The Heston stochastic volatility model has a boundary trace at zero volatility

We establish boundary regularity results in H\"older spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) $\mathbb{H} = \mathbb{R}\times (0,\infty)\subset \mathbb{R}^2$. Starting with nonsmooth initial data $u_0\in H$, we take advantage of smoothing properties of the parabolic semigroup $\mathrm{e}^{-t\mathcal{A}}\colon H\to H$, $t\in \mathbb{R}_+$, generated by the Heston model, to derive the smoothness of the solution $u(t) = \mathrm{e}^{-t\mathcal{A}} u_0$ for all $t>0$. The existence and uniqueness of a weak solution is obtained in a Hilbert space $H = L^2(\mathbb{H};\mathfrak{w})$ with very weak growth restrictions at infinity and on the boundary $\partial\mathbb{H} = \mathbb{R}\times \{ 0\}\subset \mathbb{R}^2$ of the half-plane $\mathbb{H}$. We investigate the influence of the boundary behavior of the initial data $u_0\in H$ on the boundary behavior of $u(t)$ for $t>0$.


Introduction
The Heston stochastic volatility model for pricing the European call options on stocks (S. L. Heston [25]) has been of considerable interest to economists and mathematicians for almost three decades. Numerous articles have been written about mathematical treatment and solvability of this model in a number of settings. In our present work we focus on the degenerate parabolic problem with prescribed initial and boundary conditions. The question of existence, uniqueness, and regularity of a weak solution to this problem is studied in P. M. N. Feehan and C. A. Pop [15], C. Chiarella, B. Kang, and G. H. Meyer [7], G. H. Meyer [31], and B. Alziary and P. Takáč [3,Sect. 4,, to mention only a few. The analyticity of the solution in both, space and time variables, has been established in [3,Sect. 4, Theorem 4.2, pp. [16][17]. As a consequence, the completeness of the market (cf. T. Björk [5,Sect. 8, and M. H. A. Davis and J. Ob lój [8]) described in Heston's model is verified in [3,Sect. 5,Theorem 5.2,p. 19]. Thanks to the importance of Heston's model in Mathematical Finance, there is a strong interest in efficient numerical methods applicable to computing the solution of this degenerate parabolic problem ( [7,31]). A major obstacle to an efficient numerical method is the degeneracy of the diffusion coefficient at low volatility; see e.g. B. Düring and M. Fournié [12] and S. Ikonen and J. Toivanen [26]. This degeneracy causes serious problems in formulating and justifying the correct boundary conditions on the portion of the boundary with vanishing volatility, denoted by ∂H. A numerical scheme using a finite difference method in the domain H = R × (0, ∞) ⊂ R 2 with the boundary ∂H = R × {0} has to be designed with a mesh of points much too fine near the boundary ∂H, so that it finally becomes rather inefficient and unprecise there. This is one of the reasons why in this article we investigate the limiting boundary behavior of the solution of Heston's model as the volatility approaches zero. We obtain a limiting partial differential equation of first order on the boundary ∂H, Eq. (4.4), thus specifying also the boundary conditions on ∂H. It is worth of noticing that this equation on the space-time domain ∂H × (0, ∞) is coupled with the degenerate parabolic equation (3.9) inside the domain H × (0, ∞) solely through a linear term with the partial derivative with respect to the volatility (the volatility approaching zero) that appears in Eq. (4.5). This feature of Heston's model is used in the recent work by F. Baustian, K. Filipová, and J. Pospíšil [4] with an orthogonal polynomial expansion in the spatial domain H. Orthogonal polynomial expansions have been used recently also in D. Ackerer and D. Filipović [1] for numerical approximations. Earlier, the authors [3,Sect. 11, have used orthogonal polynomial expansions with Hermite and Laguerre polynomials in Galërkin's method to approximate functions in L 2 (H) by analytic functions.
Our derivation of Eq. (4.4) on ∂H × (0, ∞) is motivated by the limiting behavior of the diffusion part (second-order partial derivatives) in Eq. (3.9). The limit, equal to zero on ∂H × (0, ∞), has been obtained in P. M. N. Feehan and C. A. Pop [15], Lemma 3.1, Eq. (3.1), on p. 4409 (see also P. Daskalopoulos and R. Hamilton [11], Prop. I.12.1 on p. 940) for the corresponding (stationary) elliptic problem with the Heston operator A given by Eq. (3.1). However, in order to fulfill the regularity hypothesis required in [15, Lemma 3.1], we need to establish a new regularity result for the weak solution u( · , · , t) : [0, ∞)] → H of the Heston model (see Proposition 4.1 with f ≡ 0) which is given by the C 0 -semigroup of bounded linear operators e −tA : H → H, t ∈ R + , determined by the homogeneous initial value problem (3.9), that is to say, u( · , · , t) ≡ u(t) = e −tA u 0 ∈ H, t ∈ R + , with an arbitrary initial value u 0 ∈ H. The underlying Hilbert space H is a weighted L 2 -type Lebesgue space H = L 2 (H; w). Our regularity result is based on the smoothing property of the holomorphic semigroup e −tA , t ∈ R + , acting on H, see Theorem 4.2. This result contains a number of local and global partial regularity results which are new, as well. We stress the main difference between the classical Hölder-type regularity treated in P. M. N. Feehan and C. A. Pop [15, Theorem 1.1 on p. 4409] and the regularity obtained by parabolic smoothing: The Hölder-type regularity in [15] assumes the same spatial regularity already for the initial value u(0) = u 0 (in a suitable weighted Hölder space). As a consequence, analogous regularity for the