概率 Random Walk 2d Recurrence

问题分析

A confused trader is trying to navigate the market's ups and downs. Imagine their portfolio value as performing a symmetric random walk on a multi-dimensional integer lattice. In a 2-dimensional market (only two assets), it is known that the trader will, with probability 1, return to their starting

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 confused trader is trying to navigate the market's ups and downs. Imagine their portfolio value as performing a symmetric random walk on a multi-dimensional integer lattice. In a 2-dimensional market (only two assets), it is known that the trader will, with probability 1, return to their starting point. This is Polya's recurrence theorem: in dimensions 1 and 2, a symmetric random walk is recurrent (returns to origin with probability 1), but in dimensions 3 and higher, it is transient (probability of return < 1).

Solution

// Simulate 2D random walk, estimate return probability
double estimateReturnProb(int max_steps, int simulations) {
    int returns = 0;
    for (int s = 0; s < simulations; ++s) {
        int x = 0, y = 0;
        for (int step = 0; step < max_steps; ++step) {
            int dir = rand() % 4;  // 4 directions in 2D
            if (dir == 0) x++; else if (dir == 1) x--;
            else if (dir == 2) y++; else y--;
            if (x == 0 && y == 0) { returns++; break; }
        }
    }
    return (double)returns / simulations;
}
// Analytical result for 2D: P(return) = 1 (recurrent)
// For 3D: P(return) ≈ 0.3405 (transient)

Complexity & Edge Cases

• Time complexity: O(simulations * max_steps)
• Space complexity: O(1)
• Edge cases: (1) In 1D and 2D, return probability converges to 1 as max_steps increases (2) In 3D+, it converges to a value < 1 (3) Expected number of steps to first return diverges even in 2D

Verification

Simulate 2D walk with increasing max_steps. Return probability should approach 1. Simulate 3D walk - probability should stabilize around 0.34, not approaching 1.

Key Considerations

Polya's theorem has implications for portfolio diversification. In a 2-asset portfolio (2D random walk), the portfolio value will eventually return to its starting point - losses are eventually recovered. In a 3+ asset portfolio, there is a nonzero probability of permanent loss. This mathematical result supports the intuition that diversification across many assets increases the risk of never recovering from a drawdown.