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Linear variance bounds for particle approximations of timehomogeneous FeynmanKac formulaeby Nick Whiteley, Nikolas Kantas and Ajay Jasra
This article establishes sufficient conditions for a linearintime bound on the nonasymptotic variance for particle approximations of timehomogeneous FeynmanKac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the nonasymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a nonnegative kernel, defined by the logarithmic potential function and Markov kernel which specify the FeynmanKac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a noncompact state space: (1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and (2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.