solution u(t) is proved (by Schauder estimates) at all times t ∈ (0, T ) in a bounded time interval. In contrast, we begin with nonsmooth initial data u 0 ∈ H at t = 0; then we apply the parabolic smoothing of the C 0 -semigroup e −tA for t ∈ (0, ∞), thus arriving at u(t) ∈ D(A k ) ⊂ H for all t ∈ (0, ∞) and every k = 1, 2, 3, . . . . Since the domain D(A k ) of the k-th power of the Heston operator A is the image (range) of the k-th power of the bounded inverse (λI + A) −1 : H → H (the resolvent of −A), the solution u(t) = e −tA u 0 has higher smoothness for all t > 0. This smoothing effect is essential for applications in Mathematical Finance where the initial data u 0 ∈ H are typically not continuously differentiable (u 0 ∈ W 1,∞ (H)\C 1 (H)). Indeed, for our derivation of the limiting equation The proof of our main result, Theorem 4.2, makes use of the factorization (λI + A) −j (λI + A) k e −tA of the bounded linear operator e −tA : H → H for t > 0; with k = 1, 2, 3 and j = 0, 1, . . . , k. Thanks to the smoothing effect, the latter factor, (λI + A) k e −tA , is a bounded linear operator on H for each t > 0, whereas the former factor, (λI + A) −j , is a bounded linear operator from H to the domain D(A j ) of the j-th power of the Heston operator A. We use the resolvent (λI + A) −1 : H → H of −A in order to describe the function space D(A j ) (endowed with the graph norm) for 1 ≤ j ≤ k ≤ 3. This factorization (in Section 6) is split into three consecutive steps in Paragraphs §6.1, §6.2, and §6.3, with the auxiliary functions f j,k ( · , · , t) ≡ f j,k (t) ∈ H for 0 ≤ j ≤ k ≤ 3 defined in eqs. (6.1) and (6.2) for a given u 0 ∈ H and t > 0. Clearly, for j = k; k = 1, 2, 3, and t > 0 we obtain f k,k (t) = u(t) which yields the desired regularity of the solution u(t) for t > 0 as stated in Theorem 4.2.
Our second theorem (Theorem 4.4) is a weak maximum principle for the initial value Cauchy problem (3.9) in the unbounded space-time domain H × (0, T ). As it is typical for parabolic problems posed in an unbounded spatial domain (the open half-plane H ⊂ R 2 in our case), the growth of the solution u(t) ≡ u(x, ξ, t) has to be limited with respect to the space variable (x, ξ) ∈ H as x → ±∞ or ξ → 0+ or ξ → +∞, uniformly for all t ∈ (0, T ). We find a positive "majorizing" function h 0 : H → (0, ∞) in Eq. (4.7) that provides the required limit on the solution u(x, ξ, t) in Theorem 4.4. This theorem has an important corollary applicable to a typical initial value problem in Mathematical Finance (see Corollary 4.5). The majorizing function U (x, ξ) def = K 1 e x+̟ξ + K 0 , for all (x, ξ) ∈ H, provides an important upper bound (independent from time) as the volatility ξ ∈ (0, ∞) approaches zero (ξ → 0+). Here, K 0 , K 1 ∈ R + are arbitrary constants, and ̟ ∈ R is another constant restricted by inequalities in (4.13). This choice of the majorizing function and the initial data covers the most typical alternatives for derivative contracts (which are determined by the choice of the initial data u 0 ); see e.g. J.-P. Fouque, G. Papanicolaou, and K. R. Sircar [20, §1.2, pp. [8][9][10][11][12]. The case of u 0 (x, ξ) ≡ u 0 (x) being independent from the volatility ξ ∈ (0, ∞) is of special interest (e.g., European call and put options); we may set ̟ = 0. Derivative contracts do not seem to include the volatility level since volatility does not produce any direct returns such as dividends or interest. Volatility does not show long term upwards trends like equities, but typically shows periods of high volatility occurring within a short period of time (i.e. volatility "jumps") and then shows a downward trend to return to the long run medium level.
This article is organized as follows. We begin with basic notations and function spaces of Hölder, Lebesgue, and Sobolev types, which involve weights. Most of these spaces were originally introduced in P. Daskalopoulos and P. M. N. Feehan [9] and [10, Sect. 2, p. 5048] and P. M. N. Feehan and C. A. Pop [17]. The mathematical problem resulting from S. L. Heston's [25] model in Mathematical Finance (described in Appendix A in "economic" terms) is formulated in Section 3. The details of this formulation, especially a justification of the boundary conditions and restrictions imposed on some important constants (e.g., the volatility σ > 0 of the volatility, the rate of mean reversion κ > 0, and the long-term variance θ > 0), such as the well-known Feller condition, can be found in our previous work [3, Sect. 2, pp. [6][7][8][9][10][11][12][13]. Our main results are collected in Section 4, in Theorems 4.2 and 4.4. In addition, also Proposition 4.1 (existence and uniqueness), Corollary 4.3 (boundary behavior), and Corollary 4.5 (maximum principle) are of importance. Our strategy of the proof of Theorem 4.2 (laid out above) is described in all details in Section 5. The first part of this strategy, obtaining Hölder regularity, is implemented in Section 6. The proof of Theorem 4.2 (and that of Corollary 4.3, as well) is completed in Section 7. The main part of this article ends up with the proofs of Theorem 4.4 and Corollary 4.5 in Section 8. We have postponed some rather technical results about weighted Sobolev spaces and boundary traces until Appendix B. Most of our regularity results gradually derived in Section 5 take advantage of difficult elliptic Schauder-type estimates for the degenerate Heston operator A in weighted Hölder spaces over the half-plane H obtained in a series of articles by P. M. N. Feehan and C. A. Pop [16,17,18]. For reader's convenience, we restate these results in Appendix C.

