Sequential Monte Carlo methods for diffusion processes

Abstract. In this paper the filtering of partially observed diffusions, with discrete-time observations, is considered. It is assumed that only biased approximations of the diffusion can be obtained, for choice of an accuracy parameter indexed by l. A multilevel estimator is proposed, consisting of a telescopic sum of increment estimators associated to the successive levels. The work associated to O(ε 2 ) mean-square error between the multilevel estimator and average with respect to the filtering distribution is shown to scale optimally, for example as O(ε −2 ) for optimal rates of convergence of the underlying diffusion approximation. The method is illustrated on some toy examples as well as estimation of interest rate based on real S&P 500 stock price data.

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“…Finally, one can use the approach in e.g. [18] to improve the stability of the particle filtering algorithm.…”

Section: Discussionmentioning

confidence: 99%

“…Note that if the variance of the weights becomes substantial, one can use the approach in [18] to deal with this issue.…”

Section: Multilevel Particle Filtersmentioning

confidence: 99%

Multilevel Particle Filters

Jasra¹,

Kamatani²,

Law³

et al. 2017

SIAM J. Numer. Anal.

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176

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“…It can also be used in the context of the estimation of marginal expectations w.r.t. laws of jump-diffusions (and hence some control problems); see for instance [17]. The basic notion of weighting functions is also applied in [2] for high-dimensional filtering problems.…”

Section: Resultsmentioning

confidence: 99%

Some contributions to sequential Monte Carlo methods for option pricing

Sen

Jasra

Zhou

2016

Journal of Statistical Computation and Simulation

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Pricing options is an important problem in financial engineering. In many scenarios of practical interest, financial option prices associated to an underlying asset reduces to computing an expectation w.r.t. a diffusion process. In general, these expectations cannot be calculated analytically, and one way to approximate these quantities is via the Monte Carlo method; Monte Carlo methods have been used to price options since at least the 1970's. It has been seen in [9,16] that Sequential Monte Carlo (SMC) methods are a natural tool to apply in this context and can vastly improve over standard Monte Carlo. In this article, in a similar spirit to [9,16] we show that one can achieve significant gains by using SMC methods by constructing a sequence of artificial target densities over time. In particular, we approximate the optimal importance sampling distribution in the SMC algorithm by using a sequence of weighting functions. This is demonstrated on two examples, barrier options and target accrual redemption notes (TARN's). We also provide a proof of unbiasedness of our SMC estimate.

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“…In particular we use the CONDENSATION algorithm, which interlaces Bayes updates with updates to move SMC particles (using drift and diffusing of the particles), to follow a stochastic process [53]. This technique has since been applied in a variety of other classical contexts [29,54] as well as in quantum information [37]. Such methods, collectively known as state-space particle filtering, are useful for following the evolution of a stochastic process observed through a noisy measurement.…”

Section: Tomographic State Trackingmentioning

confidence: 99%

Practical Bayesian tomography

2016

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In recent years, Bayesian methods have been proposed as a solution to a wide range of issues in quantum state and process tomography. State-of-the-art Bayesian tomography solutions suffer from three problems: numerical intractability, a lack of informative prior distributions, and an inability to track time-dependent processes. Here, we address all three problems. First, we use modern statistical methods, as pioneered by Huszár and Houlsby (2012 Phys. Rev. A 85 052120) and by Ferrie (2014 New J. Phys.16 093035), to make Bayesian tomography numerically tractable. Our approach allows for practical computation of Bayesian point and region estimators for quantum states and channels. Second, we propose the first priors on quantum states and channels that allow for including useful experimental insight. Finally, we develop a method that allows tracking of time-dependent states and estimates the drift and diffusion processes affecting a state. We provide source code and animated visual examples for our methods.

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Sequential Monte Carlo methods for diffusion processes

Cited by 14 publications

References 31 publications

Multilevel Particle Filters

Multilevel Particle Filters

Some contributions to sequential Monte Carlo methods for option pricing

Practical Bayesian tomography

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