36. Stochastic Integral
本章从离散网格出发，把一个连续随机过程 \(\{X_t\}\) 的增量写成漂移项 + 扩散项 (36.1)，对时间求和 (36.3) 后取 \(N\to\infty\) 的极限，得到其连续时间形式——随机积分 (stochastic integral) (36.4) \(X_T-X_0=\int_0^T m_t\,dt+\int_0^T\sigma_t\,dZ_t\)。其中漂移积分 \(\int m_t\,dt\) 是逐点 (pathwise) 收敛的普通 Riemann 积分，而 \(\int\sigma_t\,dZ_t\) 是对布朗运动的随机积分（Itô 积分）。若满足 (36.4) 的过程对一切 \(T\) 存在，则称 \(\{X_t\}\) 是随机微分方程 (SDE) (36.5) \(dX_t=m_t\,dt+\sigma_t\,dZ_t\) 的解；(36.4) 与 (36.5) 含义完全相同，(36.4) 只是 (36.5) 的形式化重述。漂移 \(m_t\)、波动 \(\sigma_t\) 允许时变，且都适应于 \(\{X_t\}\) 生成的信息流 \(\mathcal F_t\)。
Starting from a discrete grid, this chapter writes the increment of a continuous process \(\{X_t\}\) as a drift term + diffusion term (36.1), sums over time (36.3), and takes the \(N\to\infty\) limit to get its continuous-time form — the stochastic integral (36.4) \(X_T-X_0=\int_0^T m_t\,dt+\int_0^T\sigma_t\,dZ_t\). Here the drift integral \(\int m_t\,dt\) is an ordinary Riemann integral that converges pathwise, while \(\int\sigma_t\,dZ_t\) is a stochastic (Itô) integral against Brownian motion. If a process satisfying (36.4) exists for all \(T\), then \(\{X_t\}\) is a solution to the stochastic differential equation (SDE) (36.5) \(dX_t=m_t\,dt+\sigma_t\,dZ_t\); (36.4) and (36.5) mean exactly the same thing, with (36.4) being a formal restatement of (36.5). The drift \(m_t\) and volatility \(\sigma_t\) may be time-varying and are both adapted to the filtration \(\mathcal F_t\) generated by \(\{X_t\}\).
36.1 From Discrete Grid to Continuous Time
考虑一个连续随机过程 \(\{X_t\}\)，在离散网格点上由 (36.1) 刻画：
Consider a continuous stochastic process \(\{X_t\}\), characterized at discrete grid points by (36.1):
$$X_{t+\Delta t}-X_t=m_t\,\Delta t+\sigma_t\,(Z_{t+\Delta t}-Z_t)\tag{36.1}$$
其中漂移 \(m_t\)、方差（波动）\(\sigma_t\) 推广为时变；\(\{m_t\}\)、\(\{\sigma_t\}\) 都适应于 \(\{X_t\}\) 生成的信息流 \(\mathcal F_t\)；\(\{Z_t\}\) 为标准高斯过程（布朗运动）。引入等距时间网格 \(\{t_0=0,t_1=\Delta t,\dots,t_N=N\Delta t=T\}\)，把 (36.1) 改写为 (36.2)：
where the drift \(m_t\) and variance (volatility) \(\sigma_t\) are generalized to be time-dependent; \(\{m_t\}\), \(\{\sigma_t\}\) are both adapted to the filtration \(\mathcal F_t\) generated by \(\{X_t\}\); \(\{Z_t\}\) is a standard Gaussian process (Brownian motion). Introduce an equally split time grid \(\{t_0=0,t_1=\Delta t,\dots,t_N=N\Delta t=T\}\), and rewrite (36.1) as (36.2):
$$X_{t_{n+1}}-X_{t_n}=m_{t_n}(t_{n+1}-t_n)+\sigma_{t_n}(Z_{t_{n+1}}-Z_{t_n})\tag{36.2}$$
对 \(n\) 求和得 (36.3)：
Summing over \(n\) gives (36.3):
$$X_T-X_0=\sum_{n=0}^{N-1}m_{t_n}\Delta t+\sum_{n=0}^{N-1}\sigma_{t_n}\Delta Z_{t_n}\tag{36.3}$$
其连续时间对应 (36.4)，定义为 (36.3) 在 \(N\to\infty\) 时的极限：
Its continuous-time analog (36.4) is defined as the limit of (36.3) when \(N\to\infty\):
$$X_T-X_0=\int_0^T m_t\,dt+\int_0^T\sigma_t\,dZ_t\tag{36.4}$$
36.2 Interpreting the Two Terms and the SDE
第一项（漂移）：由 (36.3) 对应项的逐点收敛 (pointwise convergence) 得到——对所有 \(\omega\in\Omega\)，
$$\lim_{N\to\infty}\sum_{n=0}^{N-1}m_{t_n}(\omega)\,\Delta t=\int_0^T m_t(\omega)\,dt.$$
第二项（扩散）：若满足 (36.4) 的过程 \(\{X_t\}\) 对一切 \(T\) 存在，则称 \(\{X_t\}\) 是如下随机微分方程 (SDE) (36.5) 的解：
First term (drift): obtained from the corresponding term in (36.3) by pointwise convergence — for all \(\omega\in\Omega\),
$$\lim_{N\to\infty}\sum_{n=0}^{N-1}m_{t_n}(\omega)\,\Delta t=\int_0^T m_t(\omega)\,dt.$$
Second term (diffusion): if a process \(\{X_t\}\) satisfying (36.4) exists for all \(T\), then \(\{X_t\}\) is a solution to the following stochastic differential equation (SDE) (36.5):
$$dX_t=m_t\,dt+\sigma_t\,dZ_t,\qquad X_0\text{ given}\tag{36.5}$$
(36.4) 与 (36.5) 含义完全相同——(36.4) 只是 (36.5) 的形式化重述。SDE (36.5)/(36.4) 的解未必存在，但对大多数问题，正则性条件成立，从而存在解。
(36.4) and (36.5) mean exactly the same thing — (36.4) is just a formal restatement of (36.5). The solution to the SDE (36.5)/(36.4) may or may not exist, but for most questions of interest, regularity conditions hold so that a solution exists.
References
- He, X. (2019d). Stochastic Calculus Notes by Xindi He.
- He, X. (2020–2024). Asset Pricing (lecture notes), Ch. 36.