The comparison theorem for backward stochastic differential equations provides an ordering result: if BSDE^1 and BSDE^2 share the same noise structure but BSDE^1 has a larger driver (f^1 >= f^2) and a larger terminal condition (xi^1 >= xi^2), then Y^1(t) >= Y^2(t) for all t in [0,T], almost surely.
More precisely, consider two BSDEs: -dY^i(t) = f^i(t, Y^i, Z^i) dt - Z^i(t)^T dW(t), Y^i(T) = xi^i, i = 1, 2.
If f^1(t, y, z) >= f^2(t, y, z) for all (t, y, z) a.e. and xi^1 >= xi^2 a.s., then Y^1(t) >= Y^2(t) for all t in [0,T] a.s.
This is a central tool in mathematical finance and stochastic control. In the XVA framework of Sekine and Tanaka (2020), it is used for three purposes:
- Bid-ask ordering: Since f^+ >= f^- (the seller’s driver dominates the buyer’s by exactly the funding and repo spread terms), the comparison theorem yields Y^-(t) ⇐ Y^+(t) — the buyer’s price is below the seller’s price, establishing a well-defined bid-ask spread.
- Arbitrage-free property: Monotonicity of Y^+ in the payoff parameters (xi_T, phi_1, phi_2) is established by applying comparison to perturbed payoffs, proving that Y^+(0) equals the minimal superhedging price (Theorem 4).
- Approximation ordering: The sandwich inequality Y^- ⇐ Y^{0,-} ⇐ Y^{0,+} ⇐ Y^+ (Theorem 5) follows from comparison, since the zeroth-order drivers lie between f^- and f^+.
The comparison theorem requires the driver to be Lipschitz continuous and is closely related to the theory of g-expectations (Jiang, 2008) and nonlinear expectations.