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Stochastic Calculus / Discrete Martingales and the Gambling Blueprint

Random Walk Integrals and Doob's Maximal Inequality

A delta-hedging portfolio holds Δt\Delta_t shares of the underlying and rebalances continuously. The P&L of this strategy is an integral: ∫0TΔt dSt\int_0^T \Delta_t \, dS_t. Whether this integral is a martingale, how large it can get, and how its variance relates to the position sizes — these are the central questions of stochastic integration. In discrete time, all three have precise answers given by the random walk integral Zn=∑j=1nJjΔSjZ_n = \sum_{j=1}^n J_j \Delta S_j.
This is the closing lesson of the discrete-martingales topic, and it serves a dual purpose. First, it consolidates the three key properties of the stochastic integral in the discrete setting: the martingale property (a predictable strategy on a fair game produces a fair game), linearity (portfolio combination), and the variance rule (the discrete Itô isometry). Second, it proves Doob's maximal inequality — the bound on the running maximum of a martingale — which in continuous time becomes the Burkholder-Davis-Gundy inequality used to control the supremum of a stochastic integral. Together these results are the entire blueprint for the continuous-time theory in Topics 2–5.
Learning objectives. By the end of this note you should be able to:
  • Define the discrete stochastic integral Zn=∑j=1nJjΔSjZ_n = \sum_{j=1}^n J_j \Delta S_j and state the three properties it satisfies (martingale, linearity, variance rule).
  • Prove the variance rule from the orthogonality of martingale increments and the predictability of JjJ_j.
  • Interpret the variance rule as the discrete Itô isometry and match each term to its continuous counterpart.
  • State and prove Doob's maximal inequality for a non-negative submartingale.
  • Derive the L2L^2 maximal inequality E[Mˉn2]≤4E[Mn2]\mathbb{E}[\bar{M}_n^2] \le 4\mathbb{E}[M_n^2] for a square-integrable martingale.
  • Explain why the maximal inequality is the key estimate that makes the Itô integral extend to general adapted processes.

In this lesson

  1. The Random Walk Integral
  2. Three Core Properties
  3. Doob's Maximal Inequality
  4. Applications

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