## Biography

Alexander Novikov is Professor of Mathematics at the School of Mathematical and Pfysical Sciences, UTS.Prior to this appointment in 1999 he was Leading Research Fellow at the Steklov Mathematical Institute (Moscow, since 1970) and Senior Lecture at the University of Newcastle (Australia, from 1996 to 1999).He received a PhD in Mathematics in 1972 and his Doctor of Science degree in 1982, both from the Steklov Mathematical Institute. He has published more than 90 research papers in different areas of stochastic processes, statistics of random processes, sequential analysis, random fields and mathematical finance. He has also been invited to more than 80 visiting appointments at leading mathematical institutions.

## Professional

Bernoulli Society, Steklov Mathematical Institute,

#### Can supervise: YES

#### Research Interests

My current research interests are in stochastic analysis, mathematical finance and statistics of random processes.Particular areas of interest include option pricing, credit risk modelling, change-point analysis, boundary crossing probabilities, Monte Carlo methods, long memory processes, goodness-of-fit test.

#### Teaching Areas

Current Teaching:Stochastic Processes (37363)Probability Theory and Stochastic Calculus in Finance (25875)Advanced Stochastic Processes (37464)Past Teaching:Time SeriesRegression AnalysisMathematical StatisticsProbability TheorySurvival Analysis

#### Publications

Hinz, J, Grigoryev, I & Novikov, A 2020, 'An application of high-dimensional statistics to predictive modeling of grade variability', *Geosciences (Switzerland)*, vol. 10, no. 4.View/Download from: Publisher's site

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© 2020 by the authors. Licensee MDPI, Basel, Switzerland. The economic viability of a mining project depends on its efficient exploration, which requires a prediction of worthwhile ore in a mine deposit. In this work, we apply the so-called LASSO methodology to estimate mineral concentration within unexplored areas. Our methodology outperforms traditional techniques not only in terms of logical consistency, but potentially also in costs reduction. Our approach is illustrated by a full source code listing and a detailed discussion of the advantages and limitations of our approach.

Asatryan, AA & Novikov, A 2018, 'Anderson localization of classical waves in weakly scattering one-dimensional Levy lattices', *Physical Review B (Condensed Matter)*, vol. 98, no. 23, pp. 235144-1-235144-8.View/Download from: Publisher's site

Borovkov, K, Mishura, Y, Novikov, A & Zhitlukhin, M 2018, 'New and refined bounds for expected maxima of fractional Brownian motion', *Statistics and Probability Letters*, vol. 137, pp. 142-147.View/Download from: Publisher's site

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© 2018 Elsevier B.V. For the fractional Brownian motion BHwith the Hurst parameter value H in (0,1∕2), we derive new upper and lower bounds for the difference between the expectations of the maximum of BHover [0,1] and the maximum of BHover the discrete set of values in−1, i=1,…,n. We use these results to improve our earlier upper bounds for the expectation of the maximum of BHover [0,1] and derive new upper bounds for Pickands’ constant.

Dachian, S, Kordzakhia, N, Kutoyants, YA & Novikov, A 2018, 'Estimation of cusp location of stochastic processes: a survey', *Statistical Inference for Stochastic Processes*, vol. 21, no. 2, pp. 345-362.View/Download from: Publisher's site

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© 2018, Springer Science+Business Media B.V., part of Springer Nature. We present a review of some recent results on estimation of location parameter for several models of observations with cusp-type singularity at the change point. We suppose that the cusp-type models fit better to the real phenomena described usually by change point models. The list of models includes Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and the classical case of i.i.d. observations. We describe the properties of the maximum likelihood and Bayes estimators under some asymptotic assumptions. The asymptotic efficiency of estimators are discussed as well and the results of some numerical simulations are presented. We provide some heuristic arguments which demonstrate the convergence of log-likelihood ratios in the models under consideration to the fractional Brownian motion.

Gushchin, A, Kordzakhia, N & Novikov, A 2018, 'Translation invariant statistical experiments with independent increments', *Statistical Inference for Stochastic Processes*, vol. 21, no. 2, pp. 363-383.View/Download from: Publisher's site

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© 2018, Springer Science+Business Media B.V., part of Springer Nature. We provide a full description of the class of translation invariant experiments with independent increments. Necessary and sufficient conditions for the weak convergence and the comparison of experiments within this class are given. Finally, we prove exponential boundedness of Pitman estimators in these models.

Kordzakhia, NE, Kutoyants, YA, Novikov, AA & Hin, LY 2018, 'On limit distributions of estimators in irregular statistical models and a new representation of fractional Brownian motion', *Statistics and Probability Letters*, vol. 139, pp. 141-151.View/Download from: Publisher's site

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© 2018 Elsevier B.V. We provide new results concerning the limit distributions of Bayesian estimators (BE) and maximum likelihood estimators (MLE) of location parameters of cusp-type signals in “signal plus white noise” models. The limit distributions of BE and MLE are expressed in terms of fractional Brownian motion (fBm) with the Hurst parameter H, 0

Andréasson, JG, Shevchenko, PV & Novikov, A 2017, 'Optimal consumption, investment and housing with means-tested public pension in retirement', *Insurance: Mathematics and Economics*, vol. 75, pp. 32-47.View/Download from: Publisher's site

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© 2017 Elsevier B.V. In this paper, we develop an expected utility model for retirement behaviour in the decumulation phase of Australian retirees with sequential family status subject to consumption, housing, investment, bequest, and government-provided means-tested Age Pension. We account for mortality risk and risky investment assets, and we introduce a “health” proxy to capture the decreasing level of consumption for older retirees. Then, we find the optimal housing at retirement, the optimal consumption and optimal risky asset allocation depending on age and wealth. The model is solved numerically as a stochastic control problem, and it is calibrated using the maximum likelihood method with empirical data of consumption and housing from the Australian Bureau of Statistics 2009–2010 Survey. The model fits the characteristics of the data well to explain the behaviour of Australian retirees. The key findings are as follows. First, the optimal policy is highly sensitive to means-tested Age Pension early in retirement, but this sensitivity fades with age. Second, the allocation to risky assets shows a complex relationship with the means-tested Age Pension. As a general rule, when wealth decreases, the proportion allocated to risky assets increases, because the Age Pension works as a buffer against investment losses. Third, couples can be more aggressive with risky allocations owing to their longer life expectancy compared with singles.

