The Core Insight

In systems theory, passivity means no input can extract energy. In market microstructure, admissibility means no trading program can extract profit from transient impact. The mathematics is identical. This paper takes that coincidence seriously.

The key class: Stieltjes propagators, kernels of the form G(t)=∫e−ρtM(dρ)G(t) = \int e^{-\rho t} M(\mathrm{d}\rho). Every one is automatically admissible, admits a passive realization, and can be reduced by discretizing the measure into a finite sum of exponentials:

Gr(t)=∑k=1rAk e−ρktG_r(t) = \sum_{k=1}^{r} A_k\, e^{-\rho_k t}

The reduced kernel inherits admissibility by construction. No post-hoc verification needed. Drag the slider below to watch it happen.

Interactive Decomposition

Kernel Decomposition‖G − Gᵣ‖ L¹ = 4.774
Exponential terms (r)1 / 8
Show individual modes
ρ=0.036Long memory

The Problem in Detail

Transient cross-impact models describe how past trades in dd assets affect current prices through a matrix-valued convolution kernel GG. The impact cost of a trading program uu is:

CG(u)=⟨u, KGu⟩L2=∫0Tut⊤ ⁣∫0tG(t−s) us ds dtC_G(u) = \langle u,\, \mathcal{K}_G u \rangle_{L^2} = \int_0^T u_t^\top \!\int_0^t G(t-s)\, u_s\,\mathrm{d}s\,\mathrm{d}t

Rich kernels capture long memory and cross-asset structure, but they make optimal execution non-Markovian and high-dimensional. The single-exponential model trades expressiveness for tractability. The problem: reduce a general kernel to a finite-dimensional surrogate that is faithful enough to be useful and constrained enough to be honest.

The obstacle is structural. A kernel that admits no profitable round trips may, after naive truncation, become one that does. The approximation inherits the shape but loses the constraint. And the constraint was the whole point.

Error Guarantees

Both the optimal strategy and its cost under the reduced model converge to those under the full model:

∥ur⋆−u⋆∥L2≤∥u⋆∥L2η ∥G−Gr∥L1(0,T)\|u_r^\star - u^\star\|_{L^2} \leq \frac{\|u^\star\|_{L^2}}{\eta}\, \|G - G_r\|_{L^1(0,T)}

The error is controlled entirely by how well the reduced kernel approximates the original in L1L^1, with constants depending on the coercivity margin η\eta and the problem data.

The Energy Interpretation

The paper gives impact cost a physical grammar:

⟨u,KGu⟩L2+∫0T ⁣∫(0,∞)ρ zt(ρ)⊤M(dρ) zt(ρ) dt  =  S(T)  ≥  0\langle u, \mathcal{K}_G u \rangle_{L^2} + \int_0^T \!\int_{(0,\infty)} \rho\, z_t(\rho)^\top M(\mathrm{d}\rho)\, z_t(\rho)\,\mathrm{d}t \;=\; S(T) \;\geq\; 0

Storage is potential energy: impact that the market has absorbed but not yet forgotten, latent reversion waiting to act on future prices. Dissipation is thermal loss: energy that has already decayed into the microstructure and cannot be recovered. A round-trip strategy that appears costless in a naive model is, through this lens, one whose energy has been reclassified (stored or dissipated) but never destroyed.

Click any asset below to execute a trade. Watch the price impact ripple outward to correlated assets, then fade over time as the kernel decays. Assets in the same row share a sector, so the impact spreads faster within sectors.

Cross-Impact Propagationdecay governed by Gᵣ(t)0 trades
High impact
Low impact
Rows = sectors

Illustrative correlation structure. Same-sector assets (same row) share higher cross-impact. Values represent relative impact intensity, not calibrated to market data.

References

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