## Abstract

We study the obstacle problem for a class of degenerate parabolic operators with continuous coefficients. This problem arises in the Black–Scholes framework when considering path-dependent American options. We prove the existence of a unique strong solution *u* to the Cauchy and Cauchy–Dirichlet problems, under rather general assumptions on the obstacle function. We also show that *u* is a solution in the viscosity sense.

## 1. Introduction

We consider a class of second-order differential operators of Kolmogorov type(1.1)where *z*=(*x*, *t*)∈^{N+1}, 1≤*m*≤*N* and *b*_{ij}∈ for every *i*, *j*=1, …, *N*. We are mainly interested in degenerate operators (namely for *m*<*N*), which are usually called ultraparabolic, since only first-order derivatives with respect to *x*_{m+1}, …, *x*_{N} appear. We assume the following hypotheses:

*H1*. The coefficients *a*_{ij}=*a*_{ji} and *b*_{i} are bounded continuous functions for *i*, *j*=1, …, *m*. Moreover, there exists a positive constant *Λ*, such that

*H2*. The operator(1.2)is hypoelliptic, i.e. every distributional solution of *Ku*=*f* is a smooth solution, whenever *f* is smooth.

Hypothesis *H2* is equivalent to the classical Hörmander condition (Hörmander 1967)where denotes the Lie algebra generated by the vector fields andWe explicitly remark that uniformly parabolic operators satisfy *H1* and *H2*, with *m*=*N*. We also recall (cf. Lanconelli & Polidoro 1994) that *H2* is equivalent to the existence of a basis of ^{N} with respect to which the matrix *B*=(*b*_{ij}) assumes the following block form:(1.3)where the blocks ‘^{*}’ are constant and arbitrary; *B*_{j} is a *m*_{j−1}×*m*_{j} matrix of rank *m*_{j}; and

This paper is mainly concerned with the obstacle problem(1.4)where *a*, *f* and *g* are bounded continuous functions. The assumptions on the obstacle function *ϕ* will be specified in *H4* in §3: we require that *ϕ* is locally Lipschitz continuous and satisfies a weak convexity condition with respect to the variables *x*_{1}, …, *x*_{m}.

Apart from their obvious importance in PDE theory, obstacle problems have a natural theoretical interest in stochastic control. Moreover, they appear in several applications in physics, biology and mathematical finance. Specifically, one of the best-known problems in finance is that of determining the arbitrage-free price of American-style options. Precisely, we consider a financial model where the dynamic of the state variables is described by an *N*-dimensional diffusion process , which is a solution to the stochastic differential equation(1.5)where and *W* denotes an *m*-dimensional Brownian motion, *m*≤*N*. An American option with pay-off *ϕ* is a contract granting the holder to receive the payment of the sum *ϕ*(*X*_{t}) at a time *t*∈[0, *T*], which is chosen by the holder. Then, according to the theory of modern finance (cf., for instance, Peskir & Shiryaev 2006), the arbitrage-free price, at time 0, of the American option is given by the following optimal stopping problem:(1.6)where the supremum is taken over all stopping times *τ*∈[*t*, *T*] of *X*. The main result in (Pascucci in press) is that the function *u* in (1.6) is a solution of a problem in the form (1.4), where the obstacle function *ϕ* corresponds to the pay-off of the option and *L* is the Kolmogorov operator associated with the diffusion *X*,In the uniformly parabolic case *m*=*N*, the valuation of American options has been studied starting from the papers by Bensoussan (1984) and Karatzas (1988) using a probabilistic approach based on Snell envelopes and by Jaillet *et al*. (1990) using variational techniques. However, there are significant classes of American options, commonly traded in financial markets, whose corresponding diffusion process *X* is associated with Kolmogorov-type operators that are not uniformly parabolic. Two remarkable examples are provided by Asian-style options (cf., for instance, Barucci *et al*. (2001)) and by some recent stochastic volatility model with dependence on the past (cf. Hobson & Rogers 1998; Di Francesco & Pascucci 2004; Foschi & Pascucci in press). A general theory for these financial instruments is not available. Actually, the several papers on American–Asian options available in the literature (cf., for instance, Rogers & Shi (1995), Barraquand & Pudet (1996), Barles (1997), Hansen & Jorgensen (2000), Meyer (2000), Marcozzi (2003), Jiang & Dai (2004) and Dai & Kwok (2006)) mainly consider numerical issues.

