Why Jump Processes Do Not Add Brownian-Style HJB Terms

Summary

The post raises a question about deriving a Hamilton–Jacobi–Bellman equation for a high-frequency trading model with a limit order book. The model tracks cash wealth, stock price, and inventory: the stock price follows a diffusion, while inventory changes through a Poisson jump process. The question focuses on why a Taylor expansion of the value function appears to omit second-order terms for wealth and inventory, despite the nonzero quadratic variation of the Poisson process.

No derivation or answer is included, so the document does not resolve the issue or present evidence for a particular expansion. It identifies a useful distinction for readers to investigate: diffusion-driven variation and jump-driven changes enter dynamic programming calculations differently, and jump effects are generally represented through changes in the function across possible jump states. Further derivation is needed to explain the precise HJB equation in the referenced model.

Key ideas

• The model combines a diffusive stock price with Poisson-driven inventory changes.
• The question concerns how these different processes contribute to a value-function expansion.
• Jump effects require accounting for changes between states rather than treating inventory jumps as Brownian increments.
• The post contains no answer or full HJB derivation.

Tags

# How to derive the HJB equation under this paper's context?


# How to derive the HJB equation under this paper's context?












I'm reading this paper:High frequency trading in a limit order book. IN section 3.1, an HJB equatioin was given without any details. Could anyone show how to arrive this equation step by setp? I have basic knowledge about dynamic programming and hjb equation.

Actually I have seen some similar detailed derivation process in a Chinese website(https://zhuanlan.zhihu.com/p/161632470) but I'm still confused with it in some step. The main step I'm confused with is the expansition of J(x,s,q,t), where x(t) is the wealth in dollar at time t, s(t) is the stock price (s(t)=udt+sigma dBt), q(t) is the inventory at time t, which is a jump process represeted by a Poisson process N(t).

In this step

I wonder why there are no second order term with respect to x and q in this expansion? I thought the quadratic variation of Nt is not 0, so there should exist a second order term in this expansion, just like the s's term.

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