Comparison Principle¶
The comparison principle is the central uniqueness and ordering result for viscosity solutions. It states that if a subsolution lies below a supersolution on the boundary, then this ordering persists throughout the domain. This immediately implies uniqueness: the viscosity solution is the only function that is simultaneously a sub- and supersolution.
Statement for Parabolic Problems¶
Consider the degenerate parabolic PDE:
where \(F\) is degenerate elliptic (non-increasing in the \(D^2u\) argument) and proper (non-decreasing in the \(u\) argument, i.e., \(F(\tau, x, r, p, X) \leq F(\tau, x, s, p, X)\) whenever \(r \leq s\)).
Theorem (Comparison Principle)
Let \(u\) be a viscosity subsolution and \(v\) a viscosity supersolution of \(u_\tau + F = 0\) on \((0, T] \times \Omega\). If \(u \leq v\) on the parabolic boundary (i.e., at \(\tau = 0\) and on \(\partial\Omega\)), then:
The parabolic boundary consists of the initial data (\(\tau = 0\)) and the spatial boundary (\(x \in \partial\Omega\)).
Consequences¶
Uniqueness¶
Corollary: If the comparison principle holds, then the viscosity solution is unique.
Proof: Let \(u\) and \(v\) both be viscosity solutions with the same boundary data. Then \(u\) is a subsolution and \(v\) is a supersolution, so \(u \leq v\). By symmetry (swapping the roles), \(v \leq u\). Therefore \(u = v\). \(\square\)
Stability¶
Corollary: If \(u_n\) are viscosity solutions with boundary data \(g_n\) and \(g_n \to g\) uniformly, then \(u_n \to u\) uniformly, where \(u\) is the viscosity solution with boundary data \(g\).
This stability under perturbation of boundary data is essential for numerical analysis: it ensures that small errors in the boundary conditions do not lead to large errors in the solution.
Ordering and Bounds¶
The comparison principle gives a priori bounds. If \(\underline{u}\) and \(\overline{u}\) are explicit sub- and supersolutions (barriers) with known formulas, then the true solution satisfies:
For European options, taking \(\underline{u} = 0\) and \(\overline{u} = S\) (for calls) gives the obvious bound \(0 \leq V \leq S\).
Proof Sketch: Doubling of Variables¶
The standard proof technique is the Ishii-Lions doubling of variables method. We outline the key steps.
Setup¶
Suppose, for contradiction, that
We want to show this leads to a contradiction with the sub/supersolution properties.
Step 1: Penalized Supremum¶
The direct approach of testing at the supremum point fails because \(u\) and \(v\) may not be differentiable there. Instead, consider the doubled function:
where \(\alpha > 0\) is a large parameter. The quadratic penalty forces the supremum to occur near the diagonal \(x = y\), \(\tau = \sigma\).
Step 2: Supremum Point¶
Let \((\tau_\alpha, x_\alpha, \sigma_\alpha, y_\alpha)\) be a point where \(\Phi_\alpha\) attains its supremum. Standard arguments show:
- As \(\alpha \to \infty\): \(|x_\alpha - y_\alpha| \to 0\) and \(|\tau_\alpha - \sigma_\alpha| \to 0\)
- \(\alpha|x_\alpha - y_\alpha|^2 \to 0\)
- \(u(\tau_\alpha, x_\alpha) - v(\sigma_\alpha, y_\alpha) \to M > 0\)
For large \(\alpha\), the supremum point is in the interior (not on the parabolic boundary, where \(u \leq v\) by assumption).
Step 3: Applying the Definitions¶
At \((\tau_\alpha, x_\alpha)\), the function \((\tau, x) \mapsto v(\sigma_\alpha, y_\alpha) + \frac{\alpha}{2}|x - y_\alpha|^2 + \frac{\alpha}{2}(\tau - \sigma_\alpha)^2\) is a smooth test function touching \(u\) from above. By the subsolution property:
Similarly, at \((\sigma_\alpha, y_\alpha)\), the supersolution property gives:
Here \(X_\alpha\) and \(Y_\alpha\) are matrices satisfying certain bounds (from the Ishii lemma).