Basic notations, function spaces
We use the standard notation R = (−∞, +∞), R + = [0, ∞) ⊂ R and H = R × (0, ∞) ⊂ R 2 with the closure H = R × R + for the open and closed upper half-planes, respectively. As usual, for x ∈ R we abbreviate x + def = max{x, 0} and x − def = max{−x, 0}. The complex plane is denoted by C = R + iR. The complex conjugate of a number z ∈ C is denoted byz, so that the absolute value of z is given by |z| = (zz) 1/2 . The basic function space, H, in our treatment of the Heston model is defined as follows: We define the weight w : H → (0, ∞) by where β, γ, µ ∈ (0, ∞) are suitable positive constants that will be specified later, in Section 3 (see also Appendix B). However, it is already clear that if we want that the weight w(x, ξ) tends to zero as ξ → 0+, we have to assume β > 1. Similarly, if we want that the function u 0 (x, ξ) = K(e x − 1) + of (x, ξ) ∈ H (an initial condition in Heston's model) belongs to H, we must require γ > 2. Then H = L 2 (H; w) is the complex Hilbert space of all complex-valued Lebesgue-measurable functions f : H → C with the finite norm This norm is induced by the inner product The domain, V , of the sesquilinear form that defines the Heston operator is the weighted Sobolev space of all functions f ∈ H, such that the first-order partial derivatives (in the sense In particular, the Sobolev imbedding V ֒→ H is bounded (i.e., continuous).
We will see later that the domain of the Heston operator is contained in a local version of the following weighted Sobolev space, H 2 (H; w), of all functions f ∈ V , such that also the second-order partial derivatives (in the sense of distributions), In addition, we require that the Hilbert norm of f on H 2 (H; w), as defined below, is finite, It easy to see that the Sobolev imbeddings H 2 (H; w) ֒→ V = H 1 (H; w) ֒→ H = L 2 (H; w) are bounded (i.e., continuous).
We will make use of the following local version of the weighted Sobolev space H 2 (H; w): Let B R (x 0 , ξ 0 ) denote the open disc in R 2 with radius R > 0 centered at the point (x 0 , ξ 0 ) ∈ R 2 . If ξ 0 = 0, we define also the open upper half-disc Its closure in R 2 (hence, also in H) is denoted by We denote by H 2 (B + R (x 0 , 0); w) the weighted Sobolev space of all functions f ∈ W 2,2 loc (B + R (x 0 , 0)) whose norm defined below is finite, The half-disc B + R (x 0 , 0) being bounded in H, this norm on H 2 (B + R (x 0 , 0); w) is equivalent with the following simpler norm defined by We will employ the weighted Sobolev space H 2 (B + R (x 0 , 0); w) in Section 6. The weighted Sobolev space H 2 (B + R (x 0 , 0); w) will be imbedded into the weighted L p - Finally, the local Schauder-type regularity results near the boundary ∂H = R×{0} = H\H of the half-plane H established in Section 6 will be stated in the Hölder spaces C α s (B + R (x 0 , 0)) and C 2+α s (B + R (x 0 , 0)) over any compact half-disc B + R (x 0 , 0) with x 0 ∈ R and R ∈ (0, ∞). The Hölder norm in these spaces corresponds to the so-called cycloidal Riemannian metric s on H defined by ds 2 = ξ −1 (dx 2 + dξ 2 ). The associated cycloidal distance function on H, denoted by s cycl (P 1 , P 2 ) for two different points P i = (x i , ξ i ) ∈ H; i = 1, 2, is given by Of course, the expression |P 1 − P 1 | = |(x 1 , ξ 1 ) − (x 2 , ξ 2 )| stands for the Euclidean distance on R 2 . We will use the following equivalent metric on H introduced in H. Koch [27, p. 11], As usual, C(B + R (x 0 , 0)) denotes the Banach space of all continuous functions f : Given α ∈ (0, 1), we denote by C α The norm on this vector space is defined by We denote by C 2+α s (B + R (x 0 , 0)) its vector subspace consisting of all functions f ∈ C α s (B + R (x 0 , 0)) that are twice continuously differentiable in the open half-disc B + R (x 0 , 0) and satisfy We endow C 2+α R (x 0 ,0)) defined above. It is proved in P. M. N. Feehan and C. A. Pop [15], Lemma 3.1, Eq. (3.1), on p. 4409 (see also P. Daskalopoulos and R. Hamilton [11], Prop. I.12.1 on p. 940) that at every point stands for the Hessian matrix of f in B + R (x 0 , 0) that consists of all second-order partial derivatives of f . This means that for any function f ∈ C 2 (B + R (x 0 , 0)) the weighted Hölder norm f C 2+α s (B + R (x 0 ,0)) < ∞ forces the zero limit (2.7) which thus may be regarded as an imposed homogeneous boundary condition.