Borovkov, K, Mishura, Y, Novikov, A & Zhitlukhin, M 2017, 'Bounds for expected maxima of Gaussian processes and their discrete approximations', *Stochastics: An International Journal of Probability and Stochastic Processes*, vol. 89, no. 1, pp. 21-37.View/Download from: Publisher's site

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© 2015 Taylor & Francis The paper deals with the expected maxima of continuous Gaussian processes (Formula presented.) that are Hölder continuous in (Formula presented.)-norm and/or satisfy the opposite inequality for the (Formula presented.)-norms of their increments. Examples of such processes include the fractional Brownian motion and some of its “relatives” (of which several examples are given in the paper). We establish upper and lower bounds for (Formula presented.) and investigate the rate of convergence to that quantity of its discrete approximation (Formula presented.). Some further properties of these two maxima are established in the special case of the fractional Brownian motion.

Kordzakhia, N, Novikov, A & Ycart, B 2017, 'Approximations for weighted Kolmogorov–Smirnov distributions via boundary crossing probabilities', *Statistics and Computing*, pp. 1-11.View/Download from: Publisher's site

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© 2016 The Author(s)A statistical application to Gene Set Enrichment Analysis implies calculating the distribution of the maximum of a certain Gaussian process, which is a modification of the standard Brownian bridge. Using the transformation into a boundary crossing problem for the Brownian motion and a piecewise linear boundary, it is proved that the desired distribution can be approximated by an n-dimensional Gaussian integral. Fast approximations are defined and validated by Monte Carlo simulation. The performance of the method for the genomics application is discussed.

Novikov, A & Kaji, S 2017, 'On distibutions of first passage times of martingales arising in some gambling problems', *Japan Journal of Industrial and Applied Mathematics*, vol. 34, no. 3, pp. 859-871.View/Download from: Publisher's site

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© 2017, The JJIAM Publishing Committee and Springer Japan KK. Using a martingale technique we derive bounds and asymptotics for tail distributions of first passage times τ b associated with crossing a level b for some gambling strategies. In particilar, for the case of martingale games with so-called “Oscar strategy” we show that P{τb > n}≤Cn-3/2 for any level b > 0.

Novikov, A, Alexander, S, Kordzakhia, N & Ling, T 2017, 'Pricing of asian-type and basket options via bounds', *Theory of Probability and Its Applications*, vol. 61, no. 1, pp. 94-106.View/Download from: Publisher's site

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© 2017 Society for Industrial and Applied Mathematics. This paper sets out to provide a general framework for the pricing of average-type options via lower and upper bounds. This class of options includes Asian, basket, and options on the volume-weighted average price. The use of lower and upper bounds is proposed in response to the inherent difficulty in finding analytical representations for the true price of these options and the requirement for numerical procedures to be fast and efficient. We demonstrate that in some cases lower bounds allow for the dimensionality of the problem to be reduced and that these methods provide reasonable approximations to the price of the option.

Alexander, S, Novikov, A & Kordzakhia, N 2016, 'Bounds On Prices For Asian Options Via Fourier Methods', *ANZIAM Journal*, vol. 57, no. 3, pp. 299-318.View/Download from: Publisher's site

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The problem of pricing arithmetic Asian options is nontrivial, and has attracted much

interest over the last two decades. This paper provides a method for calculating bounds

on option prices and approximations to option deltas in a market where the underlying

asset follows a geometric L´evy process. The core idea is to find a highly correlated,

yet more tractable proxy to the event that the option finishes in-the-money. The paper

provides a means for calculating the joint characteristic function of the underlying

asset and proxy processes, and relies on Fourier methods to compute prices and deltas.

Numerical studies show that the lower bound provides accurate approximations to prices

and deltas, while the upper bound provides good though less accurate results.

Zhu, SP & Novikov, A 2016, 'Editorial: stochastic and computational methods in finance', *ANZIAM Journal*, vol. 57, no. 3, pp. 205-206.View/Download from: Publisher's site

Novikov, A & Shiryaev, AN 2014, 'Discussion on “Sequential Estimation for Time Series Models” by T. N. Sriram and Ross Iaci', *Sequential Analysis*, vol. 33, no. 2, pp. 182-185.View/Download from: Publisher's site

Novikov, A, Kordzakhia, N & Ling, T 2014, 'On Moments of Pitman Estimators: The Case of Fractional Brownian Motion', *Theory of Probability & Its Applications*, vol. 58, no. 4, pp. 601-614.View/Download from: Publisher's site

Novikov, AA & Kordzakhia, NE 2014, 'Lower and upper bounds for prices of Asian-type options', *Proceedings of the Steklov Institute of Mathematics*, vol. 287, no. 1, pp. 225-231.View/Download from: Publisher's site

Cetin, U, Novikov, A & Shiryaev, AN 2013, 'Bayesian Sequential Estimation of a Drift of Fractional Brownian Motion', *Sequential Analysis: Design Methods and Applications*, vol. 32, no. 3, pp. 288-296.View/Download from: Publisher's site

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We solve explicitly a Bayesian sequential estimation problem for the drift parameter of a fractional Brownian motion under the assumptions that a prior density of is Gaussian and that a penalty function is quadratic or Dirac-delta. The optimal stopping time for this case is deterministic. Keywords: Fractional Brownian motion; Penalty function; Sequential estimation. Subject Classifications: 62L12; 62F15; 60G22.

Novikov, A & Kordzakhia, N 2013, 'Pitman Estimators: An Asymptotic Variance Revisited', *Theory of Probability and its Applications*, vol. 57, no. 3, pp. 521-529.View/Download from: Publisher's site

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We provide an analytic expression for the variance of ratio of integral functionals of fractional Brownian motion which arises as an asymptotic variance of Pitman estimators for a location parameter of independent identically distributed observations. The expression is obtained in terms of derivatives of a logarithmic moment of the integral functional of limit likelihood ratio process (LLRP). In the particular case when the LLRP is a geometric Brownian motion, we show that the established expression leads to the known representation of the asymptotic variance of Pitman estimator in terms of Riemann zeta-function.

