I would like to compute the following tail, $$ \mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right), $$ assuming $$ \mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x), $$ and $X$ is a diffusion process, meaning that $ \mathrm{dX_t}= b(X_t) \mathrm{dW_t}+ c(X_t)\mathrm{dt} $ where $W$ is a Brownian motion, and $b$ and $c$ are given functions. Thanks in advance.

  • $\begingroup$ It depends on what your initial data X_0 is and any constraints on the SDE coefficients. Time integrals of stochastic processes often depend on the ergodicity of your process. If your process starts with an invariant measure, then the time integral will look like a scaled version of your process, as a random variable. $\endgroup$ – Kevin Yang Jan 7 '17 at 22:14

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