Modern Stochastics: Theory and Applications

We establish an exact martingale Φ-entropy balance on a Brownian–Poisson product filtration with deterministic jump compensator. For terminal variables with positive lower and finite upper bounds, the balance separates the Gaussian energy from the nonlinear jump remainder on an explicit representation domain. We formulate the Brownian–Poisson predictable representation and ${L^{2}}$-Clark projection formulas in the form needed for this identity. Marked predictable projections provide compatible versions of the add-one difference and preserve the positivity required by the entropy remainder; finite-window arguments justify the passage between integrable and square-integrable representations. This first part supplies the entropy-balance framework used for the evolution-family formulas and concentration estimates in Part II.