The Stroock-Taylor formula is a significant result in the theory of stochastic processes and stochastic differential equations (SDEs). It provides a way to express expectations of functionals involving diffusion processes, and it is closely related to Itô calculus and martingale theory. The formula is particularly valuable for analyzing the behavior of solutions to SDEs and connecting them to partial differential equations (PDEs).

1. Context and Importance

• The Stroock-Taylor formula is primarily used in the study of diffusion processes. Diffusion processes are continuous-time Markov processes characterized by stochastic dynamics that are often modeled by SDEs.
• The formula provides a bridge between probability theory and analysis, showing how expectations of stochastic processes can be represented using generators of Markov processes.

2. Mathematical Background

• Diffusion Processes: A diffusion process $X_t$ can be defined by the SDE: $dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t,$ where $b(X_t)$ is the drift term, $\sigma(X_t)$ is the diffusion coefficient, and $W_t$ is a standard Brownian motion.
• Generator $\mathcal{L}$ : The generator of a Markov process $X_t$ is an operator defined by:

$\mathcal{L}f(x) = \sum_{i} b_i(x) \frac{\partial f}{\partial x_i} + \frac{1}{2} \sum_{i, j} (\sigma \sigma^T)_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j},$

for a sufficiently smooth function \( f \).

3. The Stroock-Taylor Formula

The formula relates the expected value of a function $f(X_t)$ to the generator $\mathcal{L}$ and is expressed as:

$\mathbb{E}[f(X_t)] = f(X_0) + \mathbb{E} \left[ \int_0^t \mathcal{L}f(X_s) \, ds \right],$

where:

• $f(X_t)$ is the value of the function evaluated at the stochastic process $X_t$ ,
• $\mathbb{E}$ denotes the expectation operator,
• $\mathcal{L}f$ is the application of the generator to $f$ .

4. Interpretation

• Initial Value Contribution: The term $f(X_0)$ represents the initial value of the function applied to the process.
• Accumulated Effect: The integral $\int_0^t \mathcal{L}f(X_s) \, ds$ represents the accumulated effect of the generator $\mathcal{L}$ over the interval $[0, t]$ . This term measures how the behavior of the process influences the function $f$ through time.

5. Applications

• Stochastic Calculus: The Stroock-Taylor formula can be seen as a generalization or an alternative perspective to Itô's lemma, especially in scenarios involving expectations over time.
• Connection to PDEs: The formula is used to derive PDEs that describe the evolution of the expected value of functionals of $X_t$ . For example, if $u(x, t) = \mathbb{E}[f(X_t) \mid X_0 = x]$ , then $u(x, t)$ often satisfies the Kolmogorov backward equation:

$\frac{\partial u}{\partial t} = \mathcal{L} u.$

• Financial Mathematics: In option pricing and risk management, the formula is useful for deriving and solving PDEs associated with stochastic processes modeling asset prices.

6. Relationship to Other Results