Step 4: Contradiction¶
Subtracting the two inequalities and using the properness of \(F\) (the fact that \(F\) is non-decreasing in \(u\), so \(u > v\) can be exploited) and the regularity of \(F\) in the other variables, one obtains:
where \(\omega\) is a modulus of continuity that vanishes as its argument tends to zero. Since \(\alpha|x_\alpha - y_\alpha|^2 \to 0\), this yields \(0 \leq 0\), a contradiction. \(\square\)
Assumptions and Conditions¶
The comparison principle requires specific structural conditions on \(F\).
Degenerate Ellipticity¶
(Here \(X \geq Y\) means \(X - Y\) is positive semidefinite.) This ensures the PDE has a parabolic or elliptic character.
For the Black-Scholes operator \(F = -\frac{1}{2}\sigma^2 S^2 \text{tr}(D^2u) - rS Du + ru\), the diffusion coefficient \(\frac{1}{2}\sigma^2 S^2 \geq 0\), so degenerate ellipticity holds.
Properness¶
The operator is non-decreasing in \(u\). For Black-Scholes, the \(-rV\) term means the operator contains \(+ru\), which is indeed non-decreasing in \(u\) (provided \(r \geq 0\)).
Continuity of \(F\)¶
The function \(F\) must be continuous in all its arguments (or at least satisfy appropriate semicontinuity conditions).
Growth Conditions¶
For unbounded domains, growth conditions at infinity are needed. For Black-Scholes on \([0, \infty)\), one typically assumes solutions grow at most polynomially: \(|u(\tau, S)| \leq C(1 + S^p)\) for some \(p > 0\).
Application to Black-Scholes¶
European Options¶
The Black-Scholes PDE satisfies all conditions for the comparison principle:
- Degenerate ellipticity: \(\frac{1}{2}\sigma^2 S^2 \geq 0\)
- Properness: The \(rV\) term (after rearranging as \(F = 0\)) gives the correct monotonicity
- Continuity: All coefficients are smooth
Consequence: The Black-Scholes formula gives the unique viscosity solution for European options.
American Options¶
For the obstacle problem \(\min(-u_\tau + \mathcal{L}u,\; u - \Phi) = 0\), the comparison principle holds under the same conditions, with the additional requirement that sub- and supersolutions satisfy the obstacle constraint appropriately.
Consequence: The American option value function is the unique viscosity solution of the variational inequality, and it can be characterized as the smallest supersolution that dominates the payoff.
The Discrete Comparison Principle¶
For numerical schemes, the comparison principle has a discrete analog.
Definition¶
A numerical scheme \(S_h[u] = 0\) satisfies the discrete comparison principle if:
(given appropriate boundary ordering).
Monotone Schemes¶
A scheme is monotone if \(S_h[\cdot]\) is a non-decreasing function of each of its arguments (the values at neighboring nodes). For explicit schemes, this corresponds to non-negative stencil coefficients.
Monotone schemes automatically satisfy the discrete comparison principle, which is the discrete analog of the continuous comparison principle. This connection is the foundation of the Barles-Souganidis convergence theorem.
Example: Explicit Scheme¶
The explicit scheme \(u_j^{n+1} = a_j u_{j-1}^n + b_j u_j^n + c_j u_{j+1}^n\) is monotone if and only if \(a_j, b_j, c_j \geq 0\). This is precisely the CFL condition combined with the positivity requirement on all coefficients.
Maximum Principle Connection¶
The comparison principle generalizes the classical maximum principle for parabolic PDEs.
Classical maximum principle: If \(u_\tau - \mathcal{L}u \leq 0\) (subsolution) in a domain, then \(u\) attains its maximum on the parabolic boundary.