The aim of this paper, and of the related work (Pascucci in press), is to develop a rigorous theory for the obstacle problem (1.4) and the optimal stopping problem (1.6). The main results of this paper are the existence of a strong solution to the obstacle problem in a bounded cylindrical domain (cf. theorem 3.2) and in the strip ^{N}×]0, *T*[ (cf. theorem 4.1). We recall that, even in the standard framework of uniformly parabolic operators, problem (1.4) generally does not admit a solution in the classical sense. Three main approaches are used to tackle the existence problem: these are based on the notion of *variational solution* (cf. Kinderlehrer & Stampacchia 1980; Bensoussan & Lions 1982), *strong solution* (cf. Friedman 1975, 1988) and, more recently, *viscosity solution* (cf. Crandall *et al*. 1992; Barles 1997). Since operator (1.1) appears in non-divergence form, we adapt a classical penalization technique to find a unique strong solution to (1.4), obtained as the limit of solutions to a suitable class of nonlinear problems.

Moreover, in theorem 5.2, we show that the strong solutions are viscosity solutions and in Pascucci (in press) it is proved that the function *u* defined in (1.6) is a strong solution to the obstacle problem (1.4). As a consequence, the solutions to the American option problem provided by means of different methodologies in Barles (1997), Jiang & Dai (2004) and Dai & Kwok (2006) must coincide.

Concerning the regularity, we emphasize that any strong solution *u* is Hölder continuous with its first-order derivatives (see §2). This result, combined with the above remark, improves the regularity of viscosity solutions and solutions of the optimal stopping problem (1.6), obtained by probabilistic techniques. Starting from these results, we aim to investigate the regularity properties of the obstacle problem in a forthcoming study.

This paper is organized as follows. In §2 we set the notations and introduce the functional setting suitable for the study of the regularity properties of operator (1.1); specifically, our study is cast in the framework of analysis on Lie groups. The proofs of some of the results stated in this section are postponed to appendix A. In §§3 and 4 we prove the existence and uniqueness of the solution to the obstacle problem in bounded domains and in ^{N}, respectively. The main result of §5 states that the strong solutions are also viscosity solutions.

## 2. Functional analysis on Lie groups

Since the works by Folland (1975), Rothschild & Stein (1976) and Nagel *et al*. (1985), it has been known that the natural framework for the study of operators satisfying the Hörmander condition is the analysis on Lie groups. The Lie group structure related to Kolmogorov operators has been first studied by Lanconelli & Polidoro (1994). The explicit expression of the group law is defined by(2.1)where and *B*^{T} denotes the transpose of *B*. The solutions of the equation *Ku*=0, with *K* as in (1.2), have the remarkable property of being invariant with respect to the left translations defined by (2.1),or equivalently, if *Ku*=*f*, thenfor every *z*=(*x*, *t*), *ζ*=(*ξ*, *τ*)∈^{N+1}. Moreover, if and only if the blocks ‘^{*}’ in (1.3) are null, operator *K* is homogeneous of degree 2 with respect to the dilations defined bywhere denotes the *m*_{j}×*m*_{j} identity matrix. The numberis usually called *D*(λ)-homogeneous dimension of ^{N+1}. Then, is a homogeneous Lie group determined only by *B*.

We also recall the definition of a *D*(*λ*)-homogeneous norm: for every , we define if *ρ* is the unique positive solution ofand *q*_{1}, …, *q*_{N} are the integers, such that(2.2)Some functional spaces related to the Lie group are defined as follows. Let *Ω* be a domain of ^{N+1} and *p*≥1. We setandWe say that if for every compact . Moreover, we denote, respectively, by , and the Hölder spaces defined by the following norms:andNote that any is Hölder continuous in the usual sense since

Next, we state some results extending the usual embedding theorems and *a priori* interior estimates. In the case of homogeneous Kolmogorov operators, they have been proved in several papers (cf. Bramanti *et al*. 1996; Manfredini 1997; Manfredini & Polidoro 1998). In appendix A, we generalize these results to the non-homogeneous case. Hereafter when we claim that a constant depends on *L* we mean that it depends on *N*, *m*, *B* and the constant *Λ* in *H1*.

