Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Switching Problem

This paper deals with existence and uniqueness of a solution in viscosity sense, for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case is the Hamilton-Jacobi-Bellmann system of the Markovian stochastic optimal m-states switching problem. The switching cost functions depend on (t,x). The main tool is the notion of systems of reflected backward stochastic differential equations with oblique reflection.

We thank the two anonymous referees for their valuable comments which help us to improve the presentation of this paper.

Authors and Affiliations

1. LMM, Université du Maine, Avenue Olivier Messiaen, 72085, Le Mans, Cedex 9, France

   S. Hamadène & M. A. Morlais

Corresponding author

Correspondence to M. A. Morlais.

Appendix: Auxiliary Results on Systems of Reflected BSDES

1.1 5.1 The Multi-states Stochastic Optimal Switching Problem

1.1.1 5.1.1 Presentation of the Problem

Let \((\varOmega, \mathcal{F}, \mathbb{P})\) be a fixed probability space on which is defined a standard d-dimensional Brownian motion B=(B _t )_0≤t≤T whose natural filtration is \((\mathcal{F}_{t}^{0}:=\sigma\{B_{s}, s\leq t\})_{0\leq t\leq T}\). Let \(\mathbf{F}=(\mathcal{F}_{t})_{0\leq t\leq T}\) be the completed filtration of \((\mathcal{F}_{t}^{0})_{0\leq t\leq T}\) with all ℙ-null sets contained in \(\mathcal{F}_{0}\), hence \((\mathcal{F}_{t})_{0\leq t\leq T}\) satisfies the usual conditions, i.e., it is right continuous and complete. Furthermore, let:

• \(\mathcal{P}\) be the σ-algebra on [0,T]×Ω of F-progressively measurable sets;

• \(\mathcal{H}^{2,k}\) be the set of \(\mathcal{P}\)-measurable, ℝ^k-valued processes w=(w _t ) _t≤T such that \(\mathbb{E}[\int_{0}^{T}|w_{s}|^{2}\,ds]<\infty\);

• \(\mathcal{S}^{2}\) be the set of \(\mathcal{P}\)-measurable, continuous, ℝ-valued processes w=(w _t ) _t≤T such that \(\mathbb{E}[\sup_{t\leq T}|{w}_{t}|^{2}]<\infty\).

Problems of multi-states switching are usually encountered in the economic sphere (financial markets, energy, etc.). A strategy of switching is a pair (δ,ξ):=((τ _n ) _n≥1,(ξ _n ) _n≥1) such that:

1. (i)

   (τ _n ) _n≥1 is a non-decreasing sequence of F-stopping times (i.e. τ _n ≤τ _n+1 and τ ₀=0); τ _n is the nth time where the decision to switch is made;

2. (ii)

   (ξ _n ) _n≥0 are random variables with values in \(\mathcal{J}\) and for any n≥0, ξ _n is \(\mathcal{F}_{\tau_{n}}\)-measurable (ξ ₀ is the initial state which is assumed to be state 1).

A strategy of switching (δ,ξ):=((τ _n ) _n≥1,(ξ _n ) _n≥1) is called admissible if ℙ[τ _n <T,∀n≥0]=0 ; we denote by \(\mathcal{D}\) the set of admissible strategies. Next, with a given admissible strategy (δ,ξ), we associate a stochastic process (α _t ) _t≤T which is the indicator of the system at time t and which is given by:

(5.1)

We point out that when τ _n =T, the value of ξ _n is irrelevant since the horizon of the switching problem is already reached.

For \(i\in\mathcal{J}\), let (ψ _i (t,ω)) _t≤T be a process of \(\mathcal{H}^{2,1}\) which stands for the instantaneous profit when the system is in state i. Next for \(i\neq j \in\mathcal{J}\), let (g _ij (t,ω)) _t≤T be a process of \(\mathcal{S}^{2}\) which stands for the switching cost at time t from the current state i to another one j.