Formulation of the mathematical problem
In this section we briefly describe S. L. Heston's model [25,Sect. 1, and formulate the associated Cauchy problem as an evolutionary equation of (degenerate) parabolic type. A brief description of the "economic" model is provided in Appendix A. The reader is referred to our earlier work in B. Alziary and P. Takáč [3, Sect. 2, pp. 6-13] for a more detailed analytical treatment of Heston's model.

Heston's stochastic volatility model
We consider the Heston model given under a risk neutral measure via equations (1) − (4) in [25, pp. 328-329]. The model is defined on a filtered probability space (Ω, F, (F t ) t 0 , P), where P is a risk neutral probability measure, and the filtration (F t ) t 0 satisfies the usual conditions. After a series of standard arguments based onÎto's formula, a (terminal value) Cauchy problem for the price of a European call or put option is obtained (see [3,Eq.   Here, by r − q ≡ −q r ∈ R we have abbreviated the instantaneous drift of the stock price returns with −∞ < r ≤ q < ∞, and by θ σ ≡ θ/σ > 0 the re-scaled long term (or long-run) variance with θ, σ ∈ (0, ∞). The correlation coefficient ρ satisfies ρ ∈ (−1, 1 We make use of the Gel'fand triple V ֒→ H = H ′ ֒→ V ′ , i.e., we first identify the Hilbert space H with its dual space H ′ , by the Riesz representation theorem, then use the imbedding V ֒→ H, which is dense and continuous, to construct its adjoint mapping H ′ ֒→ V ′ , a dense and continuous imbedding of H ′ into the dual space V ′ of V as well. The (complex) inner product on H induces a sesquilinear duality between V and V ′ ; we keep the notation ( · , · ) H also for this duality. Now we define the linear operator A : V → V ′ by the sesquilinear form (cf. [3,Eq. (2.21), p. 11]), for all u, w ∈ V , All integrals on the right-hand side converge absolutely for any pair u, w ∈ V (by the proof of Prop. 6.1 in [3, pp. 21-23]).
In order to derive the right-hand side of Eq.
As we have just chosen a particular realization A : V → V ′ of the formal differential expression (3.1) defined by Eq. (3.2), we no longer need to impose the boundary conditions (3.7) and (3.8).

The Cauchy problem in the weighted L 2 -space H
The initial value Cauchy problem for the Heston model mentioned in the previous paragraph ( §3.1) takes the following abstract form in the Hilbert space H = L 2 (H; w): with the function f (x, ξ, t) ≡ 0 on the right-hand side and the initial data  (ii) the initial value u(0) = u 0 in H; (iii) the mapping t → u(t) : (0, T ) → V is a Bôchner square-integrable function, i.e., u ∈ L 2 ((0, T ) → V ); and (iv) for every function the following equation holds, Case T = +∞. Let f ∈ L 2 loc ((0, ∞) → V ′ ) (i.e., f ∈ L 2 ((0, T 0 ) → V ′ ) for every 0 < T 0 < ∞) and let u 0 ∈ H. A function u : H × [0, ∞) → R is called a weak solution to the initial value problem (3.9) with T = +∞, if it is a weak solution to the initial value problem (3.9) on every bounded time subinterval [0, T 0 ) ⊂ R + with 0 < T 0 < T = +∞, according to Case 0 < T < ∞ above.
The following remarks are in order: First, our definition of a weak solution is equivalent with that given in L. C. Evans [13, §7.1], p. 352. Here, for 0 < T < ∞, The norm is defined in the usual way; cf. L. C. Evans [13, §5.9]. The properties of V ≡ H 1 (H; w) justify the notation V ′ = H −1 (H; w). The continuity of the imbedding is proved, e.g., in L. C. Evans [13, §5.9], Theorem 3 on p. 287. We will see in Section 4 that the initial value problem (3.9) has a unique weak solution u : From now on, we use exclusively formula (3.2) to define the linear operator A : V → V ′ . This means that we no longer need the boundary conditions in (3.7) and (3.8) imposed on u ∈ V .
Remark 3.2 (Coercivity conditions.) It is important to remark at this stage of our investigation of the Heston operator A that, in order to ensure the coercivity of A + c I on V , one has to assume the well-known Feller condition ( [19,23]), However, Feller's condition (3.10) is not sufficient for obtaining the desired coercivity. We need to guarantee also cf. Ineq. (6.15) in B. Alziary and P. Takáč [3], proof of Prop. 6.2, pp. 23-27. That is, we need to assume the following coercivity condition: The last inequality is an additional condition to Feller's condition, 1 2 σ 2 − κθ < 0, both of them requiring the rate of mean reversion κ > 0 of the stochastic volatility in Heston's model to be sufficiently large. This additional condition is caused by the fact that W. Feller [19] considers only an analogous problem in one space dimension (ξ ∈ R + ), so that the solution u = u(ξ) is independent from x ∈ R. In particular, if the initial value u 0 = u( · , · , 0) ∈ H for u(x, ξ, t) permits us to take γ > 0 arbitrarily small, then inequality (3.11) is easily satisfied, provided Feller's condition 1 2 σ 2 − κθ < 0 is satisfied. This is the case for a European put option with the initial condition u 0 (x, ξ) = K (1 − e x ) + ( ≤ K) for (x, ξ) ∈ H. However, if we wish to accommodate also initial values of type u 0 (x, ξ) = K (e x − 1) + for (x, ξ) ∈ H, attached to a European call option, then we are forced to take γ > 2 to ensure that u 0 ∈ H.