*Embedding theorem*. *Let O and Ω be bounded domains of* ^{N+1}, *such that* *and p*>*Q*+2. *There exists a positive constant c, depending only on L, Ω, O and p, such that*(2.3)*for any* .

A priori *interior estimates*. *Let O and Ω be bounded domains of* ^{N+1}, *such that* . *There exists a positive constant c, depending only on L, O, Ω and p, such that*(2.4)*for every* , 1<*p*<∞.

In the sequel, we also use the following Schauder-type estimate, proved by Di Francesco & Polidoro (2006).

*Interior Schauder estimate*. *Let O and Ω be bounded domains of* ^{N+1}, *such that* . *Let the coefficients* . *For any* , *we have*(2.5)*for some positive constant c, depending only on α, Ω, O, L*, *and* .

## 3. Obstacle problem on bounded domains

In this section, we prove the existence and uniqueness of a strong solution to the obstacle problem(3.1)where *H* is a bounded domain in ^{N} anddenotes the parabolic boundary of *H*(*T*). We say that is a strong solution to problem (3.1) if the differential inequality is satisfied a.e. in *H*(*T*) and the boundary datum is attained pointwisely.

We assume that *H*(*T*) is regular in the sense that at every point of its parabolic boundary, there exists a barrier function. Precisely,

*H3*. For any , there exists a neighbourhood *V* of *ζ* and a *C*^{2} functionsuch that

*Lw*≤−1 in*V*∩*H*(*T*),*w*(*z*)>0 in and*w*(*ζ*)=0.

In §4 we solve the obstacle problem (3.1) in the cylindrical domain *H*_{n}(*T*), for *n*∈, defined as follows. Let *e*_{1}=(1,0, …, 0) be the first vector of the canonical basis of ^{N} and denote by *B*_{n}(*x*_{0}) the Euclidean ball of ^{N} with centre at *x*_{0}∈^{N} and radius *n*. We defineand, for every *T*>0,(3.2)For such a domain, a barrier function is defined at every point *z* of the parabolic boundarySpecifically, following Friedman (1964 ch. 3 §4):

if , thenis a barrier provided that

*c*_{1}and*c*_{2}are sufficiently large;if , such that

*τ*>0, then we setwhere is the centre of a ball externally tangent to*H*_{n}(*T*) at (*ξ*,*τ*) andA direct computation shows that

If is strictly positive definite, then by choosing suitably large *p*, *c*_{2} and *c*_{3}, we have that *w* is a barrier. Under assumption *H1*, *A* is generally not uniformly positive definite. However, due to the shape of the cylinder, it is possible to choose , such that does not belong to the kernel of *A* so that the same argument shows that *w* is a barrier.

Next, we state the assumption on the obstacle function *ϕ*.

*H4. ϕ* is a Lipschitz continuous function on and there exists a constant , such that(3.3)in the distributional sense, that isfor any and , *ψ*≥0.

We explicitly note that *C*^{2} functions satisfy assumption *H4* as well as the Lipschitz continuous function that are convex with respect to the first *m* variables.

The main result of this section is as follows.

*Assume H1–H4. Let* , *such that g≥ϕ, and* . *Then, there exists a strong solution u of problem* (*3.1*). *Moreover, for every p≥*1 *and O compact subset of H*(*T*)*, there exists a positive constant c, depending only on L, O, H*(*T*), *p and on the L ^{∞}-norms of f, g, ϕ and a, such that*(3.4)

*ϵ*>0,

*β*

_{ϵ}is an increasing function, bounded with its first-order derivatives, such thatandFor , we denote by

*L*

^{δ}the operator obtained from

*L*by mollifying the coefficients

*a*

_{ij}and

*b*

_{i}. We also denote by

*ϕ*

^{δ}(respectively,

*a*

^{δ}and

*f*

^{δ}) the regularization1 of

*ϕ*(respectively,

*a*and

*f*). Since

*g*≥ϕ in

*∂*

_{P}

*H*(

*T*), we havewhere

*λ*is the Lipschitz constant of

*ϕ*. Then, we consider the penalized problem(3.5)As a first step, we prove that a classical solution of (3.5) exits.