Finally when the admissible strategy (δ,ξ)=((τ _n ) _n≥1,(ξ _n ) _n≥1) is implemented, the expected total profit is given by:

In several works, the main objective is to find an optimal strategy, i.e, a strategy (δ ^∗,ξ ^∗) such that

(see e.g. [10, 19]) or at least to characterize (see e.g. [4, 15]) the optimal payoff, i.e., the right-hand side of (5.2).

1.1.2 5.1.2 Connection with Systems of Reflected BSDEs with Oblique Reflection

In order to tackle the switching problem described above, we usually relate it to systems of reflected BSDEs with oblique reflection which we introduce below in the case we need in order to deal with system of variational inequalities (2.3). So let (t,x)∈[0,T]×ℝ^k and \((X_{s}^{t,x})_{s\leq T}\) be the solution of the following stochastic differential equation:

Since b and σ verify (H1), the solution of this equation exists, is unique and satisfies:

Next let us introduce the solution of the system of reflected BSDEs with oblique reflection associated with the deterministic functions \(((f_{i})_{i\in \mathcal{J}},(g_{ij})_{i,j\in\mathcal{J}}, (h_{i})_{i\in\mathcal {J}})\). A solution consists of m triples of processes \(((Y^{i;t,x},Z^{i;t,x},K^{i;t,x}))_{i\in\mathcal{J}}\), denoted by \(((Y^{i},Z^{i},K^{i}))_{i\in\mathcal{J}} \), that satisfy: \(\forall i\in\mathcal{J}\)

We first provide an existence result of the solution of system (5.4) and some of its properties as well.

Proposition 5.1

Assume that:

1. (1)

   the functions \((f_{i})_{i\in\mathcal{J}}\) satisfy (H2)-(ii), (iii) and (iv);

2. (2)

   For any \(i,j\in\mathcal{J}\), the functions g _ij (resp. h _i ) verify (H3) (resp. (H4)).

Then the system (5.4) has a solution ((Y ⁱ,Z ⁱ,K ⁱ)) _i=1,m .

Proof

Since the above assumptions are not exactly the same as the ones of Theorem 3.2 in [18] then, for sake of completeness, we give its main steps. So let us consider the following standard BSDEs:

and

Thanks to the result by Pardoux-Peng [21], the solutions of both (5.5) and (5.6) exist and are unique. We next introduce the following sequences of BSDEs defined recursively by: for any \(i\in\mathcal{J}\), \(Y^{i,0}=\underbar{Y}\) and for n≥1 and s≤T,

By an induction argument and the result by El-Karoui et al. ([12], Theorem 5.2), we claim that the processes (Y ^i,n,Z ^i,n,K ^i,n) exist for any n≥1. Next using the comparison theorem of solutions of BSDEs (see e.g. Theorem 2.2 in [13]) we deduce that for any \(i\in\mathcal{J}\), Y ^i,0≤Y ^i,1. On the other hand, f _i satisfies the monotonicity property (H2)-(iv) and using once more the comparison of solutions of reflected BSDEs (see e.g. Theorem 4.1 in [12]) we obtain by induction that:

But the processes \(((\bar{Y}, \bar{Z},0))_{i\in\mathcal{J}}\) is a solution for the system of obliquely reflected BSDEs associated with \((([\max_{i=1,m} f_{i}](s,X^{t,x}_{s},y^{1},\dots,y^{m},z))_{i\in\mathcal{J}}, (\max_{i=1,m}h_{i}(X^{t,x}_{T}))_{i\in\mathcal{J} },(g_{ij}(s,X^{t,x}_{s}))_{i,j\in \mathcal{J}})\). Then an induction procedure and the repeated use of comparison theorem, which is justified in taking into account that f _i satisfies the monotonicity property (H2)-(iv), leads to

Using now Peng’s monotonic limit theorem (see Theorem 2.1 in [22]), we deduce that for any \(i\in\mathcal{J}\), there exist:

1. (i)

   a càdlàg (for right continuous with left limits) process Y ⁱ such that Y ^i,n↗Y ⁱ pointwisely;