Main results
Next, let us choose β ∈ R such that , and u 0 ∈ H be arbitrary. Then the initial value problem (3.9) (with u 0 ∈ H) possesses a unique weak solution in the sense of Definition 3.1. Moreover, this solution satisfies also u ∈ W 1,2 ((0, T ) → V ′ ) and there exists a constant C ≡ C(T ) ∈ (0, ∞), independent from f and u 0 , such that , and u 0 ∈ H, the same existence and uniqueness result (in the sense of Definition 3.1) is valid with H × (0, T ) → R obtained in Proposition 4.1 above for the special case f ≡ 0 in H × (0, T ). We formulate these regularity results using the C 0 -semigroup representation of the (unique) weak solution u( · , · , t) ≡ u(t) = e −tA u 0 ∈ H, t ∈ R + , to the homogeneous initial value problem (3.9) (with f ≡ 0), where we allow any 0 < T ≤ +∞ and an arbitrary initial value u 0 ∈ H. By the well-known properties of C 0 -semigroups, λI is a sufficiently large number (called the growth bound) determined by the well-known inequality (5.1) (in Section 5).
The new result in this theorem is a local Schauder-type regularity result near the boundary Theorem 4.2 (Local and global regularity.) Let ρ, σ, θ, q r , and γ be given constants in R, ρ ∈ (−1, 1), σ > 0, θ > 0, and γ > 0. Assume that γ, κ, and µ are chosen as specified in Proposition 4.1 above and u 0 ∈ H is arbitrary. Finally, in addition to Ineq. (4.2), choose β such that also β(β − 1) < 4, i.e., respectively. Then we have the following four statements for the weak solution u : (0, ∞) → H obtained in Proposition 4.1: (iii) Given 0 < T ≤ +∞ and any x 0 ∈ R, there are a radius R ∈ (0, ∞) and constants is a bounded linear operator with the operator norm (iv) Moreover, in the situation of Part (iii) above, the mapping is continuous and differentiable, with respectively, for all t ∈ (0, T ) and for all τ ∈ (0, ∞) such that t + τ < T . Here, c 1 , c ′ 1 ∈ (0, ∞) are some other constants independent from t and u 0 ∈ H.
Our proof of this theorem will be built up gradually in the next two sections (Sections 5 and 6) and completed in Section 7.
We stress that the constants c 0 , c ′ 0 ∈ (0, ∞) in Part (iii) do not depend on the choice of u 0 ∈ H or t ∈ (0, T ). However, the weighted norm on H depends on the weight function w(x, ξ) which is not translation invariant with respect to x ∈ R. This property of w means that the constants c 0 , c ′ 0 ∈ (0, ∞) may depend on x 0 ∈ R. We will see in the course of the proof of Part (iii) (in Section 6) that these constants are rendered independent from the length of the time interval, (0, T ), 0 < T ≤ +∞, thanks to the multiplicative exponential factor e λ 0 t . The constant λ 0 ∈ (0, ∞) is determined solely by Ineq. (5.1) (in Section 5). In particular, we obtain . Concerning the behavior of the weak solution u(x, ξ, t) to the Cauchy problem (3.9) in H × (0, T ) near the boundary ∂H × (0, T ), Part (iii) of Theorem 4.2 has the following important consequence. Corollary 4.3 (Boundary behavior.) Let 0 < T ≤ +∞ and t 0 ∈ (0, T ). Under the hypotheses of Theorem 4.2, we have u( · , · , t) ≡ u(t) ∈ C 1 (H) for every t ∈ (0, T ). Furthermore, the function u(x, ξ, t) verifies the following initial value Cauchy problem on ∂H × (t 0 , T ), Here, we have denoted . This transport equation for the unknown function u(x, 0, t) has a unique classical solution given by If, in addition, u ∈ C 0 (H × [0, T )), then we may take t 0 = 0 above, in Eqs. (4.4) and (4.5).
This corollary will be proved in Section 7.
Our second theorem is a weak maximum principle which, in turn, implies a pointwise bound on the weak solution u : H × (0, T ) → R obtained in Proposition 4.1 above. We begin with some auxiliary notation: First, whenever 0 < T ≤ +∞, let us denote by C 0 (H × [0, T )) the vector space of all continuous functions u : H × [0, T ) → R and by C 2,1 (H × (0, T )) the vector space consisting of all continuous functions u : H × (0, T ) → R that are continuously differentiable in H × (0, T ) and also twice continuously differentiable with respect to the space variables ( Second, let γ 0 ∈ (0, ∞) be an arbitrary constant, as large as needed. Assuming Feller's condition (3.10), i.e., σ 2 < 2κθ, we allow any constants β 0 , µ 0 ∈ (0, ∞) such that These two inequalities are motivated by conditions (8.4) and (8.6), respectively, in the proof of the theorem below. Notice that there is no upper bound on the constant µ 0 .
Third, define a "majorizing" function h 0 : H → (0, ∞) by Now we are ready to state our weak maximum principle. This is an a priori result for any strong subsolution u to the parabolic Cauchy problem (4.8), (4.9), and (4.10) as described below. As a consequence, we do not need to assume hypothesis (3.11) or (4.2) (cf. Proposition 4.1).
Our weak maximum principle in Theorem 4.4 differs from that in P. M. N. Feehan and C. A. Pop [15], Lemma 3.4 on p. 4416. Their conditions [15,Eq. (3.29)] imposed on the Heston operator A are weaker than ours. We assume that the constants σ, κ, θ ∈ (0, ∞) satisfy the Feller condition (3.10). On the other hand, we do not need that the functions u, u t , u x , u ξ , and ξ u xx , ξ u xξ , ξ u ξξ be continuous up to the boundary ∂H of the half-plane H = R × (0, ∞); cf. [15,Eq. (3.30)]. Neither do we need the boundary condition in [15,Eq. (3.31)]. In fact, we will show that this boundary condition is satisfied also by our solutions to the Heston problem by combining our growth hypothesis (4.9) with Lemma 3.1 in [15, Eq. (3.1), p. 4409].
We remark that the condition in (4.12) is satisfied for the initial value u 0 : H → R defined by u 0 (x, ξ) = K (e x − 1) + , for (x, ξ) ∈ H (the European call option). One may set ̟ = 0 together with K 0 = 0 and K 1 = 1.