Novikov, A & Shiryaev, A 2013, 'Remarks on moment inequalities and identities for martingales', *Statistics & Probability Letters*, vol. 83, pp. 1260-1261.View/Download from: Publisher's site

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We present some comments on moment inequalities and identities for martingales in the context of the paper of Langovoy

Skryabin, I, Maisano, J, Novikov, A, Gazarian, T & Radchik, A 2013, 'Ensuring long term investment for large scale solar power stations: Hedging instruments for green power', *Solar Energy*, vol. 98, no. Part B, pp. 167-179.View/Download from: Publisher's site

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There is a general consensus that solar power is one of the cleanest energy technologies available. Nevertheless, investment in large-scale Solar Power Generators (SPGs) is largely impeded by the intermittent nature of solar power. Since the electricity market has a critical responsibility to maintain the reliability of energy supply, the SPG can be registered only as the market semi-scheduled generator (AEMC, 2011). This option excludes the advantages of providing baseload supply, which in turn impedes efficient market contracting for SPGs. The existing approach relies on energy storage or co-generation facilities to be built at the same connection point as the SPG to compensate for output shortages when there is insufficient sunlight. The co-located facilities require significant additional investment in infrastructure. This paper proposes a market based financial approach that does not require an additional construction effort. The approach financially links solar or other intermittent power generation with a gas-fired station through a set of tailored swap-type instruments.

Novikov, A, Christensen, S & Irle, A 2011, 'An elementary approach to optimal stopping problems for AR(1) sequences', *Sequential Analysis*, vol. 30, no. 1, pp. 79-93.View/Download from: Publisher's site

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Optimal stopping problems form a class of stochastic optimization problems that has a wide range of applications in sequential statistics and mathematical finance. Here we consider a general optimal stopping problem with discounting for autoregressive processes. Our strategy for a solution consists of two steps: First we give elementary conditions to ensure that an optimal stopping time is of threshold type. Then the resulting one-dimensional problem of finding the optimal threshold is to be solved explicitly. The second step is carried out for the case of exponentially distributed innovations.

Hinz, J & Novikov, A 2010, 'On fair pricing of emission-related derivatives', *Bernoulli journal*, vol. 16, no. 4, pp. 1240-1261.View/Download from: Publisher's site

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Tackling climate change is at the top of many agendas. In this context, emission trading schemes are considered as promising tools. The regulatory framework for an emission trading scheme introduces a market for emission allowances and creates a need for risk management by appropriate financial contracts. In this work, we address logical principles underlying their valuation.

Mititelu, G, Areepong, Y, Sukparungsee, S & Novikov, A 2010, 'Explicit analytical solutions for the average run length of CUSUM and EWMA charts', *East-West Journal of Mathematics*, vol. special, no. 1, pp. 253-265.

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NA

Novikov, A & Shiryaev, A 2010, 'ON MARTINGALE PROOF OF THE KOLMOGOROV AND SMIRNOV DISTRIBUTIONS', *Sequential Analysis*, vol. 29, no. 4, pp. 439-443.

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this is appendix to the Shiryaev respose

Novikov, A, Liptser, R & Tartakovsky, AG 2010, 'Preface: Celebrating Albert Shiryaev's 75th Anniversary', *Sequential Analysis*, vol. 29, no. 2, pp. 107-111.

Novikov, A 2009, 'On Distributions Of First Passage Times And Optimal Stopping Of Ar(1) Sequences', *Theory of Probability and its Applications*, vol. 53, no. 3, pp. 419-429.View/Download from: Publisher's site

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Sufficient conditions for the exponential boundedness of first passage times of autoregressive (AR(1)) sequences are derived in this paper. An identity involving the mean of the first passage time is obtained. Further, this identity is used for finding a logarithmic asymptotic of the mean of the first passage time of Gaussian AR(1)-sequences from a strip. Accuracy of the asymptotic approximation is illustrated by Monte Carlo simulations. A corrected approximation is suggested to improve accuracy of the approximation. An explicit formula is derived for the generating function of the first passage time for the case of AR(1)-sequences generated by an innovation with the exponential distribution. The latter formula is used to study an optimal stopping problem.

Novikov, A 2009, 'Some remarks on distributions and expectation of exit times of AR(1) sequences', *Teoriya Veroyatnostei i ee Primeneniya*, vol. 53, no. 3, pp. 459-471.

Shiryaev, AN & Novikov, A 2009, 'On a stochastic version of the trading rule 'Buy and Hold'', *Statistics and Decision*, vol. 26, no. 4, pp. 289-302.

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The paper deals with the problem of finding an optimal one-time rebalancing strategy assuming that in the BlackâScholes model the drift term of the stock may change its value spontaneously at some random non-observable (hidden) time. The problem is studied on a finite time interval under two criteria of optimality (logarithmic and linear). The methods of the paper are based on the results for the quickest detection of drift change for Brownian motion.

Borovkov, K & Novikov, A 2008, 'On exit times of Levy-driven Ornstein-Uhlenbeck processes', *Statistics & Probability Letters*, vol. 78, no. 12, pp. 1517-1525.View/Download from: Publisher's site

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We prove two martingale identities which involve exit times of Levy-driven Ornstein-Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Levy process are exponentially distributed. © 2008 Elsevier B.V. All rights reserved.

Novikov, A & Kordzakhia, N 2008, 'Martingales and first passage times of AR(1) sequences', *Stochastics. An International Journal of Probability and Stochastic Processes*, vol. 80, no. 2-3, pp. 197-210.View/Download from: Publisher's site

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Using the martingale approach we find sufficient conditions for exponential boundedness of first passage times over a level for ergodic first order autoregressive sequences.

Schmidt, T & Novikov, A 2008, 'A Structural Model with Unobserved Default Boundary', *Applied Mathematical Finance*, vol. 15, no. 2, pp. 183-203.View/Download from: Publisher's site

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A firm-value model similar to the one proposed by Black and Cox (1976) is considered. Instead of assuming a constant and known default boundary, the default boundary is an unobserved stochastic process. Interestingly, this setup admits a default intensity, so the reduced form methodology can be applied.

Novikov, A & Shiryaev, AN 2007, 'On solution of the optimal stopping problem for processes with independent increments', *Stochastics. An International Journal of Probability and Stochastic Processes*, vol. 79, no. 3-4, pp. 393-406.View/Download from: Publisher's site

Roberts, DO & Novikov, A 2007, 'Pricing European and Discretely Monitored Exotic Options under the Levy Process Framework', *The Mathematica Journal*, vol. 10, no. 3, pp. 489-500.

