概率 Jensen 不等式应用

Prob Jensens Inequality Application

专题: Probability / 概率
难度: L2
来源: MyntBit

个人笔记

题目详情

概率 Jensen 不等式应用。投资组合经理评估潜在投资回报，认为未来价格 X 可取两种可能值（50 或 150），概率各半。经理用自然对数 ln(x) 表示给定回报的效用。

英文原题

A portfolio manager is evaluating the potential returns of an investment. They believe the investment's future price, denoted by $X$ , can take two possible values: 50 or 150, each with equal probability. The manager uses the natural logarithm, $ln(x)$ , to represent the utility of a given return.
Calculate the expected utility, $Eln(X)$ , and compare it to the utility of the expected return, $ln(EX)$ . What is the approximate numerical difference between $ln(EX)$ and $Eln(X)$ ?

解析

问题分析

解法

根据题目要求实现相应功能。核心逻辑需要：

// 核心数据结构和方法——根据题目 API 约定实现
// 1. 确定状态表示——选择支持所需操作的数据结构
// 2. 实现核心算法——确保 O(·) 时间复杂度和正确性
// 3. 处理边界条件——空输入、极值参数、并发访问

验证

用具体输入验证：构造已知输入的测试用例，确认输出匹配预期结果。

复杂度与边界

时间复杂度：取决于选用的算法
空间复杂度：取决于数据规模
关键边界条件：空输入、极值参数、并发场景下的正确性保证

英文解析

Analysis

A portfolio manager is evaluating the potential returns of an investment. The investment's future price, X, can take two possible values: 50 or 150, each with equal probability. The manager uses the natural logarithm, ln(x), to represent the utility of a given return. Jensen's inequality states that for a convex function f, E[f(X)] >= f(E[X]), and for a concave function f, E[f(X)] <= f(E[X]). Since ln(x) is concave, E[ln(X)] <= ln(E[X]).

Solution

double jensenDemonstration() {
    double E_lnX = 0.5 * std::log(50.0) + 0.5 * std::log(150.0);  // E[ln(X)]
    double ln_EX = std::log(0.5 * 50.0 + 0.5 * 150.0);             // ln(E[X])
    // E[ln(X)] = 0.5 * ln(50) + 0.5 * ln(150) = 0.5*3.912 + 0.5*5.011 = 4.4615
    // ln(E[X]) = ln(100) = 4.6052
    // E[ln(X)] < ln(E[X]) - Jensen's inequality for concave function
    return E_lnX - ln_EX;  // Should be negative (~-0.1437)
}

Complexity & Edge Cases

Time complexity: O(1)
Space complexity: O(1)
Edge cases: (1) Equality holds only when X is deterministic (2) The gap equals the risk premium for log utility (3) For convex functions (e.g., x^2), inequality reverses

Verification

Compute E[ln(X)] = 4.4615, ln(E[X]) = 4.6052. Verify E[ln(X)] < ln(E[X]). The gap (~0.1437) represents the certainty equivalent reduction due to risk.

Key Considerations

Jensen's inequality explains why geometric mean returns are always less than arithmetic mean returns for risky investments. In portfolio optimization, log utility investors prefer the geometric mean, and Jensen's inequality quantifies the cost of volatility: the more volatile the returns, the larger the gap between ln(E[X]) and E[ln(X)].