2. (ii)

   a process Z ⁱ of \(\mathcal{H}^{2,d}\) such that, at least for a subsequence, (Z ^i,n) _n≥0 converges weakly to Z ⁱ in \(\mathcal{H}^{2,d}\) and strongly in L ^p(dt⊗dP) for any p∈[1,2[;

3. (iii)

   a càdlàg non decreasing process K ⁱ such that for any stopping time τ, \((K^{i,n}_{\tau})_{n\geq0}\) converges to \(K^{i}_{\tau}\) in \(\mathbb{L}^{p}(d\mathbb{P})\) for any p∈[1,2[.

The remaining of the proof is the same as the one of Theorem 3.2 in [18]. We therefore leave it to the care of the reader. □

We now give a result related to comparison of the solutions of system (5.4) constructed in the previous proposition. Its proof is rather easy since an induction argument allows to compare the solutions of the approximating schemes and then to deduce the same property for the limiting processes (see e.g. [18] for more details).

Remark 1

(i) Let \((f'_{i})_{i\in\mathcal{J}}\) (resp. \((g'_{ij})_{i,j\in\mathcal{J}}\), resp. \((h'_{i})_{i\in\mathcal{J}}\)) be functions that satisfy (H2)-(ii), (iii), (iv) (resp. (H3), resp. (H4)) and let \(((Y'^{i},Z'^{i},K'^{i}))_{i\in\mathcal{J}}\) be the solution of the system of reflected BSDEs associated with \(((f'_{i})_{i\in\mathcal{J}},(g'_{ij})_{i,j\in\mathcal{J}}, (h'_{i})_{i\in \mathcal{J}})\) constructed as in Proposition 5.1. If for any \(i,j\in\mathcal{J}\) we have:

then for any \(i\in\mathcal{J}\),

(ii) In case of uniqueness of the solutions of those systems, this result reduces to the comparison of the solutions.

We next focus on the regularity properties of the solution of system (5.4) constructed in Proposition 5.1.

Proposition 5.2

Assume that the assumptions of Proposition 5.1 are fulfilled. Then there exist lsc deterministic functions \((v^{i})_{i\in\mathcal{J}}\), defined on [0,T]×ℝ^k, ℝ-valued and belonging to Π ^g such that:

where \(((Y^{i},Z^{i},K^{i}))_{i\in\mathcal{J}}\) is the solution of (5.4) constructed in Proposition 5.1.

Proof

Under the hypotheses of Proposition 5.1, there exist deterministic continuous with polynomial growth functions \(\bar{v}(t,x)\) and \(\underline{v}(t,x)\) with values in ℝ such that for any \(s\in[t,T], \bar{Y}_{s}=\bar{v}(s,X^{t,x}_{s})\) and \(\underline{ Y}_{s}=\underline{v}(s,X^{t,x}_{s})\) ([13], Theorem 4.1).

Next by induction and thanks to the result by El-Karoui et al. ([12], p. 729), there exist deterministic continuous functions v ^i,n(t,x) in the class Π ^g such that for any \(i\in\mathcal{J}\) and n≥0,

where Y ^i,n is the solution of (5.7) (see Step 1, Proposition 5.1). As \(Y^{i,n}\leq Y^{i,n+1}\leq\bar{Y}\) then, for fixed i, the sequence (v ^i,n) _n≥0 is non-decreasing and such that \(v^{i,n}\leq\bar{v}\). Therefore it converges pointwisely to v ⁱ which is lower semi-continuous on [0,T]×ℝ^k, of polynomial growth since \(\underline{v} \leq v^{i}\leq\bar{v}\) and finally for any \(s\in[t,T], Y^{i}_{s}=v^{i}(s,X^{t,x}_{s})\). □

Remark 2

As, for each i in \(\mathcal{J}\), v ⁱ belongs to Π ^g, then classically (see e.g. [12]) one can show that for any \(i\in\mathcal{J}\), \(\|Z^{i}\|_{\mathcal {H}^{2,d}}(t,x)\) is also of polynomial growth.