Sukparungsee, S & Novikov, A 2006, 'On EWMA procedure for detection of a change in observation via Martingale approach', *KMITL Science Journal*, vol. 6, no. 2a, pp. 373-380.

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Using martingale technique wepresent analytic approximation and exact lower bounds for the expectation of the first passage times of an Exponentially Weighted Moving Average (EWMA) procedure used for monitoring changes in distributions. Based on these results, a simple numericalprocedure for finding optimal parameters of EWMA for small changes in the means of observation processes is established.

Borovkov, K & Novikov, A 2005, 'Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process', *Journal Of Applied Probability*, vol. 42, no. 1, pp. 82-92.View/Download from: Publisher's site

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We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation

Novikov, A & Shiryaev, AN 2005, 'On an effective solution of the optimal stopping problem for random walks', *Theory of Probability and its Applications*, vol. 49, no. 2, pp. 344-354.View/Download from: Publisher's site

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We find a solution of the optimal stopping problem for the case when a reward function is an integer function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval {0, 1, ? , T} converges with an exponential rate as T approaches infinity to the limit under the assumption that jumps of the random walk are exponentially bounded

Novikov, A, Melchers, R, Shinjikashvili, E & Kordzakhia, N 2005, 'First passage time of filtered Poisson process with exponential shape function', *Probabilistic Engineering Mechanics*, vol. 20, no. 1, pp. 57-65.View/Download from: Publisher's site

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Solving some integro-differential equation we find the Laplace transform of the first passage time for filtered Poisson process generated by pulses with uniform or exponential distributions. Also, the martingale technique is applied for approximations of

Novikov, A 2004, 'Martingales and first-passage times for ornstein-uhlenbeck processes with a jump component', *Theory of Probability and its Applications*, vol. 48, no. 2, pp. 288-303.View/Download from: Publisher's site

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Using martingale technique, we show that a distribution of the first-passage time over a level for the Ornstein-Uhlenbeck process with jumps is exponentially bounded. In the case of absence of positive jumps, the Laplace transform for this passage time is found. Further, the maximal inequalities are also given when the marginal distribution is stable.

Novikov, A 2003, 'Martingales and first-exit times for the Ornstein-Uhlenbeck process with jumps', *Theory of Probability and its Applications*, vol. 48, no. 2, pp. 340-358.

Novikov, A, Frishling, V & Kordzakhia, N 2003, 'Time-dependent barrier options and boundary crossing probabilities', *Georgian Mathematical Journal*, vol. 10, no. 2, pp. 325-334.

Borovkov, K & Novikov, A 2002, 'On a new approach to calculating expectations for option pricing', *Journal of Applied Probability*, vol. 39, no. N/A, pp. 889-895.

Borovkov, K & Novikov, A 2002, 'On a new approach to calculating expectations for option pricing', *Journal of Applied Probability*, vol. 39, no. 4, pp. 889-895.View/Download from: Publisher's site

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We discuss a simple new approach to calculating expectations of a specific form used for the pricing of derivative assets in financial mathematics. We show that in the 'vanilla case', the expectations can be found by simply integrating the respective moment generating function with a certain weight. In situations corresponding to barrier-type options, we just need to carry out one more integration. The suggested approach appears to be the first (and, apart from Monte Carlo simulation, the only) one to allow the pricing of discretely monitored exotic options when the underlying asset is modelled by a general Lévy process. We illustrate the method numerically by calculating the price of a discretely monitored lookback call option in the cases when the underlying follows the geometric Brownian and variance-gamma processes.

Miyahara, Y & Novikov, A 2002, 'Geometric Levy Process Pricing Model', *Proceedings of the Stekov Institute of Mathematics*, vol. 237, no. 2, pp. 185-200.

Borovkov, K & Novikov, A 2001, 'On a Piece-Wise Deterministic Markov Process Model', *Statistics & Probability Letters*, vol. 53, pp. 421-428.View/Download from: Publisher's site

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We study a piece-wise deterministic Markov process having jumps of i.i.d. sizes with a constant intensity and decaying at a constant rate (a special case of a storage process with a general release rule). Necessary and su4cient conditions for the process to be ergodic are found, its stationary distribution is found in explicit form. Further, the Laplace transform of the 6rst crossing time of a 6xed barrier by the process is shown to satisfy a Fredholm equation of second kind. Solution to this equation is given by exponentially fast converging Neumann series; convergence rate of the series is estimated. Our results can be applied to an important reliability problem.

Le Breton, A & Novikov, AA 1999, 'On stochastic approximation procedures with averaging', *Theory of Probability and its Applications*, vol. 44, no. 3, pp. 591-605.

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Linear multidimensional stochastic approximation procedures in continuous time with martingale errors are considered. An asymptotic behavior of the estimator obtained by trajectory averaging is studied. An asymptotics of integrated squared deviations functionals of the averaged estimator is found. Some results concerning fixed-size confidence regions are presented.

Novikov, A & Valkeila, E 1999, 'On some maximal inequalities for fractional Brownian motions', *Statistics and Probability Letters*, vol. 44, no. 1, pp. 47-54.

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We prove some maximal inequalities for fractional Brownian motions. These extend the Burkholder-Davis-Gundy inequalities for fractional Brownian motions. The methods are based on the integral representations of fractional Brownian motions with respect to a certain Gaussian martingale in terms of beta kernels. © 1999 Elsevier Science B.V.

Novikov, A, Frishling, V & Kordzakhia, N 1999, 'Approximations of boundary crossing probabilities for a Brownian motion', *Journal of Applied Probability*, vol. 36, no. 4, pp. 1019-1030.View/Download from: Publisher's site

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Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

Novikov, A, Frishling, V & Kordzakhia, N 1999, 'Approximations of boundary crossing probabilities for a brownian motion', *Journal of Applied Probability*, vol. 36, no. 4, pp. 1019-1030.View/Download from: Publisher's site

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Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series. © 1999 Applied Probability Trust.

Novikov, AA 1998, 'Hedging of options with a given probability', *Theory of Probability and its Applications*, vol. 43, no. 1, pp. 135-144.