Some smoothing properties of the Heston semigroup
This section is concerned with some standard properties of the C 0 -semigroup e −tA (t ∈ R + ) of bounded linear operators e −tA : H → H on the complex Hilbert space H. This semigroup has been already mentioned in Section 4, Proposition 4.1, in connection with Theorem 4.2. It is shown in B. Alziary and P. Takáč [3], Prop. 6.1 (p. 21) and Prop. 6.2 (p. 23), respectively, that under conditions (3.10) and (3.11) the sesquilinear form We recall from Section 3, §3.2, that the (autonomous linear) operator A : V → V ′ , defined by this sesquilinear form, is bounded, by the Lax-Milgram theorem. The (unique) weak solution u(x, ξ, t) to the Cauchy problem (3.9) with f ≡ 0 in H × (0, T ) and an arbitrary initial value To be more precise, we denote by 2) for every k = 1, 2, 3, . . . from the case k = 1.
Finally, the factorization for all t > 0 .
Proof. The estimate in Ineq. (5.4) is obtained by applying (5.2) to the right-hand side of (5.3).

Smoothing properties in Hölder spaces
We apply Lemma 5.1 step by step for k = 1, 2, 3. We define the auxiliary functions f j,k (x, ξ, t) for 0 ≤ j ≤ k ≤ 3 as follows: First, for any time t > 0 we set Next, for t > 0 we introduce Clearly, for j = k; k = 1, 2, 3, and t > 0 we obtain f k,k (t) = u(t).
We conclude that, for every t > 0, 0); w is a bounded linear operator with the operator norm bounded above by C 1,1 t −1 + C 1,0 e λ 0 t .

Completion of the proof of the main regularity result
In this section we finish the proof of our main regularity result, Theorem 4.2, started in the two previous sections, Section 5 and Section 6, and prove also its Corollary 4.3.
Proof of Theorem 4.2. The regularity statement in Part (i) follows directly from the results in Section 5, Ineq. (5.2). The C ∞ -regularity in Part (ii) is a (local) interior regularity result for (local) weak solutions to a locally strictly parabolic equation established (in a more general setting) in A. Friedman [21, Chapt. 10, Sect. 4], Theorem 11 (p. 302) and its Corollary (p. 303). The complete proof of Part (iii) has been given in Section 6. The radius R ∈ (0, ∞) stands for the radius R ′ 2 ∈ (0, ∞) that appears in Eq. (6.12). Finally, we derive Part (iv) from Part (iii) as follows. The continuity and differentiability of the mapping t → u(t)| B + R (x 0 ,0) from (0, T ) to the Hölder space C 2+α s (B + R (x 0 , 0)) follow from the respective formulas for all t ∈ (0, T ) and for all τ ∈ (0, ∞) such that t + τ < T , combined with the locally uniform upper bound on the operator norm of the bounded linear operator U B + R (x 0 ,0) (t). Whereas the norm in the Hölder space C 2+α s (B + R (x 0 , 0)) of the expression in Eq. (7.1) above is estimated easily by the operator norm of U B + R (x 0 ,0) (t) from Part (iii), estimating the expression in Eq. (7.2) requires also the following estimate which follows from inequalities (5.1) and (5.2) (k = 1), The desired estimate for the norm of ∂u ∂t (t) is obtained from that for ∂u ∂t (t + τ ) in Eq. (7.2) by replacing both, t and τ , by the common value of t/2 which means that the sum t + τ is replaced by t ∈ (0, T ).