We now provide a representation result for the solutions of system (5.4) and, as a by product, we obtain a uniqueness result in some specific cases. For later use, let us fix \(\overrightarrow{u}:=(u^{i})_{i=1,m}\) in \(\mathcal{H}^{2,m}\) and let us consider the following system of reflected BSDEs with oblique reflection: \(\forall i\in\mathcal{J}\), ∀s≤T,

Let s≤T be fixed, \(i\in\mathcal{J}\) and let \(\mathcal{D}^{i}_{s}\) be the following set of admissible strategies:

where \(A^{\alpha}_{r}\), r≤T, is the cumulative switching costs up to time r, i.e.,

Therefore and for any admissible strategy α we have:

Let us now consider a strategy \(\alpha=((\tau_{n})_{n\geq0},(\xi _{n})_{n\ge 0})\in\mathcal{D}^{i}_{s}\) and let \((P^{\alpha},Q^{\alpha}):=(P^{\alpha}_{s},Q^{\alpha}_{s})_{s\leq T}\) be the solution of the following BSDE (which is not of standard type):

with

In setting up \(\bar{P}^{\alpha}:=P^{\alpha}-A^{\alpha}\), we easily deduce the existence and uniqueness of the process (P ^α,Q ^α), since A ^α is adapted and \(E[(A^{a}_{T})^{2}]<\infty\), and the generator as well as the terminal value of the transformed BSDE are standard.

We then have the following representation for the solution of (5.9) which is the main relationship between the value function of the stochastic optimal switching problem and solutions of systems of reflected BSDEs with oblique reflection. This result usually referred as the verification theorem is not new and has been already shown in several contexts and under various assumptions.

Proposition 5.3

Assume that for any \(i,j\in\mathcal {J}\):

1. (i)

   f _i satisfies (H2)-(ii), (iii);

2. (ii)

   g _ij (resp. h _i ) satisfies (H3) (resp. (H4)).

Then the solution of system of BSDEs (5.9) exists and satisfies:

Thus the solution of (5.9) is unique.

Proof

Noting that the generator f _i in system (5.9) trivially satisfies (H2)-(iv) (since it does not depend on variable \(\overrightarrow{y}\)) and relying on Proposition 5.1, we may only consider all the other assumptions on the functions \((f_{i}, g_{ij},h_{i})_{i,j\in\mathcal{J}}\). Since Y ^u,i solves system (5.9) and following the strategy \(\alpha\in \mathcal{D}_{s}^{i}\) in (5.4), we obtain:

where

As \(\tilde{K}^{\alpha}_{T}\geq0\) then we have:

Note that the right-hand side in (5.12) is not a BSDE, therefore we will rather consider the equation satisfied by Y ^u,i−P ^α where the pair (P ^α,Q ^α) satisfies (5.10). Then using an equivalent change of probability we deduce the previous inequality.

Next let \(\alpha^{*}=(\tau_{n}^{*},\xi_{n}^{*})_{n\geq0}\) be the strategy defined recursively as follows: \(\tau_{0}^{*}=0\), \(\xi_{0}^{*}=i\) and for n≥0,

and

Let us show that \(\alpha^{*}\in\mathcal{D}^{i}_{s}\) and then let us first prove that \(P[\tau_{n}^{*}<T,\forall n\geq0]=0\). We proceed by contradiction assuming that \(P[\tau_{n}^{*}<T,\forall n\geq0]>0\). By definition of \(\tau_{n}^{*}\), we have:

As \(\mathcal{J}\) is finite then there is a state \(i_{0}\in\mathcal{J}\) and a loop i ₀,i ₁,…,i _k ,i ₀ of elements of \(\mathcal{J}\) where i ₁≠i ₀ and finally a subsequence (n _q ) _q≥0 such that:

Therefore defining \(\tau:=\lim_{q\rightarrow\infty}\tau_{n_{q}}^{*}\) and taking the limit w.r.t. q to obtain:

Then

which contradicts the no-loop property. We therefore have \(P[\tau _{n}^{*}<T,\forall n\geq0]=0\).

