Abstract
The paper solves a multi-asset optimal trading problem in which each agent's trades move prices through Volterra cross-impact propagators — non-Markovian, cross-asset, decaying impact kernels — and obtains the optimal strategy as the solution of a coupled system of Fredholm integral equations. We read it as the rigorous treatment of execution when impact is both path-dependent and spills across correlated assets.
Notation / Conceptual Frame
Price S_t = S_0 + ∫_0^t G(t − s) dQ_s + noise, with G a matrix-valued propagator encoding self- and cross-impact; the optimal trading rate solves a matrix Fredholm equation of the second kind.
Commentary
The model takes impact seriously as a cross-asset, memory-bearing object rather than an instantaneous cost, and still yields a tractable Fredholm solution. It connects the propagator microstructure literature to portfolio-level execution.
Implications for Research Methodology
Cautions that execution and impact on one name are not separable from correlated names; memo-level liquidity assessments should treat cross-impact and its decay as first-order, not as a frictionless add-on.
Limitations
Propagator estimation is unstable and cross-impact terms are empirically known to change sign; the linear-propagator, quadratic-cost setting omits the documented concavity (square-root law) of impact.
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