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We consider a model of a complete market with two assets under the suggestion that an investor may hedge the payoff function with the given probability; in othei words, the investor should have capital not less than the given payoff function with probability not less than 1 - α (α is a given significance level). Under some limitations on a class of hedging strategies we find a lower bound for an option price (that it, for the initial capital of the investor) and construct a hedge (the investor strategy) for which this lower bound is achieved. For examples, we calculate the price and hedge of a European call option and also an American call option with a barrier condition.

Bastrikov, AN, Kim, AA, Koval'chuk, BM, Kremnev, VV, Kumpyak, EV, Novikov, AA & Tsoi, NV 1997, 'Low-inductance multigap spark modules', *Russian Physics Journal*, vol. 40, no. 12, pp. 1125-1134.View/Download from: Publisher's site

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We outline the design concept for low-inductance high-current spark modules at a voltage level of 100 kV and a current of 1 MA. We present the results of an investigation of the switching and operating characteristics of multichannel, multigap spark modules as a function of the design and the shape and amplitude of the beam pulse. We give a description of the designs and parameters of the developed types of spark modules. ©1998 Plenum Publishing Corporation.

Novikov, AA 1996, 'Martingales, Tauberian theorem, and strategies of gambling', *Theory of Probability and its Applications*, vol. 41, no. 4, pp. 716-729.View/Download from: Publisher's site

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Using the Tauberian theorem, we get an asymptotic relation between the tail of the distribution of the quadratic characteristic of a martingale and the expectation of its terminal value. In case of continuous martingales the following result is proven: if τ is a stopping time for a standard Wiener process Wt with integrable terminal value Wτ, then (1) lim inf t→∞ (P{τ > t}√t) ≧ √2/π|EWτ|. Using a related result for discrete time martingales, we study asymptotic characteristics of some strategies of gambling and, in particular, Oscar's strategy.

Le Breton, A & Novikov, A 1995, 'Some results about averaging in stochastic approximation', *Metrika*, vol. 42, no. 1, pp. 153-171.View/Download from: Publisher's site

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The paper presents some results concerning the averaging approach in a "general" linear regression model in one dimension under suitable conditions about the martingale structure of errors. At first asymptotics of the primary and averaged estimators are discussed. Then it is shown that variances of estimators can be consistently estimated by appropriate integrated squared deviations functionals. Finally applications to the construction of confidence regions are considered. © 1995 Physica-Verlag.

Bush, AA, Kitaev, YE, Limonov, MF, Markov, YF, Novikov, AA & Evarestov, RA 1992, 'Normal modes of BiSrCaCuO high-temperature superconductors: layer-by-layer approach', *Physica C: Superconductivity and its applications*, vol. 190, no. 4, pp. 477-482.View/Download from: Publisher's site

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The non-superconducting Bi1-x(Sr1-yCay)xO1.5- x 2 compounds and the Bi2Sr2 Can-1CunO2n+4 (n=1.2) superconductors have been synthesized and investigated. Using the method of band representations of space groups the complete group symmetry analysis of the normal vibrational modes has been made. Comparing the Raman spectra of the row of bismuth compounds and using the results of the group theory analysis all the strong RSS-lines have been attributed. © 1992.

Novikov, AA 1992, 'A test bed for testing capacitors', *Measurement Techniques*, vol. 35, no. 11, pp. 1323-1325.View/Download from: Publisher's site

Bugaev, SP, Volkov, AM, Iskol'dsky, AM, Kim, AA, Koval'chuk, BM, Kokshenev, VA, Mesyats, GA, Novikov, AA & Yakovlev, VP 1990, 'A Terawatt Pulsed-Power Generator With a Microsecond Plasma-Opening Switch', *IEEE Transactions on Plasma Science*, vol. 18, no. 1, pp. 115-118.View/Download from: Publisher's site

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A review of the first experiments with the GIT-4 generator is given. GIT-4 is based on the results obtained previously on Gamma (1985) and Marina (1986) generators [1], and has operated at the High Current Electronics Institute in Tomsk since December 1986. GIT-4 is designed to test the concept of a terawatt pulsed-power generator based on the direct charging of an inductive storage from a Marx primary store using a plasma-opening switch (POS) at currents up to several MA. Such generators need a POS with essentially microsecond conduction times that are widely investigated at present [2], as well as the POS with nanosecond conduction time [3]. In the experiments reviewed, output power up to 2.5 TW in the POS region was obtained at current levels up to 1 MA. © 1990 IEEE

Novikov, AA 1990, 'Discharger units for high-power Arkad'ev-Marx generators', *Instruments and experimental techniques New York*, vol. 32, no. 4 pt 2, pp. 882-886.

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Two- and three-electrode discharger units for high-power Arkad'ev-Marx generators consisting of parallel-connected modules are described. The units provide a module-actuation stability with a mean-square spread of 10 nsec. The breakdown strength of the discharger gaps is controlled by pressure variation.

Bastrikov, AN, Bugaev, SP, Vorob'yushko, MI, Dul'zon, AA, Kassirov, GM, Koval'chuk, BM, Kokshenev, VA, Koshelev, VI, Manylov, VI, Mesyats, GA, Novikov, AA, Podkovyrov, VG, Potalitsyn, YF, Sukhushin, KN, Timofeev, MN & Yakovlev, VP 1989, ''Gamma' high-current electron accelerator', *Instruments and experimental techniques New York*, vol. 32, no. 2 pt 1, pp. 287-293.

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The 'Gamma' microsecond accelerator of relativistic beams, which has a maximum energy of 2.8 MeV, is described. Tubular electron beams 6-12 cm in diameter with a current-pulse duration of 1-15 μsec and a stored energy of 30-140 kJ are produced. The coaxial diode, which has magnetic isolation and a sectional insulator, is supplied directly from a voltage-pulse generator. The efficiency of energy transfer from the generator to the beam is 20% for a potential of 2 MV on the diode.

Koval'chuk, BM, Kokshenev, VA, Novikov, AA & Yakovlev, VP 1989, '1-MV section for high-current pulse generators', *Instruments and experimental techniques New York*, vol. 32, no. 1 pt 2, pp. 153-156.

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The authors describe the development and testing of a high-current storage-capacitor section with an Arkad'ev-Marks circuit that has stable switching over a wide range and a stable actuation time. A generator of three parallel-connected sections has an energy capacity of 0.2 MJ, an output voltage of up to 1 MV, an inductance of 0.85 μH, a maximum current of 0.6 MA, and a specific energy capacity of 16.5 kJ/m3.