*Assume H1–H3*. *Let* *and let h*=*h*(*z*, *u*) *be a Lipschitz continuous function on* . *Then, there exists a classical solution* *of problem**Moreover, there exists a positive constant c, depending only on h and H(T), such that*(3.6)

We use a monotone iterative method. We setwhere *c* is a positive constant, such that for . Then, we recursively define the sequence by(3.7)where *λ* is the Lipschitz constant of *h*. Let us recall that the linear problem (3.7) has a unique classical solution , *α*∈]0,1], by theorem 4.1 in Di Francesco & Polidoro (2006).

Next, we prove by induction that (*u*_{j}) is a decreasing sequence. By the maximum principle, we have *u*_{1}≤*u*_{0}: indeed,and *u*_{1}≤*u*_{0} on *∂*_{P}*H*(*T*). Now, for fixed *j*∈, we assume the inductive hypothesis *u*_{j}≤*u*_{j−1}; then, recalling that *λ* is the Lipschitz constant of *h*, we haveMoreover, *u*_{j+1}=*u*_{j} on *∂*_{P}*H*(*T*), so that the maximum principle implies *u*_{j+1}≤*u*_{j}. The same argument shows that *u*_{j} is bounded from below by −*u*_{0}. In conclusion, for *j*∈, we have(3.8)Let us denote by *u* the pointwise limit of (*u*_{j}) in . Since *u*_{j} is a solution of (3.7) and by the uniform estimate (3.8), we can apply theorems 2.1 and 2.2 to conclude that, for any compact subset *O* of *H*(*T*) and *α*∈]0,1[, is bounded by a constant, depending only on *L*, *H*(*T*), *O*, *α* and *λ*. Hence, by the Schauder interior estimate (2.5), we deduce that is bounded uniformly in *j*∈. It follows that admits a subsequence (denoted by itself) that locally converges in . Thus, passing at the limit in (3.7) as *j*→∞, we haveand .

In order to prove that , we use the standard argument of barrier functions. For fixed and *ϵ*>0, let *V* be an open neighbourhood of *ζ*, such thatand a barrier function *w* as in *H3* is defined. We setwhere *k*_{ϵ} is a suitably large positive constant, independent of *j*, such thatsince *L*^{δ}*w* uniformly converges to *Lw* as *δ*→0, and *u*_{j}≤*v*^{+} on *∂*(*V*∩*H*(*T*)). The maximum principle yields *u*_{j}≤*v*^{+} on *V*∩*H*(*T*); analogously, we have *u*_{j}≥*v*^{−} on *V*∩*H*(*T*) and letting *j*→∞, we getThen,which proves the thesis since *ϵ* is arbitrary. Eventually, bound (3.6) is a direct consequence of the maximum principle and (3.8). ▪

By theorem 3.3 withthe penalized problem (3.5) has a classical solution . In the sequel, we assume *a*≤0: up to the standard transformation , this is not restrictive. We first show that(3.9)for a constant independent of *ϵ* and *δ*. Since , we have to prove only the estimate from below. Let us denote by *ζ* a minimum point of the function and assume , since otherwise there is nothing to prove. If , then

If , since *β*_{ϵ} is an increasing function, also assumes the (negative) minimum at *ζ*: then,(3.10)Now, by *H4*, is bounded from below by a constant independent of *δ*. Therefore, by (3.10), we havewith independent of *ϵ* and *δ*. This concludes the proof of (3.9).

By the maximum principle, we have(3.11)Therefore, using the interior estimates, (3.9) and (3.11), we infer that, for every and *p*≥1, the norm is bounded uniformly in *ϵ* and *δ*. It follows that the converges as *ϵ*, *δ*→0, weakly in (and in ) on compact subsets of *H*(*T*) to a function *u*. Moreover,so that *Lu*+*au*≤*f* a.e. in *H*(*T*). On the other hand, *Lu*+*au*=*f* a.e. in the set .

Finally, repeating the argument based on barrier functions at the end of the proof of theorem 3.3, we conclude that and *u*=*g* on . ▪

We close this section by proving a comparison result.

