A 0.37% simulated VaR gain over the best static schedule is a thin case for changing how a desk liquidates. Chen, Delage and Wang prove a useful result for quantile control in continuous state spaces, and one of their stopping rules has a closed form. The execution evidence carries less weight. Nothing in the paper was run on market data.
Their object is a VaR Markov decision process (MDP), which selects a policy to maximize the α-VaR of cumulative discounted reward. Dynamic programming for that objective had been confined to finite state and action spaces. Li, Zhong and Brandeau (2022) introduced it, Hau et al. (2023) corrected it, and Hau et al. (2025) gave an equivalent nested-VaR form. Chen, Delage and Wang extend those equations to Borel spaces, allowing continuous prices and inventories, across finite and infinite horizons. The trading application puts a dynamic liquidation policy into a simulated Almgren-Chriss model. Its α-VaR of revenue exceeds that of the best static schedule, fixed before trading begins.
How does the risk level move?
For a liquidation desk, α-VaR of total discounted reward is the revenue achieved in all but the worst α share of outcomes. Ordinary backward induction fails because VaR has no tower property. The authors add α to the state and let their decomposition theorem assign a risk level to each next-period state. Those levels must average to at most α. The payoff's α-VaR equals the best split of the worst conditional VaR produced by that split. Earlier results covered finite spaces; this decomposition covers any random variable on a Borel space.
The dynamic programming result requires compact action sets that vary upper-semicontinuously with the state, upper semicontinuous rewards, and a weakly continuous transition kernel. For bounded stationary rewards, the infinite-horizon version is a γ-contraction. Truncation at horizon T costs at most r̄γ^T/(1−γ).
After each period, the policy chooses the smallest next risk level whose continuation value meets the promise made at the current level. In Regime 1 of the execution model, the update is explicit: a price innovation z below the current threshold sends the level to B⁻¹(B(θ) − z). A sufficiently large positive shock sends it to 0, and the trader sells everything. A bad shock moves the trader up the distribution and leads the policy to take more risk after losses. That history dependence is the practical consequence of carrying a quantile target in the state.
The stopping threshold
The application uses discrete Almgren-Chriss: an arithmetic Gaussian walk, linear permanent impact κv, and linear temporary impact ε + ηv. The dividing line is p = κ − 2η/τ. In Regime 1, where p ≥ 0, the best static schedule is bang-bang. For α below 0.5 it sells the whole block at the first stage; for α above 0.5 it waits until the last.
The dynamic optimum in this regime reduces to a stopping rule. Sell when cumulative normalized innovations first reach B₁(α), or at the final stage. B₁(α) is the α-quantile of the running maximum of a Gaussian walk, so the trader needs no ongoing risk-level bookkeeping. For N ≥ 3, this rule strictly beats static exactly when α exceeds α₀(N) = C(2N−2, N−1)/4^(N−1). The authors obtain that cutoff from Sparre Andersen's theorem. At N = 2 the critical level is 0.5 and the strategies coincide. It first falls below 0.1 at N = 33; at N = 100 it is about 0.0566.
At α = 0.1, then, 33 or more stages make waiting preferable to the static answer of immediate liquidation. The size of the gain remains limited. Divide the dynamic-minus-static VaR difference by σ·x̄·√T: moving from 33 to 100 stages raises that normalized gain by less than 0.07.
Almgren-Chriss stays in Regime 2
Regime 1 requires N ≤ κT/(2η).
The paper's base parameters are κ = 2.5×10⁻⁷, η = 2.5×10⁻⁶, and T = 5. They put the bound at 0.25 stages. Every N ≥ 2 under that calibration is therefore in Regime 2, as the authors note. Beyond the last trading stage, no closed form exists. The authors use backward induction on an inventory grid of 1,000 intervals with 100 Gauss-Legendre nodes, then interpolate the risk level monotonically in forward simulation. At each stage, the resulting policy looks up inventory and risk level.
The discretization looks adequate. At N = 3, grids with 100, 1,000 and 10,000 intervals yield estimated VaRs within 2,000 of one another. The block has 10⁶ shares and is worth 5×10⁷ at S0 = 50.
The gains themselves are small. Across α in {0.1,..., 0.9}, the largest dynamic improvement over static is 0.37% in the base case (N = 5, σ = 0.95). At N = 3 it is 0.24%. Halving η to 1.25×10⁻⁶ lifts it to 0.62%, while σ = 2.0 takes it to 1.31%. Reaching the headline 1.93% requires both changes: more than double the base volatility and half the temporary impact.
The conclusion describes "significant potential". The abstract makes a different claim: the framework "enables the characterization of optimal dynamic strategies, resolving an open question in the field." The related-literature section identifies that question as how to characterize optimal dynamic VaR execution strategies. The paper answers it analytically in Regime 1 and numerically in Regime 2. Its characterization claim holds on those terms; small trading gains do not refute it.
What can 1,000 paths establish?
Each configuration uses 1,000 simulated paths. Static and dynamic schedules share common random numbers, and the authors report 95% bootstrap intervals. Figure 6.2 plots them, though we did not find their widths stated in the text. The largest gains occur near α = 0.5, with about 500 of the 1,000 paths below the quantile. At α = 0.1, about 100 paths fall below it, and the gains are smaller still.
The authors flag a qualification to the static benchmark: above α = 1/2, they obtain it with multi-start SLSQP, without a guarantee of global optimality. They also report that every initialization reached the same value. Outcomes beyond the quantile receive no score. We did not find a measure of losses in the worst α share of paths, where the cost of taking more risk after losses would appear. Both impact terms are linear, and whether a 0.37% VaR edge survives a nonlinear impact curve remains untested here.
Our own data cannot settle it either. Minute bars for liquid US equities and ETFs have neither order fills nor a record of our own market impact. A backtest on those bars could score only simulated execution under assumed impact coefficients. It could not establish whether the gains survive live fills; any figure it produced would depend on the same kind of model used in the paper.
A fill-level comparison would change my view if the dynamic VaR policy beat a static schedule by something near its simulated 0.37% after spread, with losses beyond the quantile reported alongside the gain.