Keywords: Stochastic Differential Equation, Verification theorem
1. Definition of SDEs
Consider a filtered probability space $(\Omega, \mathbb{F}, \mathbb{P}; \{\mathcal{F_t}\})$ on which there is a d-dimensional Brownian motion. (Note that this filtration can be larger than the filtration generated by Brownian motion, which also may include a zero measured set. Review Brownian Motion and Stochastic Calculus). The following equation systems:
$d X_t^{(i)} = \mu_i(t, X_t)dt + \sum_{j = 1}^d \sigma_{i, j}(t, X_t)dW_t^{j}$
, where $\mu(\cdot, \cdot)$ and $\sigma(\cdot, \cdot)$ are measurable functions and $\mu(t, X_t)$ and $\sigma(t, X_t)$ are \{\mathcal{F_t}\} process. Note that measurable and adapted are not equivalent since the structure of filtration is an addition to the probability space.
2. Strong Solution and Weak Solution
Strong solution: Given the filtered probability space and $\{\mathcal{F_t}\})$-adapted Brownian motion, an $\{\mathcal{F_t}\})$-adapted continuous process $X = \{X_t\}$ is called a strong solution if:
$X_t^{(i)} = X_0^{(i)} + \int \mu_i(t, X_t)dt + \sum_{j = 1}^d \int \sigma_{i, j}(t, X_t)dW_t^{j}$
Weak solution: Compared to strong solution, the probability measure $\mathbb{P}$ is also a part of the solution. There exists a probability measure $\mathbb{P}$ and an $\{\mathcal{F}_t\}$-adapted, $\mathbb{P}$-Brownian motion such that:
$X_t^{(i)} = X_0^{(i)} + \int \mu_i(t, X_t)dt + \sum_{j = 1}^d \int \sigma_{i, j}(t, X_t)dW_t^{j}$
3. Uniqueness
The uniqueness of strong solution requires Lipschitz condition. While the uniqueness in law only requires bounded $\mu$ and $\sigma$. The uniqueness in law means under different probability measure, solution $X$ and $\tilde{X}$ have the same finite-dimensional distribution. For the verification theorem in the field of Stochastic control, we should notice that most of the objective function is of the form of expectation. Therefore, the weak solution of SDE (dynamic system) is good enough to for stochastic control problem.
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