*Let u be a strong solution of* (*3.1*) *and* , *such that**Then, u≤v in H(T). In particular, the solution to* *(3.1)* *is unique*.

By contradiction, suppose that the open setis non-empty. Then, since *u*>≥*ϕ* in *D*, we haveand *u*=*v* on *∂D*. Then, the maximum principle implies *u*≥*v* in *D* and we get a contradiction. ▪

## 4. Obstacle problem on unbounded domains

In this section, we prove the existence of a unique strong solution to the obstacle problem(4.1)We say that is a (strong) super-solution of problem (4.1) if and(4.2)and that is a (strong) sub-solution if the conditions (4.2) hold with the inequalities reversed. Finally, *u* is a strong solution of problem (4.1), if *u* is super- and sub-solution. We assume

*H5*. *ϕ* is a locally Lipschitz continuous function on , such that, for every convex and compact subset *M* of , the convexity condition (3.3) holds with real constant *C* dependent on *M*.

Our main result is as follows.

*Assume H1, H2, H5 and let* , *with a*≤*a*_{0} *for some a*_{0}∈ *and* , *such that* . *If there exists a strong super-solution* *of problem* (*4.1*), *then there also exists a strong solution u of* (*4.1*), *such that* *in* .

The existence of a super-solution is ensured, for instance, if *g* and *ϕ* are bounded functions and *f*≥0. In this case, we can simply set .

We prove the theorem by solving a sequence of obstacle problems on the regular cylinders defined in (3.2). For every , we consider a cut-off function , such that *Χ*_{n}(*x*)=1 if and *Χ*_{n}(*x*)=0 if , and set

By theorem 3.2, for every , there exists a strong solution *u*_{n} ofBy proposition 3.4, it is straightforward to prove thatIn order to conclude, it is sufficient to use the same arguments as in the proofs of theorems 3.2 and 4.1, based on the *a priori* interior estimates and the barrier functions. ▪

## 5. Viscosity solutions

In this section, we prove that any strong solution to (4.1) solves the same problem in the viscosity sense as well. This is almost standard to verify using the well-known fact that viscosity solutions pass to the limit under uniform convergence. Adopting the notations of the User's Guide (Crandall *et al*. 1992), we setfor (which stands for the gradient in with respect to the variables (*x*, *t*)), *X* symmetric (*N*+1)×(*N*+1) matrix andWe denote by *u*_{δ} the classical solution of the regularized and penalized equation(5.1)subject to the initial condition in ^{N}. Moreover, let *F*^{δ} be the operator formally defined as *F*, with *A*, *a* and *f*, respectively, replaced by *A*^{δ}, *a*^{δ} and *f*^{δ}.

Then, *u*_{δ} is a viscosity solution of (5.1), i.e. *u*_{δ} is continuous and it is a sub- and super-solution of the equation in the sense that(5.2)and(5.3)In (5.2) and (5.3), and denote, respectively, the *second-order super- and sub-jet* of *u* at *z* defined byand .

For what follows, we need the following result which is contained in Crandall *et al*. (1992), proposition 4.3.

*Let* , , and . *Suppose that* (*u*_{δ}) *is a family of continuous functions, uniformly convergent as δ*→0 *to u in a neighbourhood of z. Then, there exist sequences* (*δ*_{n}) *in* , (*z*_{n}) *in* , *and* , *such that*

*Any strong solution of* (*4.1*) *is also a viscosity solution*.

Since and *u*≥*ϕ*, it suffices to show that

*Lu*≤*f*on in the viscosity sense, that is(5.4)*Lu*=*f*in the viscosity sense on .

To this end, we consider a sequence of solutions to the regularized and penalized problem, locally uniformly convergent to *u*. For fixed and , we consider a sequence as in lemma 5.1. Then, we haveand, by (5.1),and this proves (5.4).

Analogously, for fixed , such that and , by lemma 5.1 we may select a sequence , such that and then we conclude ▪

## Footnotes

↵We may suitably extend

*ϕ*,*a*and*f*by continuity in a neighbourhood of*H*(*T*).- Received June 19, 2007.
- Accepted September 25, 2007.

- © 2007 The Royal Society