June 10, 2025
Mathematical model.
T - trading times
I - instruments
\Omega - sample space of possible outcomes
(\mathcal{A}_t)_{t\in T} - information at time t is a partition of \Omega
X_t\colon\mathcal{A}_t\to\boldsymbol{{{R}}}^I - prices
C_t\colon\mathcal{A}_t\to\boldsymbol{{{R}}}^I - cash flows
\tau_0 < \cdots < \tau_n, \Gamma_j\colon\mathcal{A}_{\tau_j}\to\boldsymbol{{{R}}}^I - trading strategy
\Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_s < t} \Gamma_s - position
$V_t = (_t + _t)X_t - value
A_t = \Delta_t\codt C_t - \Gamma_t\cdot X_t - amount
X_t D_t = (\sum_{u > t} C_u D_u)|_{\mathcal{A}_t}
V_t D_t = (\sum_{u > t} A_u D_u)|_{\mathcal{A}_t}
“The Structure of the Cost of Capital under Uncertainty”
The present study represents an attempt at a slightly different approach. We postulate the existence of equilibrium prices for capital assets under uncertainty, and then proceed to analyze the properties implicit in their definition. It is shown that some results can be derived without recourse to the way individuals make decisions, their detailed preferences or their subjective assessments of probabilities
Section II. T finite times. S state space, (e_t^i)_i partition of S at t\in T. Z = \boldsymbol{{{R}}}^{\cup_{t,i} e_t^i} is prospect space.