A maximum principle and growth at low and high volatilities
According to a classical result on the weak maximum principle for a uniformly parabolic Cauchy problem in R N × (0, T ), see e.g. A. Friedman [21, Chapt. 2, Sect. 4, Theorem 9, p. 43], the weak maximum principle is valid under "very weak" restrictions on the growth of a strong solution u(x, t) as |x| → ∞, (x, t) ∈ R N × (0, T ). Consequently, one may speak of practicaly no boundary conditions being imposed on the strong solution u(x, t) as |x| → ∞, at least in contrast with classical boundary conditions of Dirichlet, Neumann, or oblique derivative (Robin) types. Nevertheless, thanks to the weak maximum principle, the uniqueness of any strong solution to the Cauchy problem with prescribed initial data is still guaranteed. Now we are ready to prove our Theorem 4.4.
We will compare the function u : H × (0, T ) → R to the smooth function h defined as follows: We calculate the partial derivatives of h(x, ξ, t) at (x, ξ) ∈ H and 0 < t < T : Similarly, we calculate the second-order partial derivatives: We plug these partial derivatives of h into formula (3.1) to calculate where we recall θ σ = θ/σ and abbreviate Our assumption on β 0 ∈ (0, ∞) in (4.6) allows us to find a constant τ ∈ (0, 1) small enough, such that or, equivalently, .
From now on we restrict ourselves to the time interval 0 < t ≤ T ω def = τ /ω with ω > 0 to be determined as follows.
In order to obtain a weak maximum principle for a strong solution u : H × (0, T ) → R of the initial value problem (3.9), such that u(x, ξ, t) ≤ const · h(x, ξ, t) for all (x, ξ) ∈ H and def = u(x, ξ, t)/h(x, ξ, t) ≤ const < ∞. Using u = wh we have and similarly We plug these partial derivatives of u into formula (3.1) to calculate (8.10) We recall that the multiplicative coefficient at w(x, ξ, t) is ≤ −ν ≤ 0, by Ineq. (8.9).
These limits follow from inequalities (8.4) and (8.6) combined with ω ≥ τ /T . From the limits above we derive analogous results for the ratio In order to complete our proof, we recall the parabolic equation (8.10) for w with the right-hand side ≥ 0, by Ineq. (4.8), and the multiplicative coefficient ≤ −ν ≤ 0 at w(x, ξ, t), by Ineq. (8.9), or equivalently, for all (x, ξ, t) ∈ H × (0, T ), Taking advantage of the initial condition (4.10) at time t = 0, which is equivalent with w(x, ξ, 0) ≤ 0 for all (x, ξ) ∈ H, in addition to the boundary behavior (8.11) and (8.12), we may apply the weak maximum principle from A. Friedman

It remains to give
Proof of Corollary 4.5. Our strategy of the proof is to verify that the weak maximum principle in Theorem 4.4 can be applied to both functions W ± (x, ξ, t) = − U (x, ξ, t) ± u(x, ξ, t) for (x, ξ, t) ∈ H × [0, T ). In Theorem 4.4, we take T ∈ (0, ∞) arbitrarily large, but finite.
The first and second partial derivatives of U are We insert them into the Heston operator (3.1), A, The last inequality follows from r 0 , K 0 ∈ R + combined with our conditions on ̟ in (4.13). Finally, we combine this inequality with ∂u ∂t + Au = 0 in H × (0, T ) to derive the desired inequality (4.8) for W ± = − U ± u in place of u.
We finish our proof by applying the weak maximum principle (Theorem 4.4) to the functions W ± : H × [0, T ) → R which guarantees W ± ≤ 0 throughout H × [0, T ).

Discussion of the boundary conditions
It is not difficult to see, as we will show below, that at any time t ∈ (0, T ) the Cauchy problem for the Heston model ( §3.1 and Appendix A) imposes on the solution u( · , t) : R N → R the "boundary" behavior at infinity (as |x| → ∞) exhibited precisely by the initial value u( · , 0) = u 0 : R N → R. More specifically, this is the case for the European call and put options, u 0 (x, ξ) = K (e x − 1) + and u 0 (x, ξ) = K (1 − e x ) + , respectively, for (x, ξ) ∈ H; cf. Eq. (A.10) (for the European call option) and J.-P. Fouque, G. Papanicolaou, and K. R. Sircar [20, Fig. 1.2 (p. 17) and Fig. 1.3 (p. 18)] (for both, European call and put options, respectively). This means that, at least in the case of European call and put options, the boundary conditions for u(x, t) as |x| → ∞ are determined by the asymptotic behavior of the initial data u 0 (x) as |x| → ∞. Hence, if any boundary conditions at infinity (independent from time t ∈ (0, T )) are to be imposed on the strong solution to the Cauchy problem, they should be obeyed also by the initial data (at time t = 0). An apparent open question is if those boundary conditions (i.e., boundary behavior ) at infinity obeyed by the initial data u 0 (x) as |x| → ∞ are inherited by the (unique) solution for all times t ∈ (0, T ) and in what sense.
Neumann boundary conditions can be treated in a similar manner using the following formula derived from Eq. (9.2) by simple differentiation: However, caution must be payed to the "weighted" Neuman boundary conditions suggested by   Table 2.1, p. 42]. We focus our attention on S. L. Heston's model [25] which has attracted significant attention of a broad community of researchers from Finance and Mathematics. We consider this model under a risk neutral measure via equations (1) − (4) in [25, pp. 328-329]. The model is defined on a filtered probability space (Ω, F, (F t ) t 0 , P), where P is a risk neutral probability measure, and the filtration (F t ) t 0 satisfies the usual conditions. Denoting by S t the stock price and by V t the (stochastic) variance of the stock market at (the real) time t ≥ 0, the Heston model requires that the unknown pair (S t , V t ) t 0 satisfies the following system of stochastic differential equations, Here, (W t ) t 0 and (Z t ) t 0 are two Brownian motions with the correlation coefficient ρ ∈ (−1, 1), a constant given by d W, Z t = ρ dt. Furthermore, q r = q − r ∈ R and σ, κ, θ ∈ (0, ∞) are some given constants whose economic meaning is explained, e.g., in B. Alziary and P. Takáč If X t = ln(S t /K) denotes the (natural) logarithm of the scaled stock price S t /K at time t ≥ 0 relative to the strike price K > 0 at maturity T > 0, then the pair (X t , V t ) t 0 satisfies the following system of stochastic differential equations, Following [8,Sect. 4], let us consider a European call option written in this market with payoffĥ(S T , V T ) ≡ĥ(S T ) ≥ 0 at maturity T , whereĥ(s) = (s − K) + for all s > 0. Recalling Heston's notation in [25,Eq. (11), p. 330], we denote by x = X t (ω) ∈ R the logarithm of the spot price of stock and by v = V t (ω) ∈ R the variance of stock market at time t. We set h(x, v) ≡ h(x) = K (e x − 1) + for all x = ln(s/K) ∈ R, so that h(x) =ĥ(s) =ĥ(Ke x ) for x ∈ R. Hence, if the instant values (X t (ω), V t (ω)) = (x, v) ∈ H are known at time t ∈ (0, T ), where H = R × (0, ∞) ⊂ R 2 , the arbitrage-free price P h t of the European call option at this time is given by the following expectation formula (with respect to the risk neutral probability measure P) which is justified in [8] and [34]: Furthermore, p solves the (terminal value) Cauchy problem with G t being the (time-independent) infinitesimal generator of the time-homogeneous Markov process (X t , V t ); cf. A. Friedman [22,Chapt. 6] where we use the instant values (s, v) = (S t (ω), V t (ω)) ∈ (0, ∞) 2 and substitute U (s, v, t) = p(x, v, t) with s = Ke x , ds/dx = s, and Hence, the function p : (x, v, t) → p(x, v, T − t) verifies a linear (initial value) Cauchy problem derived from (A.4). The functional-analytic formulation of this problem is given in Eq. (3.9) with the initial data corresponding to p(x, v, 0) = p(x, v, T ) = h(x) at t = 0.
We have replaced the meaning of the temporal variable t as real time (t ≤ T ) by the time to maturity t (t ≥ 0), so that the real time (time to maturity) has become τ = T − t. According to S. L. Heston [25,Eq. (6), p. 329], the unspecified term λ(x, v, T − t) in the second drift term on the right-hand side of Eq. (A.6) (with ∂U /∂v) represents the price of volatility risk and is specifically chosen to be λ(x, v, T − t) ≡ λv with a constant λ ≥ 0. As we have already pointed out in the Introduction (Section 1), we can treat much more general initial conditions p(x, v, 0) = h(x, v) than just those given by the terminal condition for the European call option, p(x, v, T ) = h(x) = K (e x − 1) + for (x, v) ∈ H, which does not depend on the instant value v = V T (ω) of the variance at maturity T ; see Section 4.

