Bound of a process in terms of total variation

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If we are ginen that the total variation of a stochastic proces $X_t$ is bounded by K can we get a bound on $|X_t|$ in terms of K?

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Not necessarily, consider the case of a process which is constant in time but has an unbounded initial condition.

However, it follows from the definition of total variation that $|X_t-X_0| \leq K$, so by the triangle inequality you only need to be able to control the initial condition $X_0$.