Liu's pricing formula gives curve desks a benchmark; its mild-solution hedges live in auxiliary markets. Liu's abstract distinguishes exact replication for strong solutions from asymptotic replication for mild ones. The latter is an approximation result, with no stated rate.
Pricing a vector claim
Liu takes the entire price vector X_t, an infinite collection of asset prices, as a point in a separable Hilbert space H. Its dynamics are dX = (AX + F(X))dt + B(X)dW. The operator A generates a group S(t): S(t) accumulates money as e^{rt} does, while S(-t) discounts. A claim pays the vector G(X_T), and its price is V(t,x) = S(t-T)E^Q[G(X_T^{t,x})]. If A is diagonal with rates r_i, each coordinate has its own discount rate. A cash-settled option pays one amount in one currency. Here the amounts remain separate, except when A = rI, where Liu permits real-valued payoffs such as an arithmetic basket.
Example 1 prices the coordinatewise call (x-K)^+ as Σ_i [x_i Φ(d_{1,i}) - K_i e^{-r_i(T-t)} Φ(d_{2,i})] e_i. Since 0 ≤ V_i ≤ x_i, the series converges in H. The squared gamma terms λ_i x_i^2 ∂_xx V_i sum to at most sup_i λ_i/(2π(T-t)) · Σ|x_i|^2. That bound blows up as t approaches T.
Theorem 2.4 uses the four conditions in Assumption 2.1. They require a bounded payoff with two Fréchet derivatives whose derivatives are Lipschitz, E[G(X)] in the domain of A, and B commuting with every power A^n. A call fails the payoff condition. For the diagonal model, Liu takes another route: Theorem 4.3 uses the explicit solution to relax the smoothness requirement, and Example 1 sums one-dimensional Black-Scholes equations coordinatewise. Those are the results that cover the call.
The diagonal geometric Brownian motion model in Theorem 4.1 has an equivalent risk-neutral measure if Σ(r_i - μ_i)^2/λ_i < ∞. Set λ_i = 1/i^2 and μ_i = 1/i. The condition becomes Σ(ri - 1)^2, divergent for every real r. Rates chosen coordinate by coordinate as r_i = μ_i + √λ_i a_i, with (a_i) in ℓ^2, always satisfy it. A desk observes its r_i as market data, so I read this as a test of which curves admit a measure.
Where the exact hedge ends
Exact replication requires a strong solution. Liu obtains one when A is bounded: mild and strong solutions then coincide, and dX_t can be written down. Hedge positions are operators, with risky position φ_t = D_xV. Solving for the bank position θ takes place on the two-dimensional span of B_t and AB_t. It requires Ae, and each A_n e in the approximate markets, to lie outside span{e}.
For a mild solution, Liu replaces A with Yosida approximations A_n. The hedge is exact in each n-th market; its terminal value converges to G(X_T) in L^2 (Theorem 2.6). Liu describes their status plainly: "These approximate markets are introduced solely for approximation purposes and do not represent the actual market." We did not find a convergence rate in n. A trader therefore has no indication of how close a particular n gets to the real mild market.
The completeness claim needs even more care.
Proposition 3.8 makes finite-dimensional noise necessary for the operator equation φ(t)B(X_t) = S(t)Z_t to be solvable for every claim. Liu says the markets driven by (2.10) and (2.96) are not complete in the classical sense when U0 is infinite-dimensional: some square-integrable claims cannot be attained. In the strong formulation, injectivity of B on the first m noise directions lets a noise truncation make every square-integrable claim approximately attainable. Add Yosida approximation and the result is a doubly indexed family of (n,m) markets. Theorem 3.9 gives asymptotic approximate completeness for that family, assuming each (n,m) market is complete. Liu sees no obvious necessary condition for this weaker notion. For finite-dimensional noise in the strong formulation, Theorem 3.10 equates completeness with uniqueness of the risk-neutral measure. The diagonal model supplies Liu's counterexample when noise is infinite.
Liu also flags the no-arbitrage qualification. By definition, admissible strategies have Q-martingale discounted values. The restriction matters because S(-t) need not commute with portfolio operators; self-financing alone can produce arbitrage. Proposition 3.1 goes only from a measure to no-arbitrage. Its conclusion depends, in practice, on the chosen admissible class.
Can segmented markets share a hedge?
The segmented-market example puts the formal issue in view. H is two-dimensional, with two markets, rates r1 ≠ r2, no cross-market trading, and a separate currency for each coordinate. The claim is G1(X^1,X^2)e1 + G2(X^1,X^2)e2. Independence of B_t and AB_t makes the bank position Ψ_t unique exactly when r1 ≠ r2.
From market 1 alone, Liu says G1 is "generally not attainable by trading solely in the first market," since X^2 is a non-traded risk factor. The aggregate H-valued book nevertheless replicates. Self-financing applies to the aggregate portfolio, "not separately on each market component." As Liu notes, the per-market decomposition gives holdings at each time, rather than a self-financing strategy in each market. Through the off-diagonal φ12, the e1 leg can gain value from dX^2, an exposure no market-1 desk holds. Funding each leg would require value to cross the segment boundary, despite the prohibition on cross-market trades. The vector book has a single self-financing constraint. Real segmentation would impose one on each segment, two here.
Nothing here to backtest
We could not run this. Our data consist of finite sets of listed options and dated futures on end-of-day bars; Liu's framework calls for an infinite asset collection or an entire forward curve traded continuously. The paper makes no performance claim. It has no data or calibration, and the model charges no transaction costs. Its basket example assumes independent coordinates, while the PDE verification is called "straightforward" without a derivation.
A rate for the Yosida hedge error in n would change my view. Theorem 2.6 establishes convergence, but leaves the point at which a hedge might be usable unknown.