Deriving an HJB Equation for Optimal Trade Execution

Summary

The document presents a question about deriving the Hamilton–Jacobi–Bellman equation for an optimal execution problem in the Almgren–Chriss framework. The value function minimizes expected costs over a finite horizon, combining a quadratic trading-rate penalty with inventory exposure to the asset price. The price is modeled as a driftless geometric Brownian motion, while inventory changes with the trading rate.

The proposed equation combines diffusion in the price state, the running inventory cost, and an optimization over the trading rate that accounts for its effect on future value. The document does not provide a derivation or an answer; it records the setup and the author's difficulty applying the Bellman principle. Readers should therefore treat the displayed equation as the question's asserted form, rather than a result justified within the text. The discussion also leaves details of the control and state conventions unexplored.

Key ideas

• The objective minimizes expected execution and inventory costs over a finite horizon.
• Asset price is assumed to follow driftless geometric Brownian motion.
• The proposed HJB includes price diffusion, a running inventory cost, and optimization over trading rate.
• The document poses the derivation problem but supplies no worked solution.

Tags

# How to derive this HJB equation?


# How to derive this HJB equation?












I'm reading the paper by J.Gatheral and A.Schied (2012) - "Optimal Trade Execution under Geometric Brownian Motion in the Almgren and Chriss Framework".

On page 6, the authors provide a value function, which maps initial conditions with minimal cost value ($T$ - Time remaining to expire position, $X$ - value of position, $S_0$ - asset price at first moment of time). The function can be expressed as: $$ C(T,X,S_0) = \inf_{v \in V(T,X)}\mathbb{E}\left[\int_0^Tv^2_tdt + \lambda \int_0^Tx_t^v S_tdt \right] $$

I'm struggling with deriving the HJB equation for this case, which has the form: $$ C_T = \frac{1}{2}\sigma^2S^2C_{SS} + \lambda SX + \inf_{v \in \mathbb{R}}\left(v^2 - vC_x\right) $$ It is assumed that the asset price $S$ follows a geometrical Brownian motion with no drift, $dS_t = \sigma S_t dt$. I'm familiar with the Bellman principle, but it still seems unobvious to me.

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