## Abstract

We consider an evolution non-local free boundary problem that arises in the modelling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show that the solution is convex in space, and establish several monotonicity properties of the solution and of the free boundary with respect to parameters of the problem. To study the free boundary, we use, in particular, the fact that the odd part of the solution solves a more standard obstacle problem. We show that the free boundary is and describe the asymptotics of the free boundary as *c*, the cost of transacting the asset, goes to zero.

## 1. Introduction

### (a) Main results

The goal of this paper is to study an evolution problem that arises in a model of bubbles that result from volatile differences in beliefs among speculators in a financial market. This financial model is briefly presented in §1*c*, and a more precise derivation can be found in [1]. The stationary version of this model (i.e. for infinite horizon) was introduced and solved by Scheinkman & Xiong [2]. The article [2] uses a different approach from the one presented here and provides an explicit stationary solution, based on Kummer functions. Chen & Kohn [3,4] study a stationary model that is related to the one in [2], and construct an explicit solution in terms of Weber–Hermite functions. A natural motivation for the evolution problem treated in this paper is that a finite-horizon model is necessary to deal with finite-horizon assets, such as many fixed-income securities. As will be seen, this leads to more involved mathematical problems.

In what follows, we let
1.1
denote given constants. The parameter *σ* is the volatility, *r* is the rate of interest, *c* represents the transaction cost, and *ρ* and λ are relaxation parameters. We define the following parabolic operator (possibly degenerate, when *σ*=0)
and the obstacle

We consider the following (non-local) obstacle problem
1.2
In the economic interpretation, the (indeed non-negative) quantity *u* can be seen as the speculative component of the price of an asset, owing to disagreement among investors. The larger is *u*, the larger is the financial bubble.

We also introduce the stationary problem (formally for ) with 1.3 This is the problem that was studied in [2]. This paper deals with resolution and qualitative properties of problems (1.2) and (1.3). We establish here rigorous results in the framework of viscosity solutions (see 5 for a general reference). A precise definition of viscosity solutions in our framework is given in §2.

Our first main result is the following

### Theorem 1.1 (Existence and uniqueness of a solution)

*Assume (1.1).*

(i) Evolution equation.

*There exists a unique viscosity solution u of (1.2) satisfying*(ii) Stationary equation.

*Moreover, there exists a unique viscosity solution**of (1.3) satisfying*

It is easy to see that if *σ*=0, then and . In the general case, we have inequalities as only in the next result. We also list a series of qualitative properties such as monotonicity, convexity, asymptotics and large time behaviour that are related to the economic motivation of the problem. A precise derivation of the model from assumptions on the behaviour of investors, as well as a discussion of the economic significance of these qualitative properties will be provided in our forthcoming work [1].

### Theorem 1.2 (Properties of the solution)

*Assume (1.1) and let u be the solution given in theorem 1.1. Then, u is continuous. In addition,*

(i) Asymptotics.

*There exists a function ϕ such that*1.4(ii) Monotonicity and convexity

^{1}:*u*_{t}≥0, 0≤*u*_{x}≤*α*(*t*),*u*_{xx}≥0.(iii) Convergence in long time: as

*locally uniformly in x*.(iv) Monotonicity with respect to the parameters

*r*,*c*,λ,*σ*.*The following properties hold for r,c*>0,*r*+λ>0,*σ*≥0: ∂*u*/∂*c*≤0, ∂*u*/∂*r*≤0, ∂*u*/∂λ≤0 and ∂*u*/∂*σ*≥0.(v) The limit

*c*→0:*when c*→0,*u*→*u*_{0},*where u*_{0}*is the minimal solution of (1.2) for c*=0*satisfying*on*for some constant C*>0.(vi) The w-problem.

*Set*1.5*Then, w is a viscosity solution*: 1.6^{2}

As we see (proposition 10.1), for *c*=0, the solutions of (1.2) are not unique. This is why the limit *u*_{0} of solutions *u* as *c*→0 is characterized only as the minimal solution.

We also show that *w* defined in (1.5) satisfies properties similar to those in theorem 1.2. They are stated in §7. Clearly, problem (1.6) is a free boundary problem where the exercise region is defined as the set {*w*=*ψ*}. We now make this precise and list some properties.

### Theorem 1.3 (Properties of the free boundary)

*Assume (1.1) and let u be the solution given in theorem 1.1. There exists a lower semi-continuous function* *such that for all t>0:*
*The following properties hold:*

(i) Bounds on the free boundary. For

*σ*≥0,*we have*1.7(ii) Lipschitz regularity of the free boundary.

*The lower semi-continuous function a satisfies**Moreover, if ρ*≥λ,*then**and the function a is non-increasing: a*′(*t*)≤0.(iii) Monotonicity with respect to the parameters ρ,c,r,λ,σ. The following properties hold ∂

*a*/∂*ρ*≤0, ∂*a*/∂*c*≥0 and ∂*a*/∂*σ*≥0.*Moreover, if ρ*≥λ,*then*1.8(iv) Convergence of the rescaled free boundary when c→0.

*Assume that σ*>0*and*λ≤3*r*+4*ρ. Then, the following convergence of the rescaled free boundary holds true when c*→0:*where*1.9

### Remark 1.4

In the models of equilibrium asset-pricing derived in [2] or [1] starting from assumptions on the the behaviour of investors, the condition *ρ*≥λ is always satisfied. Note that the expression of in (1.9) shows that for *c*≪1, the free boundary *a*(*t*) cannot be non-increasing in time when *ρ*<λ. Therefore, the argument proving that *a*(*t*) is non-increasing in time when *ρ*≥λ is optimal. Similarly, it is possible to see from (1.9) that the monotonicity results in (1.8) does not hold for *ρ*<λ and *c*≪1.

### (b) Comments on an alternative approach

As we have seen, the non-local problem we study here, (1.2), is closely related to a somewhat classical local obstacle problem (1.6). This problem is not straightforward either. Indeed, it is set on the whole real line, and it is seen that the free boundary starts from infinity at *t*=0. Nevertheless, it is tempting to approach the non-local *u*-problem by first solving the local *w*-problem (1.6). As a matter of fact, to derive further qualitative properties, we study this *w*-problem in §7. However, the *w*-problem does not yield the solution of the *u*-problem that is of interest in a straightforward fashion. By solving the *w*-problem, we, indeed, obtain the free boundary, but we then need to show that it is of the form {*x*=*a*(*t*)}=∂{*w*>*ψ*}. We further need to recover *u* from *w*, and this does not follow immediately from the local obstacle problem. Indeed, we have to solve the equation for *u* in the domain {*x*<*a*(*t*)} with *a*(*t*)>0, and this equation reads
1.10

Equation (1.10) is also non-local because of the boundary condition. One way to solve the *u*-problem then is to rewrite problem (1.10) with the coordinates *y*=*x*−*a*(*t*). For this, we need to first prove regularity of the free boundary *a*(*t*), what is not known in general. But, we actually derive such a property here, for *ρ*≥λ. Then, we could solve this problem by using a fixed point procedure. Furthermore, to reconstruct *u* from *w* in the region *x*>*a*(*t*), we can use the obstacle condition *u*(*x*,*t*)−*u*(−*x*,*t*)=*ψ*(*x*,*t*) for *x*>*a*(*t*).