Viscosity comparison principle: Extends this to non-smooth functions and degenerate operators, providing ordering between any subsolution and supersolution.
For monotone schemes, the discrete maximum principle states:
This is the discrete analog that ensures no new extrema are created by the scheme, and it implies \(L^\infty\) stability.
Summary¶
| Role | Detail |
|---|---|
| Uniqueness | Sub + super with same data \(\Rightarrow\) equality |
| Stability | Small boundary perturbations \(\Rightarrow\) small solution changes |
| Bounds | Explicit barriers give a priori estimates |
| Proof technique | Doubling of variables (Ishii-Lions) |
| Key assumptions | Degenerate ellipticity, properness, continuity |
| For Black-Scholes | All conditions satisfied; solution is unique |
| Discrete version | Monotone schemes preserve ordering |
The comparison principle is the cornerstone of viscosity solution theory. It guarantees that the option pricing PDE (or variational inequality for American options) has a unique answer, and it provides the theoretical foundation for proving that monotone numerical schemes converge to this unique answer.
Exercises¶
Exercise 1. State the comparison principle for viscosity sub- and supersolutions. Prove the uniqueness corollary: if \(u\) and \(v\) are both viscosity solutions with the same boundary data, then \(u = v\).
Solution to Exercise 1
Comparison Principle: Let \(u\) be a viscosity subsolution and \(v\) a viscosity supersolution of \(u_\tau + F(\tau, x, u, Du, D^2u) = 0\) on \((0, T] \times \Omega\), where \(F\) is degenerate elliptic and proper. If \(u \leq v\) on the parabolic boundary (at \(\tau = 0\) and on \(\partial\Omega\)), then
Uniqueness corollary: Let \(u\) and \(v\) both be viscosity solutions of the same PDE with the same boundary data \(g\).
Since \(u\) is a viscosity solution, it is in particular a viscosity subsolution. Since \(v\) is a viscosity solution, it is in particular a viscosity supersolution. Both satisfy the same boundary data, so \(u = g = v\) on the parabolic boundary, which gives \(u \leq v\) on the boundary. By the comparison principle:
Now swap the roles: \(v\) is also a viscosity subsolution, and \(u\) is also a viscosity supersolution, with \(v = g = u\) on the parabolic boundary. Applying the comparison principle again:
Combining both inequalities: \(u = v\) everywhere. \(\square\)
Exercise 2. For a European call, construct explicit sub- and supersolutions (barriers): \(\underline{u}(S) = 0\) and \(\overline{u}(S) = S\). Verify that \(\underline{u}\) is a viscosity subsolution and \(\overline{u}\) is a viscosity supersolution of the Black-Scholes PDE, and conclude that \(0 \leq V(t, S) \leq S\).
Solution to Exercise 2
We verify that \(\underline{u}(S) = 0\) is a viscosity subsolution and \(\overline{u}(S) = S\) is a viscosity supersolution of the Black-Scholes PDE \(V_t + \frac{1}{2}\sigma^2 S^2 V_{SS} + rSV_S - rV = 0\).
\(\underline{u} = 0\) is a viscosity subsolution: Since \(\underline{u}\) is smooth (\(C^\infty\)), it suffices to check the PDE pointwise. Substituting \(V = 0\):
So \(\underline{u} = 0\) satisfies the subsolution inequality \(\leq 0\).
\(\overline{u} = S\) is a viscosity supersolution: Since \(\overline{u}\) is smooth, substitute \(V = S\): \(V_t = 0\), \(V_S = 1\), \(V_{SS} = 0\). Then:
So \(\overline{u} = S\) satisfies the supersolution inequality \(\geq 0\).