Next it only remains to prove that \(E[(A^{\alpha^{*}}_{T})^{2}]<\infty\) and α ^∗ is optimal in \(\mathcal{D}^{i}_{s}\) for the switching problem (5.11). Following the strategy α ^∗ and since \((Y^{u,i})_{i\in\mathcal{J}}\) solves the reflected BSDE (5.9), it yields: ∀n≥1,

noting that \(K^{u,\xi^{*}_{n}}_{r}-K^{u,\xi^{*}_{n}}_{\tau_{n}}=0 \mbox{ holds for any $r$, }\tau^{*}_{n} \le r \leq\tau^{*}_{n+1}\). Taking now the limit w.r.t. n in (5.13) to obtain:

But using the assumptions (H4) and (H2)-(ii),(iii) satisfied by h _i and f _i respectively and since \(\overrightarrow{u}\in\mathcal {H}^{2,m}\) and \(Z^{a^{*}}\in\mathcal{H}^{2,d}\) and \((Y^{i})_{i\in\mathcal{J}}\in (\mathcal{S}^{2})^{m}\), we deduce from (5.14) that \(E[(A^{\alpha^{*}}_{T})^{2}]<\infty\). It follows that \(\alpha^{*}\in\mathcal{D}_{s}^{i}\) and \(Y^{u,i}_{s}=P^{\alpha ^{*}}_{s}\), thus (5.11) holds and the solution of (5.9) is unique. □

Next for \(\overrightarrow{u}:=(u^{i})_{i=1,m}\in{\mathcal{H}}^{2,m}\) let us define

where ((Y ^u,i,Z ^u,i,K ^u,i)) _i=1,m is the solution of system (5.9) which exists and is unique under the assumptions of Proposition 5.3. Note that when the processes (Y ^u,i) _i=1,m exist they belong to \(({\mathcal{S}}^{2})^{m}\) and then Φ is a mapping from \({\mathcal{H}}^{2,m}\) to \({\mathcal {H}}^{2,m}\).

The following result, established by Chassagneux et al. [6], shows that Φ is a contraction in \({\mathcal{H}}^{2,m}\) when endowed with an appropriate equivalent norm.

Proposition 5.4

Assume that for any \(i,j\in \mathcal{J}\) the following hypotheses are in force:

1. (i)

   f _i verifies (H2)-(ii), (iii);

2. (ii)

   g _ij (resp. h _i ) verifies (H3) (resp. (H4)).

Then we have:

(a) For any \(\overrightarrow{u}=(u^{i})_{i=1,m},\,\overrightarrow {v}=(v^{i})_{i=1,m} \in {\mathcal{H}}^{2,m}\),

(5.16)

(b) There exists β ₀∈ℝ such that the mapping Φ is a contraction when \({\mathcal{H}}^{2,m}\) is endowed with the following equivalent norm:

Therefore Φ has a fixed point \((Y^{i})_{i\in\mathcal{J}}\) which belongs to \((\mathcal {S}^{2})^{m}\) and which provides a unique solution for system (5.4).

Proof

We provide only its main steps since it has been already given in [6]. For \(i\in\mathcal{J}\), \(\overrightarrow{u}\) and \(\overrightarrow{v}\in\mathcal{H}^{2,m}\) let us set

and let us consider the solution, denoted by \((\tilde{Y}^{i},\tilde{Z}^{i},\tilde {K}^{i})_{i\in\mathcal{J}}\), of the system of obliquely reflected BSDEs associated with \((\varphi_{i}(r,X^{t,x}_{r},z))_{i\in\mathcal{J}}\), \((h_{i})_{i\in\mathcal{J}}\) and \((g_{ij})_{i,j\in\mathcal{J}}\) which exists and is unique by Proposition 5.1. By Proposition 5.3, the following representation holds true:

where for \(a\in \mathcal{D}^{i}_{s}\) the pair of processes \((\tilde{P}^{a},\tilde{Q}^{a})\) verifies:

