Kolmogorov Forward or Backward Equations

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Kolmogorov Backward Equation¶

对马尔可夫半群 \(\{ Q_t\}_{t\ge0}\) 和生成元 \(\mathcal{L}\)，有

\[ \frac{\mathrm{d}}{\mathrm{d}t} Q_t f = \mathcal{L}Q_t f \]

定义 \(f_t = Q_t f\)，则有 \(\frac{\mathrm{d}}{\mathrm{d}t} f_t = \mathcal{L}f_t\)

Kolmogorov Forward Equation¶

算子的共轭（conjugate）：对算子 \(P\)，其共轭 \(p^*\) 满足 \(\langle Px, y\rangle = \langle x, P^*y\rangle\)

对于连续马尔可夫链，\(\mathbf{E}[f(X_t)]\) 有两种计算方式：站在时间 \(t\) 的角度，或者站在时间 \(t=0\) 的角度，因而有 \(\mathbf{E}[f(X_t)] = \langle f, p_t \rangle = \langle Q_tf, p_0 \rangle = \langle f, Q_t^*p_0 \rangle\)，进而 \(p_t = Q_t^*p_0\)。所以 \(Q_t\) 的共轭和离散情况下 \(P^t\) 作用相同。

同理可得，\(\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{E}[f(X_t)] = \langle \mathcal{L}f, p_t \rangle = \frac{\mathrm{d}}{\mathrm{d}t}\langle f, p_t \rangle\)，即 \(\langle f, \mathcal{L}^*p_t \rangle = \langle f,\frac{\mathrm{d}}{\mathrm{d}t}p_t \rangle\)，所以有

\[ \mathcal{L}^*p_t = \frac{\mathrm{d}}{\mathrm{d}t}p_t \]

这就是 Kolmogorov Forward Equation (KFE)。对应离散情况下的等式为：\(\pi_t - \pi_{t-1} = (P - I) \pi_{t-1}\)，\(\mathcal{L}^*\) 对应 \(P - I\)

Distribution Evolution of a Diffusion¶

对于 diffusion \(\mathrm{d}X_t = \mu(X_t)\mathrm{d}t + \sigma (X_t)\mathrm{d}B_t\)，可以用 KFE 来计算概率密度分布 \(p_t(x)\)，也就是找到算子 \(\mathcal{L}^*\)

由于

\[ \mathcal{L}f(x) = f'(x)\mu(x) + \frac{1}{2}f''(x)\sigma^2 (x) \]

可以通过计算 \(\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{E}[f(X_t)]\) 得到，

\[ \mathcal{L}^*(g) = - \frac{\partial}{\partial x}\mu g + \frac{1}{2} \frac{\partial ^2}{\partial x^2}(\sigma ^2 g) \]

那么就有

\[ \frac{\partial}{\partial t}p_t(x) = - \frac{\partial}{\partial x}\mu(x) p_t(x) + \frac{1}{2} \frac{\partial ^2}{\partial x^2}(\sigma ^2(x)p_t(x)) \]

Computing Stationary Distribution¶

设 \(\pi(x)\) 为 stationary distribution，代入上节所得式子，得

\[ \frac{\partial}{\partial x}\mu(x) p_t(x) = \frac{1}{2} \frac{\partial ^2}{\partial x^2}(\sigma ^2(x)p_t(x)) \]

Evolution of High Dimension Diffusion¶

对高维情况

\[ \mathrm{d}X_t = \mu(X_t)\mathrm{d}t + \sigma (X_t)\mathrm{d}B_t \]

这里 \(X_t, \mu(X_t), \mathrm{d}B_t \in \mathbb{R}^n, \sigma (X_t) \in \mathbb{R}^{n \times n}\)

有：

Itô formula

\[ \mathrm{d}f(X_t) = \langle \nabla f(X_t), \mu(X_t) \rangle\mathrm{d}t + \langle \nabla f(X_t), \sigma (X_t)\mathrm{d}B_t \rangle + \frac{1}{2} \operatorname{tr}(\sigma (X_t)^T \nabla ^2f(X_t) \sigma (X_t)) \mathrm{d}t \]

Multi-dimensional integration by parts formula

通过计算可以得到：

\[ \langle \mathcal{L}^*p_t, f \rangle = -\langle \nabla \cdot (\mu \cdot p_t), f \rangle + \langle f, \frac{\partial ^2}{\partial x_i \partial x_j } ([\sigma \sigma^T]_{i,j} \cdot p_t)\rangle \]

进而得到：

\[ \mathcal{L}^*f(x) = - \sum_{i \in [n]} \frac{\partial }{\partial x_i} (\mu(x) \cdot f(x)) + \frac{1}{2} \sum_{i, j \in [n]} \frac{\partial ^2}{\partial x_i \partial x_j}([\sigma (x)\sigma (x)^T]_{i, j} \cdot f(x)) \]