Novikov, AA 1989, 'Devices for high-speed protection from high currents', *Instruments and experimental techniques New York*, vol. 32, no. 1 pt 2, pp. 175-179.

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The author describes a method of short-circuit protection for high-power electronic equipment. For protection, the equipment is shunted by spark dischargers with the minimum possible number of discharge gaps. High speed (<100 nsec) and a wide voltage range (6-100 kV) are achieved by the use of a trigger device that is based on an Arkad'ev-Marks generator and an air-core pulse transformer.

Novikov, AA 1984, 'Martingale identities and inequalities and their applications in nonlinear boundary-value problems for random processes', *Mathematical Notes of the Academy of Sciences of the USSR*, vol. 35, no. 3, pp. 241-249.View/Download from: Publisher's site

Glebov, BA, Novikov, AA & Shiladzhyan, AM 1983, 'BRIDGE VOLTAGE CONVERTER WITHOUT AN INPUT TRANSFORMER.', *Telecommunications and Radio Engineering (English translation of Elektrosvyaz and Radiotekhnika)*, vol. 37-38, no. 2, pp. 32-34.

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The secondary power supplies used in radio-electronic equipment, which operate from the ac power lines without an input transformer, usually have a relatively high dc input voltage and a low output voltage. This property of network secondary power supplies (in which a voltage converter is the principal stage) causes a deterioration of the switching processes and reduces the operating reliability of power switches. A voltage converter with a magnetic energy integrator in the primary circuit should be operated in the self-oscillatory mode by introducing positive feedback through the transformer. Then the stability of oscillations is guaranteed by the significant amount of energy stored in the magnetic integrator used for switching. The voltage converter is proposed which is capable of operating when the load changes from open circuit to short circuit and can therefore be used as a module when constructing modular power supply systems with an unlimited output power.

Novikov, AA 1981, 'On estimates and the asymptotic behavior of nonexit probabilities of a wiener process to a moving boundary', *Mathematics of the USSR - Sbornik*, vol. 38, no. 4, pp. 495-505.View/Download from: Publisher's site

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In this paper the asymptotic behavior, as well as upper and lower bounds, is found for the probabilities P(σ > T) = P(lW, l < f(t), 0 T) = P(W,> g(t), 0 < t < T) for large classes of functions/and g. © 1981 American Mathematical Society.

Novikov, AA 1981, 'On estimates and the asymptotic behavior of the probability of nonintersection of moving boundaries by sums of independent random variables', *Mathematics of the USSR - Izvestija*, vol. 17, no. 1, pp. 129-145.View/Download from: Publisher's site

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This paper studies estimates and the asymptotic behavior as n → ∞ for the probabilities P(Sk < f(k), m < k < n) and P(Sk > g(k), m < k < n), where (formula omitted) being independent identically distributed random variables with mean zero, and f(n) and g(n) are nonrandom functions. Under certain restrictions on the boundaries f(n) and g(n) logarithmic asymptotes of these probabilities are found in the case when the ξk satisfy (respectively) a two-sided or a one-sided Cramer condition. The method is based on an absolutely continuous substitution for the original probability measure. © 1981 American Mathematical Society.

Novikov, AA 1981, 'Small deviations of Gaussian process', *Mathematical Notes of the Academy of Sciences of the USSR*, vol. 29, no. 2, pp. 150-155.View/Download from: Publisher's site

ZELYAKH, EV & NOVIKOV, AA 1981, 'ACTIVE WIDE-BAND PIEZOELECTRIC FILTERS.', *TELECOMMUN RADIO ENG*, vol. V 35-36, no. N 10, pp. 47-51.

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A METHOD IS DESCRIBED FOR OBTAINING WIDE-BAND FILTERS WHICHIS BASED ON THE USE OF NEGATIVE CAPACITANCE FOR WIDENING THE RESONANCE INTERVAL OF THE PIEZOELECTRIC RESONATOR.

Novikov, AA 1980, 'Active Network Realization of Operating Transfer Factor of a Bridge Circuit.', *Izvestiya Vysshikh Uchebnykh Zavedenij. Radioelektronika*, vol. 23, no. 7, pp. 53-59.

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A synthesis method is proposed for active networks whose transfer factor is equivalent to that of a bridge two-port. The method yields a set of solutions which may be optimized with reprieve to a specific problem. An example of an active low-frequency filter is included.

Akodis, MM, Shipitsyn, VV, Luzgin, VI, Rudnyi, VV, Novikov, AA, Rukhman, AA & Antonova, VN 1979, 'Investigation and Comparison of Series-type Inverters Connected to Oscillatory Circuit with Variable Parameters.', *Izvestiya Vysshikh Uchebnykh Zavedenij i Energeticheskikh Ob''edinenij Sng. Energetika*, no. 8, pp. 94-97.

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Analog modeling was carried out of electromagnetic processes on series-type inverters with different types of recuperation of excessive reactive power accumulated in commutating circuits.

Novikov, AA 1979, 'Method of normal forms for wave systems', *Radiophysics and Quantum Electronics*, vol. 22, no. 1, pp. 77-79.View/Download from: Publisher's site

Novikov, AA 1979, 'On conditions for absolute continuity of probability measures', *Mathematics of the USSR - Sbornik*, vol. 35, no. 5, pp. 697-707.View/Download from: Publisher's site

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A number of sufficient conditions are obtained for the absolute continuity of probability measures under the assumption that their restrictions to some increasing system of Σ-algebras are absolutely continuous. An example is given which shows that these conditions are close to being best possible. Bibliography: 7 titles. © 1979 IOP Publishing Ltd.

Novikov, AA 1979, 'Optimum operating modes of distributed self-excited oscillators', *Radiophysics and Quantum Electronics*, vol. 22, no. 1, pp. 47-50.View/Download from: Publisher's site

Novikov, AA, Sinitsyn, VV & Tager, AS 1979, 'Dynamics of a two-circuit oscillator employing an avalanche transit-time diode', *Radiophysics and Quantum Electronics*, vol. 22, no. 3, pp. 266-267.View/Download from: Publisher's site

Novikov, AA 1978, 'Self-oscillations in a segment of a line with discrete active elements', *Radiophysics and Quantum Electronics*, vol. 21, no. 6, pp. 643-645.View/Download from: Publisher's site

Zelyakh, EV & Novikov, AA 1978, 'ACTIVE PIEZOELECTRIC REJECTION FILTERS.', *Telecommunications and Radio Engineering (English translation of Elektrosvyaz and Radiotekhnika)*, vol. 32-33, no. 8, pp. 40-44.

