Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance

Abstract

The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order degenerate elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. With the aid of weighted Sobolev spaces, we prove existence, uniqueness, and global regularity of solutions to stationary variational inequalities and obstacle problems for the elliptic Heston operator on unbounded subdomains of the half-plane. In mathematical finance, solutions to obstacle problems for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

“…When the operator, L, is given in divergence form, so one can define a weak solution, u ∈ W 1,2 (Q), to a boundary value problem, one can also obtain uniqueness of solutions with partial Dirichlet data when the Fichera sign condition holds along / ∂ 0 Q [19,20,34,38,39]. However, the Fichera weak maximum principle does not take into account a more modern view of the appropriate function spaces in which uniqueness is sought, such as those used by P. Daskalopoulos and the author [6], Daskalopoulos, R. Hamilton, and E. Rhee [7,8], E. Ekström and J. Tysk [10], C. L. Epstein and R. Mazzeo [11], C. A. Pop and the author [18], and H. Koch [26].…”

“…Indeed, the Fichera weak maximum principles lead to the imposition of additional Dirichlet boundary conditions which are not necessarily motivated by the underlying application, whether in biology, finance, or physics. These additional Dirichlet boundary conditions, usually for certain ranges of parameters defining the operator, L, are often less natural than the physically-motivated regularity properties suggested by choices of appropriate weighted Hölder spaces [7,8,11,18] or Sobolev spaces [6,16,26], which automatically encode enough regularity up to the portion, / ∂ 0 Q, of the parabolic boundary where the operator, L, becomes degenerate.…”

“…, n d ) is the inward-pointing unit normal vector field along / existence and uniqueness of solutions to the Cauchy problem on H T , where H := R d−1 × R + . On the other hand, if we ask for too little regularity, such as C 0 up to / ∂ 0 Q, the examples of the Heston stochastic volatility model [6,5] in mathematical finance, the porous medium equation [7], Wright-Fisher diffusion model in mathematical biology [11], and interest rate models in mathematical finance [10] indicate that this usually leads to the imposition of an unphysical Dirichlet boundary condition. In particular, this has the unintended consequence that the unique solutions selected by the Fichera weak maximum principle can be no more than continuous up to the boundary; a detailed example illustrating this point with the aid of the Kummer equation is provided by the author in [15, §1.1].…”

A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps

“…By extending the methods of Koch [26] and restricting to the case where A is the Heston operator (1.24) and d = 2, Daskalopoulos, Pop, and the author succeeded in proving existence, uniqueness, and regularity os solutions to the equation (1.1) and obstacle problem (1.3) with partial Dirichlet boundary condition (1.2) by solving the associated variational equation and inequality for solutions, u, in weighted Sobolev spaces [8]; appealing to [13] to obtain uniqueness; proving that the solutions are continuous up to the boundary [15] using a Moser iteration technique; proving Schauder regularity when f ∈ C ∞ 0 (O) ∩ C b (O) using a variational method [16]; proving the expected Schauder regularity,…”

“…Recall that by a "domain" in R d , we always mean a connected, open subset 7. For example, the coefficient b d has this property if it is continuous and positive along ∂0O 8. The Riemannian metric is called cycloidal in[9] since its geodesics, when d = 2, are the standard cycloidalcurve, (x1(t), x2(t)) = (t − sin t, 1 − cos t) for t ∈ R, curves obtained from the standard cycloid by translations (x1, x2) → (x1 + b, x2), b ∈ R, or dilations (x1, x2) → (cx1, x2), c ∈ R+, or are vertical lines, x1 = a, a ∈ R [9, Proposition I.2.1].…”