However, even if we succeed with this procedure, the best we can obtain is the existence of one solution *u* to the *u*-problem. It does not solve the question of uniqueness of the solution (and more generally, the question of the comparison principle). In particular, if equality were to hold in the obstacle condition for some *x*<0, that is *u*(*x*,*t*)−*u*(−*x*,*t*)=*ψ*(*x*,*t*), then *w*(*x*,*t*)=*u*(*x*,*t*)−*u*(−*x*,*t*) would not satisfy the local obstacle problem (1.6). Rather, in this case, at least formally, it would satisfy a double obstacle problem with *ψ*(*x*,*t*)≤*w*(*x*,*t*)≤−*ψ*(−*x*,*t*). Such a situation therefore has to be ruled out.

Lastly, our aim here is to establish several qualitative properties of the solution *u* related to the economic motivation of the problem (see [1]). We also derive some properties of *w*, but the properties for *u* do not follow immediately from *w*. Our direct approach of proving a comparison principle for the *u*-problem, in the framework of viscosity solutions, allows us to prove the properties of *u* (uniqueness, comparison, convexity in *x*, monotonicity in *t*) that are of interest.

### (c) A brief description of the economic model

We refer the reader to [1,2] for a detailed derivation of the model, starting with postulates on the behaviour of investors. Here, we present a self-contained and heuristic introduction to the evolution model.

We consider a market with a single risky asset, which provides dividends up to a maturity *T*>0. There are two groups of investors *A* and *B*, who disagree about the future evolution of the cumulative dividends *D*_{t}. Under the belief of investors in group *C*∈{*A*,*B*}, the process of dividends is given by the following pair of diffusions
1.11
and
1.12
where for each *C*∈{*A*,*B*}, *W*^{C,D} and are Brownian motions (under *C*′s beliefs) that are possibly contemporaneously correlated, λ≥0 is a rate of mean reversion. When investors in group *A* are relatively optimistic about the future growth of dividends.

To complete the model, we need to consider the views that investors in group *C*∈{*A*,*B*} have of the evolution of beliefs of the investors in the complementary group. We write the complementary group of investors (i.e. if *C*=*A*, and if *C*=*B*), and We assume that from the viewpoint of agents in group *C*, *g*^{C} satisfies
1.13
where *ρ*>λ, *σ*>0, and *W*^{g} is a Brownian from the point of view of *both* groups of investors, and the future (past) increments to *W* are independent of the past (future) values of *W*^{C,D} and for *C*∈{*A*,*B*}. Assuming that investors agree on the evolution of differences in beliefs amounts to assuming that investors in each group know the model used by the other group and agree to disagree.

The model developed in [2] postulates a particular information structure and derives equations (1.11)–(1.13) using results on optimal filtering (see also [6]).

All investors are *risk-neutral*—that is they value payoffs according to their expected value—and discount the future at a continuously compounded rate *r*>0. Short-sales are not allowed, that is every investor must hold a non-negative amount of the asset. We assume that the supply of the asset is finite and that each group of investors is large. Competition guarantees that buyers must pay their *reservation price*; the maximum price they are willing to pay.

Write for the price that investors in *C* are willing to pay for the asset at *t*. Assets are traded *ex-dividend*, that is a buyer of the asset at time *s* gains the right to the flow of dividends after time *s*. Because there are no dividends after time *T*, We assume that there is a cost *c*>0 per unit for any transaction. We also assume that if an investor holds the asset to *T*, he can dispose of the (worthless) asset for free. Note that because transaction costs are positive, every transaction must involve a seller in a group and a buyer in the complementary group which values the asset more.

Write for the expected value calculated using the beliefs of agents in group *C*. Then,
1.14
The first term represents the discounted payoff of a sale at time *τ*; the second term the discounted cumulative dividends over the period (*t*,*τ*]. The price is computed by maximizing the expected value of the buyer over random selling times.

Given the assumptions concerning the laws of motion (1.11)–(1.13), one can rigorously show that there is a solution to (1.14) given by Furthermore, the function satisfies 1.15 (See [2] or [1] for a derivation.)

Note that equation (1.15) is similar to the equation for an American option, except that the exercise price is related to the value of the option. Standard dynamic programming arguments suggest that if *q* solves (1.15) then *u*(⋅,*t*):=*q*(⋅,*T*−*t*) satisfies (1.2). In [1], we establish that if *q* solves (1.15), then *u* is a viscosity solution to (1.2).

The quantity is the amount that an investor in group *C* is willing to pay for the asset, in addition to her valuation of future dividends. This amount reflects the option value of resale and is a result of *fluctuating* differences in beliefs among investors. Because a buyer of the asset is a member of the most optimistic group, the amount by which the purchase price exceeds his valuation, *q*, can be legitimately called a *bubble*.

### (d) Organization of the paper

In §2, we recall the definition of viscosity solutions and the stability properties of these solutions for the evolution problem, the stationary problem and the *w*-problem. In §3, we prove the comparison principle for the *u*-problem. Section 4 is devoted to the proof of theorem 1.1 which states existence and uniqueness of the solution *u*. In §5, we prove some properties of the solution *u*, and we establish further properties of *u* in §6, by introducing a modified problem (problem (6.3)) which allows us to show that *w* solves an obstacle problem. As a consequence, we give the proof of theorem 1.2 at the beginning of §6. In §7, we study the *w*-problem, following the lines of proof used previously for the *u*-problem. In §8, we establish a Lipschitz estimate for the free boundary. We study the asymptotics of the free boundary in the limit *c*→0 in §9. As a consequence, we obtain the proof of theorem 1.3. In §10, we show that the comparison principle does not hold for *c*=0 (and *σ*>0).

To shorten the paper, we consigned to an electronic supplementary material, appendix some additional material. In §A.1 of the electronic supplementary material appendix, we give precise definitions of viscosity solutions for equations (1.3) and (1.6). In §A.2, we provide a more elaborate statement, lemma A.3, and a proof of the Jensen–Ishii lemma for our obstacle problem. We show in §A.3 that the antisymmetric part of *u* is a viscosity solution to the *w*-problem. Section A.4 establishes a comparison principle for the *w* problem, and in §A.5, we construct subsolutions and supersolutions for the *w*-problem. Sections A.6 and A.7 contain proofs of the convexity and monotonicity properties of solutions to the *w*-problem, as well as the proof of corollary 7.4. We complete the proof of our claim that the free boundary is in §A.8. This is an adaptation of a proof in [7], and we actually provide an argument for a more general problem, because this result may be of interest in other applications. The last section of electronic supplementary material, appendix provides the proof for lemma 9.4, which is used to establish the asymptotics of the free boundary.

## 2. Definition of viscosity solutions

### (a) Viscosity solutions for the *u*-problem

#### (i) The evolution problem

### Definition 2.1 (Viscosity sub/super/solution of equation (1.2)).

Let .