Boundary verification: For a European call with payoff \(\Phi(S) = (S - K)^+\):
- At \(\tau = 0\): \(0 \leq (S - K)^+ \leq S\) for all \(S \geq 0\)
- At \(S = 0\): \(V(t, 0) = 0\), and \(0 \leq 0 \leq 0\)
- As \(S \to \infty\): \(V \sim S\) and \(0 \leq V \leq S\)
Since \(\underline{u} \leq V\) and \(V \leq \overline{u}\) on the parabolic boundary, the comparison principle gives
This is the well-known no-arbitrage bound for European call prices.
Exercise 3. The stability corollary states that small perturbations in boundary data produce small changes in the solution. If the terminal payoff is perturbed by \(\epsilon\) (i.e., \(\Phi_\epsilon(S) = (S - K)^+ + \epsilon\)), bound the change in the option price using the comparison principle.
Solution to Exercise 3
Let \(V\) be the viscosity solution with terminal data \(\Phi(S) = (S - K)^+\) and \(V_\epsilon\) the viscosity solution with perturbed data \(\Phi_\epsilon(S) = (S - K)^+ + \epsilon\).
Define \(w = V_\epsilon - \epsilon e^{-r(T-t)}\). We claim \(w\) satisfies the Black-Scholes PDE with terminal data \((S-K)^+\).
Since the Black-Scholes PDE is linear, and \(\epsilon e^{-r(T-t)}\) satisfies \(\frac{\partial}{\partial t}(\epsilon e^{-r(T-t)}) - r(\epsilon e^{-r(T-t)}) = r\epsilon e^{-r(T-t)} - r\epsilon e^{-r(T-t)} = 0\), which means \(\epsilon e^{-r(T-t)}\) is itself a solution of the PDE. By linearity, \(w = V_\epsilon - \epsilon e^{-r(T-t)}\) is a viscosity solution with terminal data \(\Phi_\epsilon - \epsilon = (S-K)^+\).
By uniqueness (from the comparison principle), \(w = V\), so
Therefore, the change in the option price is bounded by
This shows that the option price depends continuously on the terminal data: an \(\epsilon\)-perturbation in the payoff produces at most an \(\epsilon\)-change in the price (discounted), confirming the stability guaranteed by the comparison principle.
Exercise 4. Verify the four assumptions required for the comparison principle to hold for the Black-Scholes operator: degenerate ellipticity, properness, continuity, and growth conditions. For the properness condition, explain why \(r \geq 0\) is essential.
Solution to Exercise 4
The comparison principle for the Black-Scholes operator requires four conditions. Writing the PDE as \(F(\tau, S, V, V_S, V_{SS}) = -\frac{1}{2}\sigma^2 S^2 V_{SS} - rSV_S + rV = 0\) (after the change to backward time \(\tau = T - t\), so \(V_\tau = F\)):
1. Degenerate ellipticity: We need \(F(\tau, S, r, p, X) \leq F(\tau, S, r, p, Y)\) whenever \(X \geq Y\) (i.e., \(X - Y \geq 0\)). The second-derivative term in \(F\) is \(-\frac{1}{2}\sigma^2 S^2 V_{SS}\). If \(X \geq Y\), then
since \(\frac{1}{2}\sigma^2 S^2 \geq 0\). Thus \(F(\ldots, X) \leq F(\ldots, Y)\). The condition holds because the diffusion coefficient is non-negative.
2. Properness: We need \(F(\tau, S, r, p, X) \leq F(\tau, S, s, p, X)\) whenever \(r \leq s\). The \(u\)-dependent term in \(F\) is \(+rV\) (where \(r\) is the interest rate). So
If \(r_1 \leq r_2\) and \(r \geq 0\) (non-negative interest rate), then \(r(r_1 - r_2) \leq 0\), confirming properness. This is why \(r \geq 0\) is essential: if the interest rate were negative, the \(+rV\) term would be decreasing in \(V\), violating properness and potentially destroying uniqueness.
3. Continuity: The coefficients \(\frac{1}{2}\sigma^2 S^2\), \(rS\), and \(r\) are all smooth (indeed polynomial or constant) functions of \(S\). Thus \(F\) is continuous in all its arguments.
