This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of the ergodic SDEs with a drift which is not globally Lipschitz over an infinite time interval. If the timestep is bounded appropriately, we show not only the stability of the numerical solution and the standard strong convergence order, but also that the bound for moments and strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo. Numerical experiments support our analysis.Several different methodologies have been developed to estimate it.First, we can compute the probability density function ρ(x) of π by solving the corresponding stationary Fokker-Planck equation, see [14]. However, the stationary Fokker-Planck equation is a partial differential equation (PDE) MSC 2010 subject classifications: 60H10, 60H35, 65C30
This paper, based on two main papers [2,3] which contains the full details of the literature review, numerical analysis and numerical experiments, aims to give an overview of the adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift in a concise structure without any proof. It shows that if the timestep is bounded appropriately, then over a finite time interval the numerical approximation is stable, and the expected number of timesteps is finite. Furthermore, the order of strong convergence is the same as usual, i.e. order 1 2 for SDEs with a non-uniform globally Lipschitz volatility, and order 1 for Langevin SDEs with unit volatility and a drift with sufficient smoothness. For a class of ergodic SDEs, we also show that the bound for the moments and the strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo method to compute the expectations with respect to the invariant measure. The analysis is supported by numerical experiments.
This paper proposes a new multilevel Monte Carlo (MLMC) method for the ergodic SDEs which do not satisfy the contractivity condition. By introducing the change of measure technique, we simulate the path with contractivity and add the Radon-Nikodym derivative to the estimator. We can show the strong error of the path is uniformly bounded with respect to T. Moreover, the variance of the new level estimators increase linearly in T, which is a great reduction compared with the exponential increase in standard MLMC. Then the total computational cost is reduced to O(ε −2 | log ε| 2 ) from O(ε −3 | log ε|) of the standard Monte Carlo method. Numerical experiments support our analysis.becomes O(ε −2 | log ε|), where the additional O(| log ε|) comes from the length of simulation time T. In [15], by simulating different time intervals T ℓ across different levels ℓ, we further reduce the computational cost to O(ε −2 ).However, a larger class of SDEs satisfying the dissipativity condition (2) does not satisfy the contractivity condition and instead only satisfies the one-sided Lipschitz condition:for some λ > 0. The major benefit of the contractivity is that two solutions to the SDE starting from different initial data but driven by the same Brownian 20 motion, will converge exponentially, which means the discretization error from previous time steps will decay exponentially, and then we can prove a uniform bound for the strong error. Without the contractivity, the strong error may increase exponentially with respect to T. Then multilevel correction variances V ℓ also increase exponentially, which, as shown in Theorem 5, increases the total 25 computational cost to O(ε −2− κ 2λ * | log ε|), where κ is the Lyapunov exponent of the system. For some SDEs with a chaotic property, the Lyapunov exponent κ can be sufficiently large such that κ 2λ * ≥ 1 and MLMC loses its advantage over the standard Monte Carlo method.In this paper, a change of measure technique is employed to deal with SDEs 30 satisfying the one-sided Lipschitz condition (6). We provide the numerical analysis only for the case of a globally Lipschitz drift but this scheme works well for SDEs with non-globally Lipschitz drift such as the stochastic Lorenz equation which is only locally one-sided Lipschitz.The key feature of this class of SDEs, especially the chaotic SDEs, is that the 35 behaviour of solutions is highly sensitive to initial conditions and the difference between the fine path and coarse path will increase exponentially. An intuitive way to avoid this kind of divergence is by adding a "spring" between the fine path and coarse path to draw them closer to each other.
Objectives. Determine the optimal, licensed, first-line anticoagulant for prevention of ischemic stroke in patients with non-valvular atrial fibrillation (AF) in England and Wales from the UK National Health Service (NHS) perspective and estimate value to decision making of further research. Methods. We developed a cost-effectiveness model to compare warfarin (international normalized ratio target range 2–3) with directly acting (or non–vitamin K antagonist) oral anticoagulants (DOACs) apixaban 5 mg, dabigatran 150 mg, edoxaban 60 mg, and rivaroxaban 20 mg, over 30 years post treatment initiation. In addition to death, the 17-state Markov model included the events stroke, bleed, myocardial infarction, and intracranial hemorrhage. Input parameters were informed by systematic literature reviews and network meta-analysis. Expected value of perfect information (EVPI) and expected value of partial perfect information (EVPPI) were estimated to provide an upper bound on value of further research. Results. At willingness-to-pay threshold £20,000, all DOACs have positive expected incremental net benefit compared to warfarin, suggesting they are likely cost-effective. Apixaban has highest expected incremental net benefit (£7533), followed by dabigatran (£6365), rivaroxaban (£5279), and edoxaban (£5212). There was considerable uncertainty as to the optimal DOAC, with the probability apixaban has highest net benefit only 60%. Total estimated population EVPI was £17.94 million (17.85 million, 18.03 million), with relative effect between apixaban versus dabigatran making the largest contribution with EVPPI of £7.95 million (7.66 million, 8.24 million). Conclusions. At willingness-to-pay threshold £20,000, all DOACs have higher expected net benefit than warfarin but there is considerable uncertainty between the DOACs. Apixaban had the highest expected net benefit and greatest probability of having highest net benefit, but there is considerable uncertainty between DOACs. A head-to-head apixaban versus dabigatran trial may be of value.
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