(i)

*Viscosity sub/supersolution on*. A function is a viscosity subsolution (resp. supersolution) of (1.2) on , (i.e. of the first equation in (1.2)), if*u*is upper semi-continuous (resp. lower semi-continuous), and if for any function and any point such that*u*(*P*_{0})=*φ*(*P*_{0}) and then(ii)

*Viscosity sub/supersolution on*. A function is a viscosity subsolution (resp. supersolution) of (1.2) on , (i.e. of the initial value problem), if*u*is a viscosity subsolution (resp. supersolution) of (1.2) on and satisfies moreover*u*(*x*,0)≤0 (resp.*u*(*x*,0)≥0) for all .(iii)

*Viscosity solution on*. A function is a viscosity solution of (1.2) on , if and only if*u** is a viscosity subsolution and*u*_{*}is a viscosity supersolution on where^{3}

The notion of discontinuous viscosity solution using the upper/lower semi-continuous envelopes was introduced by Barles & Perthame in [8]. Our definition is in the same spirit. A key property of the viscosity sub/supersolutions is their stability:

### Proposition 2.2 (Stability of sub/supersolutions)

*For any ε*∈(0,1), *let* *be a non-empty family of subsolutions (resp. supersolutions) of (1.2) on* . *Let*
*If* *(resp.* *then* *is a subsolution (resp*. *is a supersolution) of (1.2) on* .

### Proof of proposition 2.2.

The proof of proposition 2.2 is classical, except for the new term *u*(*x*,*t*)−*u*(−*x*,*t*). In fact, Barles and Imbert give a related definition of viscosity solution and established stability results for a general class of non-local operators in [9]. Here, we simply check this property, proving that if for all functions , we have
2.1
in the viscosity sense, then still satisfies (2.1) (the proof being similar for ).

Indeed, by definition of , there exists (*y*_{ε},*s*_{ε},*ε*)→(*x*,*t*,0) and such that and *v*_{ε}(*y*_{ε},*s*_{ε})−*v*_{ε}(−*y*_{ε},*s*_{ε})−*ψ*(*y*_{ε},*s*_{ε})≤0. Because *ψ* is continuous,
which ends the proof. ▪

In parallel to the definition above, we may define viscosity sub/supersolutions for the stationary problem, and for the *w* problem. (For a precise definition, see the electronic supplementary material, appendix.)

## 3. Comparison principle for the *u*-problem

### (a) Comparison principle for the original *u*-problem

We consider the following non-local obstacle problem (see equation (1.2)): 3.1

### Theorem 3.1 (Comparison principle for the evolution problem)

*Assume (1.1), in particular that c>0. Let u (resp. v) be a subsolution (resp. supersolution) of (3.1) on* *for some T>0, satisfying for some constant C*_{T}*>0:*
*Then, u≤v on* .

We show in §10 that the comparison principle does *not* hold when *c*=0. We start by explaining the heuristic idea that underlies the proof.

*Quick heuristic proof of the comparison principle.* Let *u* be a subsolution and *v* a supersolution of (3.1). If
then, formally, at the point (*x*_{0},*t*_{0}):
3.2
and
3.3

(i) Case . We obtain the usual comparison principle using .

(ii) Case . In this case, we have 3.4

Subtracting the second line of (3.3) from this inequality, we deduce that *M*=(*u*−*v*)(*x*_{0},*t*_{0})≤(*u*−*v*)(−*x*_{0},*t*_{0})=*M* and we can apply the same reasoning at the point (−*x*_{0},*t*_{0}). Again, case (i) for (−*x*_{0},*t*_{0}) is excluded, and it remains case (ii) for (−*x*_{0},*t*_{0}), i.e. *u*(−*x*_{0},*t*_{0})−*u*(*x*_{0},*t*_{0})−*ψ*(−*x*_{0},*t*_{0})≤0. Summing this inequality to (3.4), we obtain
which yields a contradiction.

We now turn to the rigorous proof of the comparison principle. In this proof, we use the following adaptation of the (parabolic) Jensen–Ishii Lemma.

### Lemma 3.2 (Jensen–Ishii lemma for the obstacle problem)

*Let u (resp. v) be a subsolution (resp. a supersolution) of (3.1) on* *for some T*>0, *satisfying*
*Let for* *and ε*,*β*,*η*>0 *and δ*≥0:
*and*
*Assume that there exists a point* *such that*

*Then, we have*
3.5
*where*
3.6

The proof of this lemma is technical and rests on an adaptation of the doubling variable techniques (see lemma 8 in [10]). We provide it in the electronic supplementary material, appendix where we actually state and prove a more precise version of the Jensen–Ishii lemma.

### Proof of theorem 3.1.

We use the doubling of variables technique in the proof.

*Step 1:* *preliminaries.* Let
3.7
and let us assume by contradiction that *M*>0. Then, for small parameters *ε*,*β*,*η*>0 and *δ*≥0, let us consider
with
for a point to be fixed later. Clearly, *Φ*_{δ} satisfies
which shows that the supremum in *M*_{ε,β,η,δ} is reached at some point . Because of the zero initial data, it must be the case that . Moreover, for *β*,*η*,*δ* small enough, we have
and we see, in particular, that the following penalization terms are bounded

*Step 2:* *viscosity inequalities.* Let , *φ*_{δ} and *Φ*_{δ} be as defined above and in lemma 3.2. We now analyse the various possibilities in the lemma.

*Case* *A*_{0}≤0 *and* *δ*≥0. From (3.6) and the fact that , we deduce that
which gives a contradiction for *β*>0 small enough and *δ*≥0 small enough with *δ*≤*δ*_{0}(*β*,*z*_{0}).

*Case* *B*≤0,*B*_{1}≤0 *and* *δ*≥0 In this case, we have
3.8
In the limit *ε*→0 and up to extracting a subsequence, we have for
It is also classical that
3.9
Passing to the limit in (3.8), using (3.9) and the semi-continuities of *u* and *v*, we obtain
3.10
For the special *case δ=0*, and from the fact that
we deduce that
i.e.
We also recall (from (3.10)) that
3.11

*Case* *B*≤0,*B*_{1}≤0 *and* *δ*>0 *with the choice*

Note that which shows that and . Then from (3.10), we obtain and from (3.11), we obtain Summing these two inequalities, we obtain which gives the desired contradiction. The proof of theorem 3.1 is thereby complete. ▪

A similar proof yields

### (b) Comparison principle for a modified *u*-problem

We now consider the following modified problem for some positive constant *ε*_{0}>0:
3.12

Similar to §2, we can introduce the notion of viscosity sub- and supersolutions. Then, adapting the proof of theorem 3.1, we obtain easily the following result:

## 4. Existence by sub/supersolutions

The goal of this section is to prove theorem 1.1 on existence and uniqueness of the solution to the *u*-problem. This result will be proven by the method of sub- and supersolutions. We start with two lemmata.

### Lemma 4.1 (Subsolution)

*The function* *is a subsolution of (1.2)*.

### Proof.

We have in the region {*ψ*<0} and in the region {*ψ*≥0}. ▪

### Remark 4.2

Note that is the solution of the problem for *σ*=0.

The obstacle *ψ* depends on *t*, and for this reason, the function is not a natural supersolution of the evolution problem (indeed is not a supersolution for *x*<0, because *ψ*(*x*,*t*) has the wrong monotonicity in time for *x*<0). Actually, a direct computation shows that the function
is a supersolution of the evolution problem (1.2), where is the stationary solution of (1.3). We could thus use the result of [2] which proves existence of In order to keep a self-contained proof, we indicate in the following lemma a direct construction of a supersolution (figure 1). We also use this explicit supersolution in the proof of lemma A.8 in the electronic supplementary material, appendix to derive the initial bound (7.4) on the free boundary. This bound allows us to establish properties of the free boundary in theorem 7.2.