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A method of designing active rejection filters based on active R-circuits with resonant two-poles containing piezoelectric resonators is described. Models of the filters with direct and indirect compensation of the static capacitance of the resonant two-pole are examined. Expressions are obtained for the transmission loss and phase constants of the circuits. Experimental data for a rejection filter are given.

Zelyakh, EV & Novikov, AA 1977, 'ACTIVE REJECTION FILTERS WITH PIEZOELECTRIC RESONATORS.', *Telecommunications and Radio Engineering (English translation of Elektrosvyaz and Radiotekhnika)*, vol. 31-32, no. 1, pp. 45-50.

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Multichannel communications equipment uses highly stable rejection crystal filters, constructed on the basis of bridge or ladder networks. The presence of inductive coils and of matching transformers in these circuits hinders the microminiaturization of equipment containing such filters. In this respect circuits constructed from a combination of piezoelectric resonators, capacitors, resistors and active elements are very promising. A method of constructing active rejection filters with piezoelectric resonators is proposed. The method is based on an amplifier circuit with a single negative feedback loop. Models of such filters, containing one or two resonators, are investigated and a method for their design is given. Experimental results are presented.

Novikov, AA 1976, 'Application of the method of coupled waves to an analysis of nonresonance interaction', *Radiophysics and Quantum Electronics*, vol. 19, no. 2, pp. 225-227.View/Download from: Publisher's site

Novikov, AA 1972, 'Sequential estimation of the parameters of diffusion processes', *Mathematical Notes of the Academy of Sciences of the USSR*, vol. 12, no. 5, pp. 812-818.View/Download from: Publisher's site

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For the parameter λ of a diffusion processξ(t), satisfying the stochastic differential equation dξ(t)=λf (t,ξ)dt+dw(l), we propose an effective sequential estimation plan with an unbiased and normally distributed estimate. The proposed sequential plan is discussed in detail for the example of a process ξ(t) having a linear stochastic differential. © 1973 Consultants Bureau.

Kordzakhia, N & Novikov, A 2017, 'Bounds and approximations for distributions of weighted Kolmogorov-Smirnov tests' in *From Statistics to Mathematical Finance: Festschrift in Honour of Winfried Stute*, pp. 235-250.View/Download from: Publisher's site

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© Springer International Publishing AG 2017. The paper is motivated by the use of weighted Kolmogorov-Smirnov (wKS) tests in Gene Set Enrichment Analysis where the key requirements are speed and accuracy of computations. We reduce the problem of finding of distributions of one-and two-sided wKS statistics to the nonlinear boundary crossing problem for a Brownian motion. Theoretical estimates of accuracy of the approximations using piecewise linear boundaries are derived. The approximations with 2-knot piecewise linear boundaries are discussed for the one-sided wKS. In the numerical example the estimates of tail probabilities obtained with the use of upper and lower bounds were validated using Monte-Carlo simulation.

Novikov, A, Kordzakhia, N & Ling, T 2014, 'Pricing of Volume-Weighted Average Options: Analytical Approximations and Numerical Results' in Kabanov, Y, Rutkowski, M & Zariphopoulou, T (eds), *Inspired by Finance*, Springer, London, pp. 461-474.View/Download from: Publisher's site

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The volume weighted average price (VWAP) over rolling number of days in the averaging period is used as a benchmark price by market participants and can be regarded as an estimate for the price that a passive trader will pay to purchase securities in a market. The VWAP is commonly used in brokerage houses as a quantitative trading tool and also appears in Australian taxation law to specify the price of share-buybacks of publically-listed companies. Most of the existing literature on VWAP focuses on strategies and algorithms to acquire market securities at a price as close as possible to VWAP. In our setup the volume process is modeled via a shifted squared Ornstein-Uhlenbeck process and a geometric Brownian motion is used to model the asset price. We derive the analytical formulae for moments of VWAP and then use the moment matching approach to approximate a distribution of VWAP. Numerical results for moments of VWAP and call-option prices have been verified by Monte Carlo simulations.

Kordzakhia, N, Novikov, A & Tsitsiashvili, G 2012, 'On ruin probabilities in risk models with interest rate' in Sibillo, M & Perna, C (eds), *Mathematical and Statistical Methods for Actuarial Sciences and Finance*, Springer, Milano, pp. 245-253.View/Download from: Publisher's site

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An explicit formula for ruin probability in a discrete time risk model with interest rare is found under the assumption that claims follow a hyperexponential distribution.

Kordzakhia, N, Novikov, A & Tsitsiashvili, G 2012, 'On ruin probabilities in risk models with interest rate' in *Mathematical and Statistical Methods for Actuarial Sciences and Finance*, Springer Milan, pp. 245-253.View/Download from: Publisher's site

Borovkov, K, Downes, AN & Novikov, A 2010, 'Continuity Theorems in Boundary Crossing Problems for Diffusion Processes' in Chiarella, C & Novikov, A (eds), *Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen*, Springer, Germany, pp. 335-368.View/Download from: Publisher's site

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Computing the probability for a given diffusion process to stay under a particular boundary is crucial in many important applications including pricing financial barrier options and defaultable bonds. We discuss results on the accuracy of approximations for both the Brownian motion process and general time-homogeneous diffusions and also some contiguous topics.

Kordzakhia, N & Novikov, A 2008, 'Pricing of Defaultable Securities under Stochastic Interest' in Sarychev, A, Shiryaev, A, Guerra, M & Grossinho, M (eds), *Mathematical Control Theory and Finance*, Springer, Berlin, pp. 251-263.

Lipster, R & Novikov, A 2006, 'Tail distributions of supremum and quadratic variation of local Martingales' in Kubanov, Y, Lipster, R & Stoyanov, J (eds), *From Stochastic Calculus to Mathematical Finance*, Springer, Heidelberg, Germany, pp. 421-432.

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We extend some known results concerning the distribution tails of supremum and quadratic variation of a continuous local martingale tothe case of locally square integrable martingales with bounded jumps. The predictable and optional quadratic vairations are involved inthe main result.

