the Kunita-Watanabe inequality
/ koo-NEE-tah wah-tah-NAH-bay /
The Kunita-Watanabe inequality is the Cauchy-Schwarz inequality of stochastic calculus: it bounds the quadratic covariation between two semimartingales (and, more usefully, the covariation of two stochastic integrals) by the geometric mean of their quadratic variations. It is the basic estimate that controls cross terms throughout the theory, and it is what makes the bilinear form [ , ] behave like an inner product, so that the stochastic integral really is a Hilbert-space construction.
In its pointwise form: for two continuous local martingales M and N, the total-variation measure of the covariation [M, N] is dominated by the geometric mean of the variation measures, d|[M, N]| <= sqrt(d[M]) sqrt(d[N]) in the sense of measures on [0, t]. The integrated and most-used form: for predictable processes H and K, the absolute integrated covariation is bounded by integral_0^t |H_s| |K_s| d|[M, N]|_s <= ( integral_0^t H_s^2 d[M]_s )^{1/2} ( integral_0^t K_s^2 d[N]_s )^{1/2}. In words, the covariation of the integral integral H dM and the integral integral K dN is controlled by the L^2(d[M]) norm of H times the L^2(d[N]) norm of K — exactly Cauchy-Schwarz with d[M] and d[N] playing the role of the reference measures. The proof is the classical Cauchy-Schwarz argument applied to the positive-semidefinite bilinear structure of covariation: for any reals a, b the process a^2 [M] + 2ab [M, N] + b^2 [N] = [aM + bN] is increasing, which forces the discriminant inequality that is exactly Kunita-Watanabe.
Its role is to make covariation an honest inner product and to control error and cross terms. It shows the bilinear map (H, K) -> integral H K d[M, N] is bounded, which is what lets you define stochastic integrals against general continuous martingales coherently and lets you estimate the difference of two integrals. It underlies the orthogonal (Kunita-Watanabe / Galtchouk-Kunita-Watanabe) decomposition used in quadratic hedging of incomplete markets, where one projects a claim onto the space of attainable integrals and the inequality guarantees the projection is well-defined. The honest caveats: (1) The inequality is about covariation measures, and the integrated form requires H, K predictable and the relevant integrals finite. (2) Like Cauchy-Schwarz it is an inequality, not an identity; equality holds only in the degenerate aligned case (K proportional to H along the support of the covariation). (3) It is stated for local martingales / their integral parts; for full semimartingales one applies it to the martingale components, since bounded-variation parts have zero covariation with everything.
Bound the covariation of two Ito integrals against the same Brownian motion B: M = integral H dB and N = integral K dB. Here d[M] = H^2 dt, d[N] = K^2 dt, d[M, N] = HK dt, so KW reads | integral_0^t H_s K_s ds | <= ( integral_0^t H_s^2 ds )^{1/2} ( integral_0^t K_s^2 ds )^{1/2} — which is just the ordinary Cauchy-Schwarz inequality in L^2[0, t], recovered as the special case of the stochastic statement.
Kunita-Watanabe is Cauchy-Schwarz for covariation; against a common Brownian motion it reduces to ordinary L^2 Cauchy-Schwarz.
It is an inequality, not an equality, and is stated for the martingale parts (finite-variation parts have zero covariation); the integrated form needs predictable integrands and uses d[M], d[N] as the reference measures, not Lebesgue time.