### Lemma 4.3 (Supersolution)

*Set*
*where for A*>0:
*Choose the positive constants b*, *B and q to satisfy the following inequalities*:
4.1
4.2
4.3
4.4
*Then, for σ*≥0, *the function* *is a supersolution of* (1.2).

### Proof of lemma 4.3.

We first note that and *ϕ* is *C*^{1} except for ∣*y*∣=*B*, and *C*^{2} except for ∣*y*∣=*B*,*A*. We also check that condition (4.2) implies that *ϕ* is non-decreasing, which also implies that *ϕ*≥0, because .

On the one hand, we have with *y*=*x*−*d*(*t*)
where we have used in the second line the fact that *ϕ* is non-decreasing. On the other hand, we want to check that
4.5
Note that this inequality is automatically satisfied in the viscosity sense at points corresponding to ∣*y*∣=*B* (because there is no test functions from below at those points). Outside that set, we have
because *ϕ*≥0, *ϕ*′≥0 and
Therefore, it is enough to show that , which means
i.e.
Using the fact that , it is enough to show that , i.e.
4.6

*Case 1:* ∣*y*∣<*A*.

Then, (4.6) means
i.e.
4.7
The minimum of *f* is reached for
and then (4.7) is satisfied if and only if (4.1) holds true, which also implies *h*=*b*−*A*/4≥0.

*Case 2:* *A*<∣*y*∣<*B*.

Then, (4.6) means and which are obviously true.

*Case 3:* *B*<∣*y*∣.

*Case 3.1:* *y*<−*B*.

Then, (4.6) means which is true if (4.3) holds true.

*Case 3.2:* *y*>*B*.

Then, (4.6) means
i.e.
which is implied by (because *q*<*r*/*ρ*)
which is true if
i.e. if (4.4) holds true.

Thus, (4.5) holds in all the previous cases and then by continuity also for ∣*y*∣=*A*. Therefore, (4.5) holds in the viscosity sense everywhere, which concludes the proof of the lemma. ▪

### Proof of theorem 1.1.

We prove only the (i), (the proof of (ii) for the stationary problem being similar, replacing and , respectively by and ).

*Step 1:* *definition of* *S*. We easily check that
with and respectively, defined in lemmata 4.1 and 4.3.

Indeed, *ϕ*≥0, and then it is sufficient to check that *ϕ*(*y*)≥*y* for *y*≥0. Moreover, for *B*>*A*, *ϕ* is *C*^{1} and convex on (−*B*,*B*), and then it is easy to check that *ϕ*(*y*) is above ∣*y*∣ on this interval. It is also straightforward to check that this is true on its complement. By continuity, it stays true in the limit case *B*=*A*. We define the set of functions

*Step 2:* *existence by Perron's method.* We now define
From the stability property (proposition 2.2), we can deduce that *u* is automatically a subsolution. We now check that *u*_{*} is a supersolution. Because
we only have to check that if
then
If , we obtain a contradiction with the optimality of *u* as usual (see Ishii [11], or for instance Chen *et al.* [12]). If *u*_{*}(*x*_{0},*t*_{0})−*u*_{*}(−*x*_{0},*t*_{0})−*ψ*(*x*_{0},*t*_{0})<0, we can write it as follows for some *η*>0:
As usual, up to replacing *φ* by *φ*(*x*,*t*)−∣(*x*,*t*)−(*x*_{0},*t*_{0})∣^{4}, we can assume that
We then check that
satisfies
with *R*_{δ}→0 as *δ*→0. And if at some point (*y*,*s*)∈*B*_{Rδ}(*x*_{0},*t*_{0}), then we have
for *δ*>0 small enough, where the last equality holds (for *δ*>0 small enough) because *x*_{0}≠0. This implies that is a subsolution for *δ*>0 small enough, i.e. . On the other hand, it is classical to check that we do not have everywhere, which gives a contradiction with the optimality of *u*. This shows that *u* is a viscosity solution of (1.2).

*Step 3:* *uniqueness.*

We just apply the comparison principle (theorem 3.1), which proves the uniqueness of *u* among solutions satisfying This completes the proof of the theorem. ▪

## 5. First properties of the solution *u*

The main result of this section is the following.

### Theorem 5.1 (Properties of the solution)

*Assume (1.1) and let u be the solution given in theorem 1.1. Then, u is continuous. There exists a function ϕ such that*
*and the following properties hold*

(i) asymptotics: 5.1

(ii) monotonicity and convexity: 0≤

*u*_{x}≤*α*(*t*),*u*_{xx}≥0;(iii) convergence in long time: as

(iv) monotonicity with respect to the parameters c,σ: For

*c*>0,*σ*≥0,*we have*∂*u*/∂*c*≤0,*and*∂*u*/∂*σ*≥0;(v) the limit c→0:

*u*→*u*_{0}*as c*→0,*where u*_{0}*is the minimal solution of (1.2) for c*=0*satisfying**on**for some constant**C*>0.

### Proposition 5.2 (Convexity of the solution)

*The solution u of (1.2) given by theorem 1.1 (i) is convex in x, for all time t*≥0.

This and several of the other results here also hold for more general obstacles *ψ*, provided *ψ* is convex. More generally, PDEs could also be addressed using the methods proposed by Imbert in [11].

### Proof of proposition 5.2.

In the literature, we find a few proofs of the convexity of solutions (see for instance Alvarez *et al.* [13], Imbert [14], Giga [15] and Rapuch [16]), but none of these approaches seems to apply directly to our problem. For this reason, we provide a new approach—our proof is based on a scheme obtained by an implicit discretization in time of the problem. This allows us to come back (at each timestep) to a stationary problem that we can analyse more easily.

*Step 1:* *the implicit scheme.*

Given a timestep *ε*>0, consider an approximation *u*^{n}(*x*) of *u*(*x*,*nε*) defined for as a solution of the following implicit scheme:
5.2

*Step 2:* *subsolution* .

As in the proof of lemma 4.1, we check that
is a subsolution of the scheme (5.2), distinguishing for the regions *ψ*^{n+1}≥0 and *ψ*^{n+1}<0 with *ψ*^{n+1}(*x*)=*ψ*(*x*,(*n*+1)*ε*) (and using the fact that *ψ* is non-decreasing in time).

*Step 3:* *supersolution* .

Set
and as in the proof of lemma 4.3, we easily check that is a supersolution of the scheme (5.2). To this end, we have in particular to note that *u*^{n+1}−*u*^{n}≥0 and we already checked that which implies

*Step 4:* *existence of a unique solution for the scheme.*

We can then apply Perron's method as in step 2 of the proof of theorem 1.1 and also prove a comparison principle similar to theorem 3.1. This shows that there exists a unique solution (*u*^{n})_{n} to the scheme. Moreover, the comparison principle implies that for each *n*, the function *u*^{n} is continuous.

