Discounted wealth equation in discrete time
= Discounted wealth equation in discrete time
Relative to a positive <numéraire>, a predictable self-financing strategy with risky holdings $\theta_t$ and discounted risky prices $X_t$ has discounted value
$$
V_t=V_0+\sum_{s=1}^t\theta_s\mathbin{\cdot}(X_s-X_{s-1}).
$$
Thus its gains are the discrete stochastic integral of its holdings against discounted price increments.