Market completion with derivative securities
研究在不完备市场中加入衍生品合约后市场是否完备，给出了模型系数上容易验证的条件，允许雅可比矩阵处处奇异，涵盖随机波动率模型用欧式期权完备化的经典例子。
Let $S^F$ be a $\\mathbb{P}$-martingale representing the price of a primitive\nasset in an incomplete market framework. We present easily verifiable\nconditions on model coefficients which guarantee the completeness of the market\nin which in addition to the primitive asset one may also trade a derivative\ncontract $S^B$. Both $S^F$ and $S^B$ are defined in terms of the solution $X$\nto a $2$-dimensional stochastic differential equation: $S^F_t = f(X_t)$ and\n$S^B_t:=\\mathbb{E}[g(X_1) | \\mathcal{F}_t]$. From a purely mathematical point\nof view we prove that every local martingale under $\\mathbb{P}$ can be\nrepresented as a stochastic integral with respect to the\n$\\mathbb{P}$-martingale $S := (S^F\\ S^B)$. Notably, in contrast to recent\nresults on the endogenous completeness of equilibria markets, our conditions\nallow the Jacobian matrix of $(f,g)$ to be singular everywhere on\n$\\mathbf{R}^2$. Hence they cover, as a special case, the prominent example of a\nstochastic volatility model being completed with a European call (or put)\noption.\n