*Step 5:* *convexity of u^{n+1}*. We prove by recurrence that

*u*

^{n+1}is convex, assuming that

*u*

^{n}is convex (and noting that

*u*

^{0}=0 is obviously convex).

*Substep 5.1:* *definition of the convex envelope U^{n+1}.*

Define the convex envelope of *u*^{n+1} as
with the set *E* of affine functions below *u*^{n+1} defined as
By construction, we have
Our goal is to show that *U*^{n+1} is a supersolution. Then, the comparison principle implies
which shows that *u*^{n+1} is convex.

*Substep 5.2:* *U*^{n+1} *is a supersolution.*

Consider a test function *φ* such that
We want to show that
5.3
Because *u*^{n+1} is continuous, we see that the set *E* is closed, and then the supremum defining *U*^{n+1}(*x*_{0}) is a maximum, i.e. there exists *l*_{0}∈*E* such that we have
Let us write
and
the extremal affine functions below *u*^{n+1} with *p*^{+} maximal and *p*^{−} minimal. Then, we have
If *U*^{n+1}(*x*_{0})=*u*^{n+1}(*x*_{0}), then *φ* is a test function for *u*^{n+1} which implies that (5.3) is satisfied. Let us therefore assume that *U*^{n+1}(*x*_{0})<*u*^{n+1}(*x*_{0}). This implies that *p*^{+}=*p*^{−}=*p* and then . Hence,
5.4
and
5.5
and moreover
5.6
Because of the asymptotics given by the inequalities
5.7
we deduce that
5.8
We distinguish several cases.

*Case 1:* *x*_{−} *and* *x*_{+} *finite*. Note that *l*_{0} is a test function from below for (the supersolution) *u*^{n+1} both at *x*=*x*_{−} and *x*=*x*_{+}. This implies that
5.9
We can write *x*_{0}=*ax*_{−}+(1−*a*)*x*_{+} for some *a*∈(0,1). Using the fact that *l*_{0} and *ψ*(⋅,(*n*+1)*ε*)) are affine, we deduce that
where we used the convexity of *U*^{n+1} to obtain the last inequality. This implies
We also compute
which is affine in *x*. Using the convexity of *u*^{n}, we then see that (5.9) implies
5.10
Finally, we see that this implies (5.3), because
follows from the fact that *φ* is tangent from below to the affine function *l*_{0} (because of (5.6)).

*Case 2:* *x*_{−} *finite and* . We consider a sequence of points . We first compute for *δ*>0
for *k* large enough depending on *δ* (using the asymptotics (5.7)). This shows that
This implies as in case 1 that
5.11
Similarly, we compute (using the asymptotics (5.7) at the level *n*):
for *x* large enough with
Therefore,
for *k* large enough. As in case 1, this implies (5.10). Taking the limit *δ*→0 in (5.11), this implies (5.3) as in case 1.

*Case 3:* *and* *x*_{+} *finite*. This case is similar to case 2, and we omit the details.

*Case 4:* *and* . This case is excluded by (5.8).

This ends step 5 and shows that *U*^{n+1} is a supersolution. We then conclude that *u*^{n+1}=*U*^{n+1} is convex.

*Step 6:* *convergence towards* *u* *as* *ε* *tends to zero*. We set
Using the asymptotics (5.7) and adpating the stability property (proposition 2.2) to this framework, it is then standard to show (see Barles & Souganidis [17]) that is a subsolution of (1.2) and is a supersolution of (1.2). The comparison principle then implies that
and *u* is convex in *x* as a limit of convex (in *x*) functions. This concludes the proof of the proposition. ▪

### Proof of theorem 5.1.

The continuity of *u* follows from the comparison principle.

*Proof of (i)*

Estimate (5.1) follows from inequality 5.12 with and given in lemmata 4.1 and 4.3.

*Proof of (ii)*

The convexity follows from proposition 5.2, and the asymptotics (5.12) implies

*Proof of (iii)*

Locally, in *x*, *u* is uniformly bounded in time (because of the asymptotics (5.12)) and is non-decreasing in time. Therefore, we have
and *U* is a viscosity solution of the stationary problem (1.3). Moreover, we have
Then, the comparison for the stationary problem (theorem 3.3) implies that i.e.
which shows in particular that is also convex.

*Proof of (iv)*

We start by showing that ∂*u*/∂*c*≤0. Let *c*_{2}>*c*_{1}>0 and the corresponding solutions *u*^{2}, *u*^{1}. Note that *u*_{2} is a subsolution for the problem satisfied by *u*_{1}. The comparison principle implies that *u*^{2}≤*u*^{1},

Next, we show that ∂*u*/∂*σ*≥0. Suppose *σ*_{2}≥*σ*_{1}≥0 and let the corresponding solutions *u*^{2}, *u*^{1}. Because *u*_{2} is a supersolution for the problem solved by *u*_{1} and thus *u*^{2}≥*u*^{1}.

*Proof of (v).* For *c*>0, consider the solution *u* given by theorem 1.1. Choose any solution *u*^{0} of (1.2) for *c*=0 satisfying for some constant *C*>0. Then, *u*^{0} is a supersolution of the equation satisfied by *u*. The comparison principle implies that *u*^{0}≥*u*≥0. The monotonicity of *u* with respect to *c* implies that *u* has a limit *u*_{0} as *c* goes to zero, which satisfies
5.13
Using the stability of viscosity solutions and (5.13), it is straightforward to show that *u*_{0} is a viscosity solution of (1.2) for *c*=0. Therefore, *u*_{0} is the minimal solution.

This completes the proof of the theorem. ▪

## 6. Further properties of the solution *u*

The main result of this section is the following.

### Theorem 6.1

*Assume (1.1) and let u be the solution given in theorem 1.1. Then, in the standard viscosity sense,*
6.1
*Moreover, u*_{t}*≥0 and the following monotonicities with respect to the parameters r>0 and λ>−r hold: ∂u/∂r≤0 and ∂u/∂λ≤0. Set w(x,t):=u(x,t)−u(−x,t). Then, in the viscosity sense, w solves*
6.2

### Proof of theorem 1.2.

Theorem 1.2 just combines theorems 5.1 and 6.1.

To obtain further properties of the solution *u* stated in theorem 6.1 (including the monotonicity with respect to the parameter *r*), it is convenient to consider the following *modified equation*:
6.3

Similar to definition 2.1, we can introduce a notion of viscosity solution for this equation. The only difference is that for a viscosity subsolution *u* such that
we require both
and
▪

### Proposition 6.2 (Existence and uniqueness for the modified equation)

*Assume (1.1). Then, there exists a unique solution u of (6.3). Moreover, this solution u is the same as the one given by theorem 1.1*.