Novikov, A & Chiarella, C 2009, 'Contemporary Quantitative Finance, Essays in Honour of Eckhard Platen', *Quantitative Mathematical Finance*, Quantitative Mathematical Finance, Springer, Sydney, Australia, pp. 1-410.

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The contributors to this volume write a series of articles outlining contemporary advances in a number of key areas of mathematical finance such as, optimal control theory applied to finance, interest rate models, credit risk and credit derivatives, use of alternative stochastic processes, numerical solution of equations of mathematical finance, estimation of stochastic processes in finance. The list of authors includes many of the researchers who have made the major contributions to these various areas of mathematical finance. This volume addresses both researchers and professionals in financial institutions, as well as regulators working in the above mentioned fields.

Novikov, A 2007, 'Pricing of Defaultable Securities under Stochastic Interest', *Workshop on Mathematical Control Theory and Finance*, Workshop on Mathematical Control Theory and Finance, Instituto Superior de Economia e GestÃ£o, Lisbon, Portugal.

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This is a CD edtion of conference papers. An extended version of the paper is accepted for publication in Mathematical Control Theory and Finance, Springer, Editores: A. Sarychev, A. Shiryaev, M. Guerra e M. R. Grossinho, 2008.

Roberts, DO & Novikov, A 2005, 'Pricing European and discretely monitored exotic options under the Levy process framework', *International Mathematica Symposium 2005*, International Mathematic Symposium, Wolfram Research, Perth, Australia, pp. 1-11.

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We shall consider both European and idscretely monitored Exotic options (Bermudan and Discrete Barrier) in a market where the underlying asset follows a Geometric Levy process. First we shall briefly introduce this extended framework, then using the Variance Gamma model we shall show how toprice European Options and then we will proceed to demonstrate the application of the recursive quadrature method to Bermudan and Discrete Barrier Options

Kordzakhia, N, Melchers, R & Novikov, A 2000, 'First passage analysis of a 'square wave' filtered Poisson process', *Applications of Statistics and Probability*, A.A. Balkeme,, Sydney, Australia, pp. 35-43.

Kim, AA, Kovalchuk, BM, Kremnev, VV, Kumpjak, EV, Novikov, AA, Etlicher, B, Frescaline, L, Leon, JF, Roques, B, Lassalle, F, Lample, R, Avrillaud, G & Kovacs, F 1997, 'Multi gap, multi channel spark switches', *Digest of Technical Papers-IEEE International Pulsed Power Conference*, pp. 862-867.

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The results of the multigap, multichannel switch development performed in High Current Electronics Institute in order to reach the requirements of the SYRINX project in France are presented. The requirements include: 90 kV operating voltage; >500 kA switching current; <1 μs current rising time; and <10 nH switch inductance. Different switches developed on consecutive steps of the collaboration are described. In these switches, the multichannel discharge is realized in multigap spark switch configuration, where the voltage during charging is uniformly distributed between the few gaps connected in series. The triggering occurs due to disturbance of voltage distribution between the gaps when trigger pulse is applied.

Novikov, A & Kordzakhia, N 1997, 'Stochastic and statistical analysis of long-range dependent processes with `Mathematica'', *Proceedings of the International Mathematica Symposium*, pp. 369-376.

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Mathematical models of stationary long-range dependent processes are more complicated then ordinary autoregressive models as they involve fractional difference equations (or, even fractional differential equations in continuous time case). The explicit representation of solutions of these equations requires special functions like hypergeometric or Gegenbauer polynomials. This paper demonstrates that Mathematica capability doing symbolic calculations makes both stochastic and statistical analysis of stationary processes with long memory easier.

Bugaev, SP, Volkov, AM, Iskoldsky, AM, Kim, AA, Kovalchuk, BM, Kokshenev, VA, Mesyats, GA, Novikov, AA & Yakovlev, VP 1988, 'Terawatt pulsed power generator with plasma opening switch', p. 77.

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GIT-4 (SNOP4) has been developed using pulsed-power generators with intermediate energy storage and the plasma opening switch. GIT-4 has a primary capacitive energy storage with thirty-six parallel Marx generators located in four oil-filled tanks containing nine generators each. The energy from the Marx generators is transferred to the vacuum insulator through four oil-insulated lines. The intermediate inductive energy storage as well as the plasma opening switch and the load are located in a vacuum chamber of the generator. The discharge circuit has a 4.8-μF capacitance in the primary store and a 0.2-μH inductance in the intermediate store. At 40-80-kV charging voltage the energy in the primary store is 0.54-2.16 MJ, the output voltage is 480-960 kV, and the current rises to 2.35-4.7 MA in 1.54 μs. GIT-4 is being tested with current levels up to 2 MA. The output power of approximately 1.6 TW and voltage of approximately 1.2 MV are achieved at a charging voltage of 40 kV.

Kordzakhia, N, Kutoyants, Y, Novikov, A & Hin, L-Y 2017, 'On a representation of fractional Brownian motion and the limit distributions of statistics arising in cusp statistical models'.

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We discuss some extensions of results from the recent paper by Chernoyarov et

al. (Ann. Inst. Stat. Math., October 2016) concerning limit distributions of

Bayesian and maximum likelihood estimators in the model "signal plus white

noise" with irregular cusp-type signals. Using a new representation of

fractional Brownian motion (fBm) in terms of cusp functions we show that as the

noise intensity tends to zero, the limit distributions are expressed in terms

of fBm for the full range of asymmetric cusp-type signals correspondingly with

the Hurst parameter H, 0

Hinz, J & Novikov, A 2009, 'On fair pricing of emission-related derivatives', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 257 Abstract: The climate rescue is on the top of many agendas. In thisc ontext, emission trading schemes are considered as promising tools. The regulatory framework of an emission trading scheme introduces a market for emission allowances and creates need for risk management by appropriate financial contracts. In this work, we address logical principles underlying their valuation.

Novikov, A & Kordzakhia, N, 'On lower and upper bounds for Asian-type options: a unified approach'.

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In the context of dealing with financial risk management problems it is

desirable to have accurate bounds for option prices in situations when pricing

formulae do not exist in the closed form. A unified approach for obtaining

upper and lower bounds for Asian-type options, including options on VWAP, is

proposed in this paper. The bounds obtained are applicable to the continuous

and discrete-time frameworks for the case of time-dependent interest rates.

Numerical examples are provided to illustrate the accuracy of the bounds.

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