4. Growth conditions: On the unbounded domain \(S \in [0, \infty)\), we need to control the behavior at infinity. For European options, the solution grows at most linearly: \(|V(\tau, S)| \leq C(1 + S)\) for some constant \(C > 0\). This polynomial growth condition (here with \(p = 1\)) ensures that the doubling-of-variables argument in the comparison principle proof can be carried out, as the penalization terms dominate the solution growth at infinity.
Exercise 5. The discrete comparison principle states: if \(S_h[u] \leq S_h[v]\) at all nodes, then \(u \leq v\) at all nodes. Show that the explicit scheme satisfies this property when all stencil coefficients are non-negative (monotone scheme).
Solution to Exercise 5
The explicit scheme is
with \(a_j, b_j, c_j \geq 0\) (monotone scheme). We show that the discrete comparison principle holds: if \(S_h[u] \leq S_h[v]\) at all nodes (meaning \(u\) is a discrete subsolution and \(v\) is a discrete supersolution) and \(u \leq v\) on the boundary, then \(u \leq v\) everywhere.
Write the scheme as \(S_h[u]_j^{n+1} = u_j^{n+1} - a_j u_{j-1}^n - b_j u_j^n - c_j u_{j+1}^n\). Define \(w_j^n = v_j^n - u_j^n\). Then
More directly, suppose \(u_j^n \leq v_j^n\) for all \(j\) at time level \(n\). Then at time level \(n+1\), the time-stepped values satisfy:
Since \(v_k^n - u_k^n \geq 0\) for all \(k\) and \(a_j, b_j, c_j \geq 0\), each term on the right is non-negative. Therefore
for all \(j\). By induction over time levels (starting from the boundary where \(u \leq v\) by assumption), we conclude \(u_j^n \leq v_j^n\) for all \(j\) and all \(n\). This is precisely the discrete comparison principle, and it holds because the non-negative coefficients ensure that the ordering at one time level is propagated to the next.
Exercise 6. The "doubling of variables" proof technique introduces a penalized supremum \(\Phi_\alpha(\tau, x, \sigma, y) = u(\tau, x) - v(\sigma, y) - \frac{\alpha}{2}|x-y|^2 - \frac{\alpha}{2}(\tau - \sigma)^2\). Explain the role of the quadratic penalty terms: why do they force the supremum to occur near the diagonal \(x = y\), \(\tau = \sigma\)?
Solution to Exercise 6
The penalized function is
The quadratic penalty terms \(\frac{\alpha}{2}|x - y|^2\) and \(\frac{\alpha}{2}(\tau - \sigma)^2\) force the supremum to occur near the diagonal for the following reason.
Let \((\tau_\alpha, x_\alpha, \sigma_\alpha, y_\alpha)\) be a point where \(\Phi_\alpha\) attains its supremum. By definition:
for any \((\tau, x)\). In particular, taking the supremum over \((\tau, x)\):
where \(M = \sup_{(\tau,x)}(u - v)(\tau, x) > 0\). Since \(u\) and \(v\) are bounded (say \(|u|, |v| \leq C\)):
This gives
As \(\alpha \to \infty\), the right-hand side tends to zero, forcing \(|x_\alpha - y_\alpha| \to 0\) and \(|\tau_\alpha - \sigma_\alpha| \to 0\). Thus the supremum points converge to the diagonal \(x = y\), \(\tau = \sigma\).
The purpose of this construction is to convert the problem into one where the sub- and supersolution properties can be applied. On the diagonal, the penalized function reduces to \(u(\tau, x) - v(\tau, x)\), so the supremum of \(\Phi_\alpha\) converges to \(M\). The penalty terms create smooth test functions that can be used to invoke the viscosity sub/supersolution definitions at the supremum point, which is the core of the contradiction argument.