### Proof of proposition 6.2.

We can check that the notion of viscosity solution for (6.3) is stable (as in proposition 2.2). It is straightforward to verify that the function given in lemma 4.1 is a subsolution of (6.3). Because the definition of a supersolution is unchanged for (6.3) in comparison with (1.2), the function given in lemma 4.3 is still a supersolution of (6.3). Thus, we can apply Perron's method that shows the existence of a solution of (6.3). Finally, note that any viscosity solution of (6.3) is also a viscosity solution of (1.2). Therefore, we can apply the comparison principle for equation (1.2) which shows that the solution is the same as the one given by theorem 1.1. This ends the proof of the proposition. ▪

### Proof of theorem 6.1.

The first part of the theorem, viz. (6.1), follows from proposition 6.2. To show the monotonicity in time of *u*, we simply check that *u*_{h}(*x*,*t*):=*u*(*x*,*t*+*h*) is a supersolution of (3.12) for *h*>0, because *u*_{h}(*x*,0)≥0= *u*(*x*,0) and the obstacle satisfies *ψ*_{t}≥0 for *x*>0. Then, the comparison principle (theorem 3.4) yields *u*_{h}≥*u* for any *h*>0. This implies that *u*_{t}≥0. ▪

*Proof of monotonicity with respect to parameters* *r* *and* λ. For *r*>0, define the set
where the notation is explicit of the dependence on *r*:
and set
and note the dependence in *r* by writing
We have
6.4
Let *r*^{2}>*r*^{1}>0 and the corresponding solutions *u*^{2} and *u*^{1} of (1.2) (or equivalently (6.3)).

Because of (6.4),
and
On the other hand,
because *u*^{2}≥0. Because *u*^{2} is a solution of (6.3) for *r*=*r*^{2}, for any test point (*x*,*t*) (tested from above), either
or
This implies that
which shows that *u*^{2} is a subsolution for the equation satisfied by *u*^{1}. Therefore, *u*^{2}≤*u*^{1} which implies the expected monotonicity in *r* of the solution. The proof of monotonicity with respect to the parameter λ is similar.

*Equation satisfied by* *w*

Set
The fact that *w* solves (6.2) in the viscosity sense follows from lemma A.3 in the electronic supplementary material, appendix.

## 7. The obstacle problem satisfied by *w*

Recall that solves the problem: 7.1 and define the stationary problem (for ) with : 7.2

We now state the main results for the solution of the *w*-obstacle problem. The proof of these results, including the relevant comparison principle, is detailed in electronic supplementary material, appendix.

### Theorem 7.1 (Properties of the solution *w*)

*Assume (1.1). Then, there exists a unique solution w to equation (7.1) satisfying*
*Moreover, w is continuous, and there exists a function* *satisfying*
*such that the following properties hold*

(i) asymptotics: if

*d*(*t*):=*c*/*α*(*t*) 7.3(ii) monotonicity and convexity:

*w*_{t}≥0, 0≤*w*_{x}≤*α*(*t*),*w*_{xx}≥0;(iii) convergence in long time: as where is the unique solution of (7.2) satisfying on

(iv) monotonicity with respect to the parameters c,ρ,r,λ,σ: ∂

*w*/∂*c*≤0, ∂*w*/∂*ρ*≤0, ∂*w*/∂*r*≤0, ∂*w*/∂λ≤0 and ∂*w*/∂*σ*≥0.

Note that if *σ*=0.

### Theorem 7.2 (Properties of *a* and )

*Assume (1.1) and let w be the solution given in theorem 7.1. Then, there exists a lower semi-continuous function* *such that for all t>0:*
*Let*
*Then, the following properties hold*

(

*i*) bounds when σ≥0 7.4 and 7.5(

*ii*) time monotonicity*If ρ≥λ, then*7.6(

*iii*) monotonicity with respect to the parameters*ρ,c,r,λ*,σ 7.7*and*7.8*Moreover, if ρ≥λ, then*7.9

In the electronic supplementary material, appendix (§A.7), we show that is the solution of the equation: 7.10 with We also establish in electronic supplementary material, appendix further properties of .

### Remark 7.3

The condition *ρ*≥λ is always satisfied for the model derived in [2].

### Corollary 7.4 (The exercise region for *u* is on the right)

*Assume (1.1). Let u be the solution given in theorem 1.1. Then*

## 8. Regularity of the free boundary

### (a) Lipschitz regularity

### Theorem 8.1 (Lipschitz regularity of the free boundary)

*With the notation of theorem 7.2, the map t↦a(t)α(t) is non-decreasing and*
8.1
*As a consequence, in view of (7.6), if ρ≥λ, then the function a is locally Lipschitz.*

### Proof of theorem 8.1.

*Step 1:* *change of function.*

Define *v* by
where is the solution of (7.10). Writing *y*=*xα*(*t*)
with

*Step 2:* *monotonicity of the coefficients and v^{h} supersolution.* Recall that satisfies −1≤

*v*

_{y}≤0. Then, for −1≤

*z*≤0, we compute 8.2 Here, we used the fact that (

*α*′/

*α*)′≤0. For any

*h*>0, let Then,

*v*

^{h}satisfies where we used in the second line, the properties

*v*

_{yy}≥0,

*α*non-decreasing, and because of (8.2). Therefore, because,

*v*is a supersolution of the equation , we deduce that

*v*

^{h}is also a supersolution of the same equation.

*Step* 3: *supersolution* .

As a consequence, is also a supersolution of the first line of (7.10). Moreover, we have 8.3 and thus satisfies This shows that is a supersolution of (7.10) (now also including the boundary conditions).

*Step 4*: *conclusion.*

We can now apply the comparison principle and deduce that for any *h*>0:
Fix *t*_{0}>0 and for any *ε*>0 (small enough), set
Then
This shows that
Because this holds for any *ε*>0 small enough, we deduce that
which shows that *t*↦*a*(*t*)*α*(*t*) is non-decreasing.

Therefore, which implies (8.1). This concludes the proof of the theorem. ▪

### (b) Further regularity

We prove now the following result, which is very much in the spirit of Kinderlehrer & Nirenberg [7].

### Proposition 8.2 (Smoothness of the free boundary for *σ*>0)

*Let r*,*c*,*σ*>0 *and ρ*≥λ≥0. *Then*, .

### Proof.

*Step 1*: *from the viscosity formulation to the variational formulation.*

Set
Recall that *w*(*x*,*t*)=*ψ*(*x*,*t*)−*ψ*(−*x*,*t*) solves
with
If moreover *ρ*≥λ, then we have
and . Because we have
we proceed as in §5.3 of [18] to deduce that
8.4
and then almost everywhere and in the distributional sense, for
8.5
Note that
Because *a*(*t*)≥*c*/*α*(*t*),
8.6

*Step 2*: *preliminary regularity theory.*

We can then apply theorem 1.3 from [19] (with [20]) to deduce that is continuous up to the free boundary *x*=*a*(*t*). We also deduce from (8.4) that
Therefore, from the continuity of *a*, we deduce the continuity of up to the free boundary. Finally, from the PDE (8.5), we deduce the continuity of on the set {*x*≤*a*(*t*)}, and then
which shows that the standard non-degeneracy condition is satisfied for this obstacle problem.

*Step 3*: *higher regularity theory.* This is an adaptation of theorem 3 in Kinderlehrer & Nirenberg [7]. The details are provided in electronic supplementary material, appendix. With this result, we conclude that the free boundary is smooth, i.e. that it is . The proof of the proposition is thereby compete. ▪

## 9. Convergence of the free boundary as *c*→0

The main result of this section is the following.

### Theorem 9.1 (Convergence of the rescaled free boundary when *c**→*0)

*Assume σ>0 and λ≤3r+4ρ. Then, the following convergence of the rescaled free boundary holds:* *locally uniformly on any compact sets of* *as c→0, where*

As a corollary, we can deduce theorem 1.3.

