Central limit theorem for the multilevel Monte Carlo Euler method

This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607-617] which is significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg-Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [Ann. Probab. 26 (1998) 267-307], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. A complexity of the multilevel Monte Carlo algorithm is carried out.Comment: Published in at http://dx.doi.org/10.1214/13-AAP993 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing

We study the approximation of $\mathbb{E}f(X_T)$ by a Monte Carlo algorithm, where $X$ is the solution of a stochastic differential equation and $f$ is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps $\delta$ and $\delta^{\beta},0<\beta<1$. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal $\beta$ (which turns out to be $\beta=1/2$), we establish a central limit type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.Comment: Published at http://dx.doi.org/10.1214/105051605000000511 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model

We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian L\'evy process of not necessarily bounded variation with a L\'evy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. We also present some numerical illustrations to confirm our results.Comment: 51 pages, 3 figure

Central Limit Theorem for the Multilevel Monte Carlo Euler Method and Applications to Asian Options

This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [8] and significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg Feller type for the multilevel Monte Carlo method associated to the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [15], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. We investigate the application of the Multilevel Monte Carlo method to the pricing of Asian options, by discretizing the integral of the payoff process using Riemann and trapezoidal schemes. In particular, we prove stable law convergence for the error of these second order schemes. This allows us to prove two additional central limit theorems providing us the optimal choice of the parameters with an explicit representation of the limiting variance. For this setting of second order schemes, we give new optimal parameters leading to the convergence of the central limit theorem. Complexity analysis of the Multilevel Monte Carlo algorithm were processed

Parameter Estimation for the Square-root Diffusions : Ergodic and Nonergodic Cases

22 pagesThis paper deals with the problem of parameter estimation in the Cox-Ingersoll-Ross (CIR) model $(X_t)_{t\geq 0}$. This model is frequently used in finance for example as a model for computing the zero-coupon bound price or as a dynamic of the volatility in the Heston model. When the diffusion parameter is known, the maximum likelihood estimator (MLE) of the drift parameters involves the quantities : $\int_{0}^{t}X_sds$ and $\int_{0}^{t}\frac{ds}{X_s}$. At first, we study the asymptotic behavior of these processes. This allows us to obtain various and original limit theorems on our estimators, with different rates and different types of limit distributions. Our results are obtained for both cases : ergodic and nonergodic diffusion. Numerical simulations were processed using an exact simulation algorithm

Kinetics of Mosquito-Injected Plasmodium Sporozoites in Mice: Fewer Sporozoites Are Injected into Sporozoite-Immunized Mice

Malaria is initiated when the mosquito introduces sporozoites into the skin of a mammalian host. To successfully continue the infection, sporozoites must invade blood vessels in the dermis and be transported to the liver. A significant number of sporozoites, however, may enter lymphatic vessels in the skin or remain in the skin long after the mosquito bite. We have used fluorescence microscopy of Plasmodium berghei sporozoites expressing a fluorescent protein to evaluate the kinetics of sporozoite disappearance from the skin. Sporozoites injected into immunized mice were rapidly immobilized, did not appear to invade dermal blood vessels and became morphologically degraded within several hours. Strikingly, mosquitoes introduced significantly fewer sporozoites into immunized than into non-immunized mice, presumably by formation of an immune complex between soluble sporozoite antigens in the mosquito saliva and homologous host antibodies at the proboscis tip. These results indicate that protective antibodies directed against sporozoites may function both by reducing the numbers of sporozoites injected into immunized hosts and by inhibiting the movement of injected sporozoites into dermal blood vessels

Intradermal immunization of mice with radiation-attenuated sporozoites of Plasmodium yoelii induces effective protective immunity

<p>Abstract</p> <p>Background</p> <p>Intravenous injection of mice with attenuated <it>Plasmodium berghei </it>sporozoites induces sterile immunity to challenge with viable sporozoites. Non-intravenous routes have been reported to yield poor immunity. Because intravenous immunization has been considered to be unacceptable for large scale vaccination of humans, assessment was made of the results of intradermal immunization of mice with <it>Plasmodium yoelii</it>, a rodent malaria parasite whose infectivity resembles that of human malaria.</p> <p>Methods</p> <p>Mice were immunized with two injections of isolated, radiation-attenuated <it>P. yoelii </it>sporozoites, either by intravenous (IV) or intradermal (ID) inoculation. In an attempt to enhance protective immunogenicity of ID-injections, one group of experimental mice received topical application of an adjuvant, Imiquimod, while another group had their injections accompanied by local "tape-stripping" of the skin, a procedure known to disrupt the stratum corneum and activate local immunocytes. Challenge of immunized and non-immunized control mice was by bite of sporozoite-infected mosquitoes. Degree of protection among the various groups of mice was determined by microscopic examination of stained blood smears. Statistical significance of protection was determined by a one-way ANOVA followed by Tukey's <it>post hoc </it>test.</p> <p>Results</p> <p>Two intravenous immunizations produced 94% protection to mosquito bite challenge; intradermal immunization produced 78% protection, while intradermal immunization accompanied by "tape-stripping" produced 94% protection. There were no statistically significant differences in degree of protective immunity between immunizations done by intravenous versus intradermal injection.</p> <p>Conclusions</p> <p>The use of a sub-microlitre syringe for intradermal injections yielded excellent protective immunity. ID-immunization with large numbers of radiation-attenuated <it>P. yoelii </it>sporozoites led to levels of protective immunity comparable to those achieved by IV-immunization. It remains to be determined whether an adjuvant treatment can be found to substantially reduce the numbers of attenuated sporozoites required to achieve a strong protective immunity with as few doses as possible for possible extension to immunization of humans.</p

Quantifying Uncertainty with a Derivative Tracking SDE Model and Application to Wind Power Forecast Data

We develop a data-driven methodology based on parametric It\^{o}'s Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of forecast errors. Our SDE framework features time-derivative tracking of the forecast, time-varying mean-reversion parameter, and an improved state-dependent diffusion term. Proofs of the existence, strong uniqueness, and boundedness of the SDE solutions are shown under a principled condition for the time-varying mean-reversion parameter. Inference based on approximate likelihood, constructed through the moment-matching technique both in the original forecast error space and in the Lamperti space, is performed through numerical optimization procedures. We propose another contribution based on the fixed-point likelihood optimization approach in the Lamperti space. All the procedures are agnostic of the forecasting technology, and they enable comparisons between different forecast providers. We apply our SDE framework to model historical Uruguayan normalized wind power production and forecast data between April and December 2019. Sharp empirical confidence bands of future wind power production are obtained for the best-selected model.Comment: 28 pages and 11 figure