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Difficulty: Hard
Category: Stochastic Calculus
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Topics: stochastic-calculus, kolmogorov-equation, diffusion-process, partial-differential-equation
Consider a diffusion process $dX_t = \mu(X_t) dt + \sigma(X_t) dW_t$, where $X_t$ is the state variable at time $t$, $\mu(X_t)$ is the drift, $\sigma(X_t)$ is the diffusion coefficient, and $W_t$ is a standard Brownian motion. Let $u(x, t) = Ef(X_T) | X_t = x$ be the expected value of a function $f(X_T)$ of the state variable at a future time $T$, given that the state variable is currently at $x$ at time $t$. Which of the following partial differential equations (PDEs) does $u(x, t)$ satisfy?
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