### Proof of theorem 1.3.

Theorem 1.3 follows from theorems 7.2, 8.1 and 9.1. ▪

### (a) Preliminary results

### Lemma 9.2 (Global subsolution and bound from below on the free boundary)

*Assume σ*>0 *and* λ≤3*r*+4*ρ*. *Consider the function*
*with*
9.1
*and*
*Then*, *is a subsolution of equation (7.10). In particular, we have (for each c>0)*:
9.2

### Remark 9.3

Note that for all *t*≥0
if *r*>0 and *ρ*≥0, λ+*r*>0.

So we have 9.3

### Proof of lemma 9.2.

Our goal is to build a subsolution for close to the axis *x*=0.

*Step 1:* *Change of function.*

Set
with
Set
and recall from (7.10) that
This implies that
9.4
Because satisfies (7.10), *v* satisfies
9.5

*Step 2*: *Constructing a candidate subsolution.*

If we neglect completely the term *c*^{2/3}*G*_{0}[*v*] in (9.5), as a first guess, for each fixed *t*>0, then we can look for a stationary solution *v*^{0} of
9.6
with
We can solve this equation explicitly, recalling that for such an obstacle problem, we have
Then, we obtain successively in
9.7
Finally, the condition *v*^{0}(0,*t*)=1 implies
which is exactly (9.1). Define *ϕ*∈*C*^{1,1} by
which satisfies the following properties:
Then, we have

*Step 3:* *Checking the subsolution property.*

Set By construction,

*Case 1:* *ρ*≥λ.

Compute on :
In the last inequality, we used the properties , and if *ρ*≥λ.

*Case 2:* λ≤3*r*+4*ρ*.

When λ<*ρ*, we must use a different estimate. Write and
Setting
we obtain
i.e.
with
which is an affine function. It satisfies *g*(0)=(3(*r*+*ρ*)^{2})/(*r*+λ)≥0 and
This implies that *g*(e^{−(r+λ)t})≥0 and then .

*Step 4:* *Conclusion.* This implies that on
This shows that is a subsolution of (9.5). The comparison principle implies the result. This ends the proof of the lemma. ▪

For some constant , consider the following problem 9.8

Even if there is no zero-order term in the PDE part of problem (9.8), we are able to show the following result (see the proof given in electronic supplementary material, appendix):

### Lemma 9.4 (Comparison principle for a stationary obstacle problem without zero-order terms)

*Assume σ*>0. *If u (resp. v) is subsolution (resp. supersolution) of (9.8), satisfying*
9.9
*Then*

### (b) Convergence as *c*→0

### Proposition 9.5 (Convergence of the rescaled solution as *c*→0)

*Assume σ*>0 *and* λ≤3*r*+4*ρ*. *Consider the solution* *of (7.10) on* *and set*
9.10
*Then,*
9.11
*where*
*with* *ϕ* *and* *defined in lemma 9.2.*

### Proof of proposition 9.5.

*Step 1:* *The relaxed semi-limits.*

We know that *v*^{c} satisfies on :
with *F*_{0},*G*_{0} defined in (9.4). Note that the condition *v*^{c}≤1 follows from (7.5) and *v*^{c}≥*v*^{0} follows from lemma 9.2. We define the relaxed semi-limits:
By construction,
9.12
From the stability of viscosity solutions, we deduce that (resp. ) is a subsolution (resp. supersolution) of
and (9.12) implies that
9.13

*Step 2:* *Sub/supersolutions of the stationary problem.*

We claim that for any fixed *t*_{0}>0, (resp. ) is a subsolution (resp. supersolution) of
9.14
with
We check it for (the reasoning being similar for ).

*Step 2.1:* *preliminaries.*

The boundary condition is obvious because of (9.13). Recall that is upper semi-continuous, and then for any *δ*>0 small enough, there exists *r*_{δ}>0 such that
where *P*_{0}=(*y*_{0},*t*_{0}) and
Consider now a test function *φ* satisfying
Up to adding *η*∣*y*−*y*_{0}∣^{2} to *φ* (with *η* large enough), we can assume that
and
9.15

*Step 2.2:* *the* *ε*-*penalization*.

For *ε*>0, define
Up to choosing an *ε* small enough (*ε*≤*ε*_{δ}), we have
Therefore,
and
9.16
for some point *P*_{ε}=(*x*_{ε},*t*_{ε})∈*Q*_{rδ}(*P*_{0}). This implies that we have
9.17
Because *t*_{ε}→0 as *ε*→0, and up to some subsequence, we have , we deduce from (9.16) that
Hence, (9.15) implies that
and then
and passing to the limit in (9.17), we obtain
9.18
Indeed, either we have for a subsequence, and we define the along that subsequence which implies that , or we have for a subsequence, and this implies (9.18).

*Step 3*: *conclusion.*

We can now apply the comparison principle (lemma 9.4), to deduce that
Because we have the reverse inequality by construction, we deduce that
where *v*^{0}(⋅,*t*_{0}) is the explicit solution of (9.14). This implies (9.11).

This ends the proof of the proposition. ▪

In order to conclude to the convergence of the free boundary itself, we need the following result, which is adapted from Caffarelli [21]:

### Lemma 9.6 (Non-degeneracy)

*Assume σ*>0 *and* λ≤3*r*+4*ρ*. *Let t*_{0}>0 *and*
*For d*>0, *define*
*Let δ*_{1}>0 *such that*
*Let v*^{c} *defined in* (9.10). If
*then*

### Proof of lemma 9.6.

*Step 1:* *auxiliary function.*

Recall that *v*^{c} satisfies on
with *F*_{0},*G*_{0} defined in (9.4).

In particular, we have
Given a point *P*_{0}=(*y*_{0},*t*_{0})∈*Ω*, consider the auxiliary function
For any *d*>0 small enough, define
We have
and

*Step 2*: *non-degeneracy.*

Assume (and we prove it in the next step) that *ξ* is a supersolution of the linear parabolic operator, namely assume
9.19
We now apply a non-degeneracy argument owing to Caffarelli (see [21,22]). Define
From the local comparison principle (the proof being similar to the usual proof of the comparison principle), we have
and
Now
and therefore
This implies that
This shows that (9.19) implies the following non-degeneracy property
9.20

*Step 3:* *proof of (9.19)*.

For the reader's convenience, we recall that
Compute
with
Let *δ*_{1}>0 such that
Then, the previous computation shows that for (*y*,*t*)∈*Q*^{−}_{d}(*P*_{0})
for *y*_{0}−*d*≥*δ*_{1} and , and
i.e.
9.21

*Step 4:* *conclusion.*

Consider now
which satisfies because of (9.2). We now consider a sequence of points *y*_{n}
and a sequence *d*_{n} such that for any *d*∈(0,*δ*_{1}]:
Then, assuming (9.21), and applying the previous steps at the point (*y*_{n},*t*_{0}), we obtain
Passing to the limit in *n*, this implies
This ends the proof of the lemma. ▪

### Proof of theorem 9.1.

We already know from (9.2) that Assume by contradiction that the statement is false.

