution properties
Proposition 3.2 The function v * is a viscosity supersolution of (3.11).
Proof. The fact that v * ≥ g on {T } ×Ō is a direct consequence of Proposition 3.1 and the continuity of g. Fix (t 0 , x 0 ) ∈ [0, T ) ×Ō and ϕ ∈ C 1,2 ([0, T ] ×Ō) such that and work toward a contradiction. Under the above assumption, we can find (a 0 , e 0 ) ∈ A × E and δ > t 0 for which Observe that we can assume, without loss of generality, that (t 0 , x 0 ) achieves a strict local minimum so that It then follows from Itô's Lemma, (3.13) and (3.14) that where we used the fact that β k (θ k ) = 1 on {θ k < ϑ k }. Let c > 0 be such that |ρ| ≤ c onŌ × E and observe that which leads to a contradiction to Lemma 3.2 for k large enough, recall (3.15).

2.
The case where (t 0 , x 0 ) ∈ D is treated similarly. It suffices to take δ small enough so that B(x 0 , δ) ⊂ O and therefore θ k < ϑ k . 2

Proposition 3.3
The function v * is a viscosity subsolution of (3.11).
The case where (t 0 , x 0 ) ∈ [0, T ) ×Ō is treated by similar arguments as in the proof of Proposition 3.2, see also below. We therefore assume that t 0 = T .

1.
We first consider the case where x 0 ∈ ∂O. We assume that min (a,e)∈A×E min {ϕ − g , H e ϕ} := 2ε > 0 . Set x) → −∞ as t → T and observe that (T, x 0 ) also achieves a maximum for v * − φ. Without loss of generality, we can therefore assume that (∂/∂t)ϕ(t, x) → −∞ as t → T and that we can find Observe that we can assume, without loss of generality, that (t 0 , x 0 ) achieves a strict local maximum so that It follows from Itô's Lemma, (3.16), (3.17) and the identity v(T, Arguing as in 1. of the proof of Proposition 3.2, this implies that for some ν > 0. By arbitrariness of (α, ) and (3.18), this leads to a contradiction to Lemma 3.2 for k large enough.

2.
The case where x 0 ∈ O is treated similarly, it suffices to take δ small enough so that B(x 0 , δ) ⊂ O and therefore θ k < ϑ k . 2

Remark 3.5
The smoothness assumptions on ρ and γ are only used either to construct (X α, t,x , L α, t,x ) or to prove the dynamic programming principle of Lemma 3.2. We shall see through an example in Section 4.3 below how they can be relaxed.

Application to the pricing of barrier options under constraints
As already stated in the introduction, the main motivation comes from applications in mathematical finance. More precisely, [1] provides a PDE characterization of the super-hedging price of barrier options under portfolio constraints which is very similar to the equation Kϕ = 0 up to an additional term inside the domain O which imposes a constraint on the gradient of the logarithm of the solution. The aim of this section is to show that the super-hedging price of barrier options under portfolio constraints can actually admit a dual formulation in terms of an optimal control problem for a reflected diffusion in which the direction of reflection is controlled. Due to the additional term which appears in the PDE of [1], we can not expect this result to be general and we shall restrict to a Black and Scholes type model, see below.
In order to simplify the presentation, we shall work under quite restrictive conditions, assuming for instance that the equation Kϕ = 0 admits a sufficiently smooth solution for a suitable choice of parameters. The general case is left for further research.

Problem formulation
We briefly present the hedging problem. Details can be found in [1] and the references contained in this paper.
We consider a financial market which consists of one non-risky asset, whose price process is normalized to unity, and d risky assets where Σ is a d-dimensionnal invertible matrix. A financial strategy is described by a d-dimensional predictable process π = (π 1 ,...,π d ) satisfying the integrability condition where π i (s) is the proportion of wealth invested at time s in the risky asset S i t,x . To an initial capital y ∈ R and a financial strategy π, we associate the induced wealth process Y π t,y which solves on [t, T ] In this paper, we restrict to the case where the proportion invested in the risky asset are constrained to be bounded from below. Given m i > 0, i ≤ d, we set and denote by Π K the set of financial strategies π satisfying π ∈ K dt × dP − a.e. We consider an up-and-out type option. More precisely, we take O such that The "pay-off" of the barrier option is a continuous map g defined on R d + satisfying In order to apply the general results of [1], we assume that the mapĝ defined bŷ Here, δ is the support function of K, see Remark 3.1. We also assume thatĝ is almost everywhere differentiable onŌ * and we denote by Dĝ its gradient, when it is well defined. The super-replication cost of the barrier option is then defined as the minimal initial dotation y such that Y π t,y (T ) ≥ g(S t,x (T ))1 T <τt,x for some suitable strategy π ∈ Π K . This leads to the introduction of the value function defined on [0, T ] ×Ō * by w(t, x) := inf y ∈ R : Y π t,y (T ) ≥ g(S t,x (T ))1 T <τt,x for some π ∈ Π K . (4.5)

Theorem 4.1 ([1]) The value function w is the unique viscosity solution in the class of bounded functions on
In the above theorem, the notion of viscosity solution has to be taken in the classical sens.
When the equation (4.6)-(4.7)-(4.8) below admits a sufficiently smooth solution, the above equation can be simplified as follows.

Proposition 4.1 Assume that there is a bounded non
ψ =ĝ on {T } ×Ō * (4.8) Proof. In view of Theorem 4.1, it suffices to show that ψ is a solution of Gϕ = 0.
With this construction, we can now consider the control problem v(t, x) := sup where E n := BV F (E n ). It follows from the previous discussion that we can apply Lemma 3.2 to v n . Since, v = sup n≥1 v n = lim n→∞ ↑ v n , a monotone convergence argument shows that the dynamic programming principle of Lemma 3.2 holds for v. Following the arguments used in Proposition 3. where ρ(x, e) = δ(e)/|diag [x] e|, τ ϑ t,x is the first exit time of S ϑ t,x from O * and we use the convention 0/0 = 0.
Sinceĝ ≥ 0, we should seek for a control ϑ such that τ ϑ t,x > T , i.e. which "causes reflection" of S ϑ at the boundary ∂O * . Moreover, the "reflection" should be optimal so that the right hand-side of (4.13) is maximal. If d = 1 andĝ is non-decreasing on O * , the action of ϑ should be minimal since it decreases the value of S ϑ t,x (T ) and ρ(x, e) > 0 if e = 0. This phenomenon, which was already observed in [9] in the one dimensional case, naturally leads to the formulation (4.12).