B Appendix: Weighted Sobolev spaces and boundary traces
We denote by H 1 (B + R (x 0 , 0); w) the weighted Sobolev space of all functions f ∈ W 1,2 loc (B + R (x 0 , 0)) whose norm defined below is finite, Let us recall that the weighted Sobolev space H 2 (B + R (x 0 , 0); w) has been defined by its norm in Eq. (2.2). As we will see later, in Lemma B.5, functions from the weighted Sobolev spaces H j (B + R (x 0 , 0); w); j = 1, 2, must satisfy certain homogeneous boundary conditions as ξ → 0+, i.e., near the boundary We will see in the proof of this result (Lemma B.5), as well, that these boundary conditions, if satisfied by a function, imply that this function belongs to a particular weighted Sobolev space.
We will follow a similar procedure in treating the case f ∈ H 2 (B + R (x 0 , 0); w). More precisely, we wish to show that if we take a weaker norm, · ♭ Eq. (B.4) below, the restriction mapping defined as follows: It is easy to see thatH 2 (B + R (x 0 , 0); w) consists of all functions f ∈ W 2,2 loc (B + R (x 0 , 0)) that satisfy f ♭ In the course of the proof of this restriction imbedding, we will obtain also certain boundary conditions (i.e., trace results as ξ → 0+) on the boundary ∂H ∩ B + R/ √ 2 (x 0 , 0). Keeping in mind that some of the constants in our estimates below may depend on the choice of x 0 ∈ R, we suppress the dependence on x 0 in the notation for the half-disc B + R (x 0 , 0) ⊂ H and, thus, write only B + R ≡ B + R (x 0 , 0). We further denote by (a "half-square") with side lengths 2r and r ∈ (0, ∞). Its closure in R 2 is denoted by Q + r .
Our first lemma is an essential estimate for obtaining the boundary trace as ξ → 0+. Given a function u ∈ W 1,1 loc (B + ), we abbreviate the gradient ∇u = (u x , u ξ ).
Lemma B.1 (ξ-derivative inequalities.) Let β > 0 and R > 0, and set r = R/ √ 2. Assume that u ∈H 2 (B + R ; w). Then Q + r ⊂ B + R and the following inequalities hold at almost every point (x, ξ) ∈ Q + r , Proof. The following partial derivatives exist almost everywhere in B + R ; we first calculate with the scalar product ∇u · ∂ ξ ∇u = u x u xξ + u ξ u ξξ in R 2 ⊂ C 2 , then estimate the scalar product on the right-hand side by Cauchy's inequality, for a.e. (x, ξ) ∈ B + R . We apply Ineq. (B.8) to estimate the right-hand side of Eq. (B.7), thus arriving at (B.6) as desired.
Finally, we integrate all equations and inequalities (B.11) -(B.14) with respect to x ∈ (−r, r) to derive the following corollary of Lemma B.2.

C Appendix: Some known elliptic regularity results
In this appendix we collect a few known results on the local regularity of a weak solution u ∈ V to the degenerate elliptic problem (λI + A)u = f ∈ H, i.e., for u = (λI + A) −1 f ∈ V . Recall that λ > λ 0 with the constant λ 0 > 0 determined by Ineq. (5.1).