Additionally an optimal strategy \(\tilde{a}\) exists i.e. \(\tilde {Y}^{i}_{s}=\tilde{P}^{\tilde{a}}_{s}\). Note here that the dependence of \(P^{\tilde{a}}_{s}\) on i is made through the strategy \(\tilde{a}\) which belongs to \({\mathcal{D}}^{i}_{s}\). Now since for any r≤T and z∈ℝ^d, \(\varphi_{i}(r,X^{t,x}_{r},z)\geq f_{i}(r,X^{t,x}_{r}, \overrightarrow{u_{r}},z)\) and \(\varphi_{i}(r,X^{t,x}_{r},z)\geq f_{i}(r,X^{t,x}_{r}, \overrightarrow{v_{r}},z)\) then by comparison and uniqueness (see Remark 1) we have:

Next for \(a\in\mathcal{D}_{s}^{i}\), let \((P^{a}_{r},Q^{a}_{r})_{r\leq T}\) be the solution of the non-standard BSDE (5.10) and let \((P'^{a}_{r},Q'^{a}_{r})_{r\leq T}\) be the solution of the same non-standard BSDE with generator \(f_{a}(r,X^{t,x}_{r},\overrightarrow{v_{r}},z)\). Then we have:

which implies:

But for any η≤T we also have:

and a similar equation is valid for \(\tilde{P}^{\tilde{a}}_{\eta}-P'^{\tilde{a}}_{\eta}\). Using Itô’s formula with \(|\tilde {P}^{\tilde {a}}_{\eta}-P^{\tilde{a}}_{\eta}|^{2}\) and the inequality |x∨y−y|≤|x−y|, for any x,y∈ℝ, to obtain:

We now classically show the existence of a real constant C≥0 such that:

In the same way considering \(|\tilde{P}^{\tilde{a}}_{\eta}-P'^{\tilde{a}}_{\eta}|^{2}\) we obtain a similar inequality as (5.19) where \(P^{\tilde {a}}\) is replaced by \(P'^{\tilde{a}}\). Finally going back to (5.18), squaring and taking the expectation, we get (5.16).

Next in considering \(e^{\beta\eta}|\tilde{P}^{\tilde{a}}_{\eta}-P^{\tilde{a}}_{\eta}|^{2}\) (β>0) and arguing as previously we obtain a similar inequality as (5.16) where the term e ^βη appears. Finally it is enough to integrate w.r.t. η and to choose β=β ₀ appropriately in order to get that Φ is a contraction in the Banach space \(({\mathcal{H}}^{2,m},\|.\|_{\beta_{0}})\). Therefore it has a fixed point (Y ⁱ) _i=1,m which can be chosen continuous since \(\varPhi((Y^{i})_{i=1,m})\in(\mathcal{S}^{2})^{m}\). Thus the system of reflected BSDEs with interconnected obstacles (5.4) has a unique solution. □

Remark 3

Let \((Y^{i,0})_{i\in\mathcal{J}}\) be fixed processes of \({\mathcal {H}}^{2,m}\) and for n≥1 let us set \((Y^{i,n})_{i\in\mathcal{J}}=\varPhi ((Y^{i,n-1})_{i\in \mathcal{J}})\). Then the sequence \(((Y^{i,n})_{i\in\mathcal{J}})_{n\geq0}\) converges in \(({\mathcal{H}}^{2,m},\|.\|)\) to the unique solution of the system of reflected BSDEs associated with \(((f_{i})_{i\in\mathcal {J}},(g_{ij})_{i,j\in\mathcal{J}}, (h_{i})_{i\in\mathcal{J}})\) since Φ is a contraction in \(({\mathcal{H}}^{2,m},\|.\|_{\beta_{0}})\) and the norms \(\|.\|_{\beta_{0}}\) and ∥.∥ are equivalent.

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