Then, for any *δ*>0, there exists *η*>0 and a sequence of times *t*_{c}∈[*δ*,1/*δ*], such that
Applying lemma 9.6, we obtain for *c*>0 small enough that for any *d*∈[0,*δ*_{1}]
From proposition 9.5, we know that *v*^{c}→*v*^{0} locally uniformly. Moreover, extracting a subsequence if necessary, we can assume that
This implies that for any *d*∈(0,*δ*_{1}] and *K*>0
For 0<*d*≤*η*, this gives a contradiction, because
This ends the proof of the theorem. ▪

## 10. No comparison principle for *c*=0 and *σ*>0

Here, we discuss the existence of multiple solutions for *c*=0, i.e. solutions *u*^{0} of
10.1
with *ψ*^{0}(*x*,*t*)=*α*(*t*)*x*. Indeed, for *c*=0, we can reduce the construction of solutions to a more classical problem.

### Proposition 10.1 (Family of solutions *u*^{0} for *c*=0 and *σ*>0)

*For c=0, there exists an infinite family of viscosity solutions u*^{0} *of (10.1), such that for each u*^{0}, *there exists a constant C*>0 *such that* . *More precisely, when c*=0, *all viscosity solutions u*^{0}(*x*,*t*) *of*
10.2
*are viscosity solutions of (10.1)*.

### Remark 10.2 (No comparison principle for *c*=0 and *σ*>0)

This shows that the condition *c*>0 for our comparison principle is sharp, because the comparison principle is not valid when *c*=0.

Before giving a rigorous proof of proposition 10.1, let us explain heuristically why we expect to have non-uniqueness of solutions of (10.1) in case *c*=0. First, when *c*<0 we expect that , because we gain −*c* in each transaction and there are no limits to the amount of trades one can do in any interval. As a consequence, for *c*=0, we expect to have a transition family of solutions *u*^{0} between the two limit cases for *c*<0 and *u*=*u*_{0} for *c*=0^{+}. Thus, we loose the comparison principle when *c*=0. Another heuristic argument starts from observing that the functions *w*^{0}(*x*,*t*)≡*u*^{0}(*x*,*t*)−*u*^{0}(−*x*,*t*) and *ψ*^{0}(*x*,*t*) are odd in , and thus the inequality *u*^{0}(*x*,*t*)−*u*^{0}(−*x*,*t*)≥*ψ*^{0}(*x*,*t*) on all of implies that *u*^{0}(*x*,*t*)−*u*^{0}(−*x*,*t*)≡*ψ*^{0}(*x*,*t*). This equality holds pointwise. Hence, the system of complementary inequalities (10.1) reduces to in the viscosity sense, and one should expect the existence of many solutions.

### Proof of proposition 10.1.

Because we deal with viscosity solutions, a solution of (10.2) is obviously a solution of (10.2). It is clear that system (10.2) admits a large set of solutions. First, we note that it is a convex set. Then, to construct solutions of (10.2), we can reduce the problem to a problem on the half-line in the following manner.

### Proposition 10.3

*Let v be a* *solution of the following system*:
10.3
*Then, u*^{0} *defined by u*^{0}(*x*,*t*)=*v*(*x*,*t*) *for* *and* *u*^{0}(*x*,*t*)=*v*(−*x*,*t*)+*ψ*^{0}(*x*,*t*) *for* *x*≥0 *is a (classical) solution of (10.2)*.

It is straightforward to check that the construction of *u*^{0} from *v* yields a function of class C^{2,1}. Further, the function *u*^{0} recovered from a solution *v* of (10.3) satisfies . Indeed, this is true on by the inequality for *v*. Now, for a function *z*=*z*(*x*,*t*) denote *z*^{♯} the function defined by *z*^{♯}(*x*,*t*)=*z*(−*x*,*t*). Observe that . Thus, on we obtain
We know that on , and
Therefore, we see that on as well. Note also that

For instance, a solution (10.3) can be obtained in the form
where *κ* is a Kummer function (see [2] for the construction and properties of Kummer functions in our setting) satisfying:
We know that and *κ*′(0)>0 (see [2]). The function *z* satisfies because *α*′(*t*)>0.

We thus obtain a solution *z* of (10.3) such that . As we have seen, such a solution yields a solution *u*^{0} of our original problem (10.1) such that . In fact, it satisfies .

By using the method of super- and subsolution, we can construct another solution *v* that satisfies equality in the first line of (10.3) rather than an inequality. Because the set of solutions of (10.3) is convex, it is clear that because of the inequality in (10.3), there is a very large indeterminacy.

As a further example, we can construct a one parameter family of solutions by considering the operators
Let *κ*_{s} be a Kummer function associated with : on . Then, define
For each *s* with 0<*s*<*r*, we obtain . Thus, the functions *z*_{s} for 0<*s*<*r* is a one parameter family of (pairwise distinct) solutions of (10.3). Likewise, they yield a one parameter family of solutions of our original problem (10.1) each of which satisfies .

This concludes the proof of the proposition. ▪

### Remark 10.4 (An explicit supersolution)

In the proof of proposition 10.1, we can also use in place of the Kummer function *κ*(*x*), the function where *ϕ* is the function constructed in lemma 4.3 (for supersolutions of the *u*-problem). One can verify that is a supersolution for the stationary problem (1.3) in the case *c*=0. Proposition 10.3 holds for the function , because *v* is *C*^{2,1} in a neighbourhood of *x*=0. This produces one viscosity solution *u*^{0} for (10.2) for .

### Remark 10.5 (A particular solution *u*^{0})

In a recent work, Tian [23] constructed a particular solution *u*^{0} of (10.1) from a function *v* that satisfies (10.3) with equality in the first line (i.e. on the negative half-line). One can show that there is a unique function *v* which satisfies (10.3) and for This can be verified by setting *z*(*x*,*t*)=*v*(*x*,*t*)−*x*(*α*(*t*)/2)*g*(*x*), where *g* is a smooth function with compact support such that *g*′(0)=1. Then, *z*_{x}(0,*t*)=0 and one can check that for *x*<0. Extending *f* and *z* as even functions for positive *x* (still using the same notation for the extension), we see that *z* now solves the same PDE for all real *x*. This can be checked also for *x*=0 using the boundary regularity of the solution on the half-line. We can now apply a comparison principle on the whole real line to derive the uniqueness of the function *v*. (The proof of this comparison principle is similar to the one of the *u*-problem, but is much simpler here.)

## Funding statement

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC grant agreement no. 321186: ‘reaction–diffusion equations, propagation and modelling’ held by Henri Berestycki. Part of this work was done while H. Berestycki was visiting the University of Chicago. He was also supported by an NSF FRG grant no. DMS-1065979.

## Acknowledgements

We thank the editor and referees for reading carefully a previous version and providing excellent suggestions. R. Monneau thanks N. Forcadel for references to the literature.

## Footnotes

One contribution of 13 to a Theme Issue ‘Partial differential equation models in the socio-economic sciences’.

↵1 Monotonicity and convexity inequalities in this paper should be understood in either the viscosity sense or distributional sense.

↵2 For a precise definition of viscosity solution to this problem, see the electronic supplementary material, appendix.

↵3 In this paper, we reserve the notations

*f** and*f*_{*}for the sup and inf-envelopes of a function*f*.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.