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Professor Alex Novikov

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

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Steklov Mathematical Institute

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Image of Alex Novikov
Professor of Mathematics, School of Mathematical and Physical Sciences
Core Member, QFRC - Quantitative Finance
MAppM (Steklov MI), DSc (Steklov MI)
 
Phone
+61 2 9514 2242

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.

Can supervise: Yes
Gabriel Mititelu Change-point analysis for hyperexponential distributions Qi Nan Zhai Pricing of barrier options and defaultable bonds under stochastic interest Timothy Ling Pricing of barrier options with Monte-Carlo technique using parallel computations

Current Teaching:
Stochastic Processes (37363)
Probability Theory and Stochastic Calculus in Finance (25875)
Advanced Stochastic Processes (37464)

Past Teaching:
Time Series
Regression Analysis
Mathematical Statistics
Probability Theory
Survival Analysis

Chapters

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 Cham Heidelberg New York Dordrecht London, London, pp. 461-474.
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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-Verlag Italia, Milano, pp. 245-253.
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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.
Borovkov, K., Downes, A.N. & 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.
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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.
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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.

Conferences

Kordzakhia, N., Novikov, A. & Tsitsiashvili, G. 2012, 'On ruin probabilities in risk models with interest rate', Mathematical and Statistical Methods for Actuarial Sciences and Finance, pp. 245-253.
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An explicit formula for the finite-time ruin probability in a discrete-time collective ruin model with constant interest rate is found under the assumption that claims follow a generalised hyperexponential distribution. The formula can be used for finding approximations for finite-time ruin probabilities in the case when claim sizes follow a heavy-tailed distribution e.g. Pareto. We also provide theoretical bounds for the accuracy of approximations of the finite-time ruin probabilities in terms of a distance between the distribution of claims and its approximation. Results of numerical comparisons with asymptotic formulas and simulations are presented. © Springer-Verlag Italia 2012.
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.
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 Gestão, Lisbon, Portugal.
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, D.O. & 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.
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.
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.
Kim, A.A., Kovalchuk, B.M., Kremnev, V.V., Kumpjak, E.V., Novikov, A.A., Etlicher, B., Frescaline, L., Leon, J.F., 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.
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 s 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.

Journal articles

Borovkov, K., Mishura, Y., Novikov, A. & Zhitlukhin, M. 2016, 'Bounds for expected maxima of Gaussian processes and their discrete approximations', Stochastics, pp. 1-17.
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&copy; 2015 Taylor & Francis The paper deals with the expected maxima of continuous Gaussian processes (Formula presented.) that are H&ouml;lder 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.
Novikov, A., ALEXANDER, S. & KORDZAKHIA, N. 2016, 'BOUNDS ON PRICES FOR ASIAN OPTIONS VIA FOURIER METHODS', ANZIAM Journal.
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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&acute;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, S.P. & Novikov, A. 2016, 'EDITORIAL: STOCHASTIC and COMPUTATIONAL METHODS in FINANCE', ANZIAM Journal, vol. 57, no. 3, pp. 205-206.
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Novikov, A.A. & Kordzakhia, N.E. 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.
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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.
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Novikov, A. & Shiryaev, A.N. 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.
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Novikov, A. & Shiryaev, A. 2013, 'Remarks on moment inequalities and identities for martingales', Statistics & Probability Letters, vol. 83, pp. 1260-1261.
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We present some comments on moment inequalities and identities for martingales in the context of the paper of Langovoy
Novikov, A. & Kordzakhia, N. 2013, 'Pitman Estimators: An Asymptotic Variance Revisited', Theory of Probability and its Applications, vol. 57, no. 3, pp. 521-529.
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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.
Cetin, U., Novikov, A. & Shiryaev, A.N. 2013, 'Bayesian Sequential Estimation of a Drift of Fractional Brownian Motion', Sequential Analysis: Design Methods and Applications, vol. 32, no. 3, pp. 288-296.
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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.
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.
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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 atopping problems for AR(1) sequences', Sequential Analysis, vol. 30, no. 1, pp. 79-93.
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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.
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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.
Novikov, A., Liptser, R. & Tartakovsky, A.G. 2010, 'Preface: Celebrating Albert Shiryaev's 75th Anniversary', Sequential Analysis, vol. 29, no. 2, pp. 107-111.
Novikov, A. & Shiryaev, A. 2010, 'ON MARTINGALE PROOF OF THE KOLMOGOROV AND SMIRNOV DISTRIBUTIONS', Sequential Analysis, vol. 29, no. 4, pp. 439-443.
this is appendix to the Shiryaev respose
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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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, A.N. & 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 Black&acirc;Scholes 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.
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.
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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. & 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.
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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.
Borovkov, K. & Novikov, A. 2008, 'On exit times of Levy-driven Ornstein-Uhlenbeck processes', Statistics & Probability Letters, vol. 78, no. 12, pp. 1517-1525.
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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. &copy; 2008 Elsevier B.V. All rights reserved.
Schmidt, T. & Novikov, A. 2008, 'A Structural Model with Unobserved Default Boundary', Applied Mathematical Finance, vol. 15, no. 2, pp. 183-203.
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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, A.N. 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.
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Roberts, D.O. & 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.
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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, A.N. 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.
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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.
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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.
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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.
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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.
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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.
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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. 2002, 'On a new approach to calculating expectations for option pricing', Journal of Applied Probability, vol. 39, no. 4, pp. 889-895.
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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 L&eacute;vy 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.
Borovkov, K. & Novikov, A. 2001, 'On a Piece-Wise Deterministic Markov Process Model', Statistics & Probability Letters, vol. 53, pp. 421-428.
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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.
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.
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.
Le Breton, A. & Novikov, A.A. 1999, 'On stochastic approximation procedures with averaging', Theory of Probability and its Applications, vol. 44, no. 3, pp. 591-605.
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.
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. &copy; 1999 Elsevier Science B.V.
Novikov, A.A. 1998, 'Hedging of options with a given probability', Theory of Probability and its Applications, vol. 43, no. 1, pp. 135-144.
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 - ( 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, A.N., Kim, A.A., Koval'chuk, B.M., Kremnev, V.V., Kumpyak, E.V., Novikov, A.A. & Tsoi, N.V. 1997, 'Low-inductance multigap spark modules', Russian Physics Journal, vol. 40, no. 12, pp. 1125-1134.
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. &copy;1998 Plenum Publishing Corporation.
Novikov, A.A. 1996, 'Martingales, Tauberian theorem, and strategies of gambling', Theory of Probability and its Applications, vol. 41, no. 4, pp. 716-729.
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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 is a stopping time for a standard Wiener process Wt with integrable terminal value W, then (1) lim inf t (P{ > t}t) 2/|EW|. 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.
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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. &copy; 1995 Physica-Verlag.
Novikov, A.A. 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.
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Novikov, A.A. 1981, 'Small deviations of Gaussian process', Mathematical Notes of the Academy of Sciences of the USSR, vol. 29, no. 2, pp. 150-155.
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Novikov, A.A. 1979, 'Optimum operating modes of distributed self-excited oscillators', Radiophysics and Quantum Electronics, vol. 22, no. 1, pp. 47-50.
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Zelyakh, E.V. & Novikov, A.A. 1978, 'ACTIVE PIEZOELECTRIC REJECTION FILTERS.', Telecommunications and Radio Engineering (English translation of Elektrosvyaz and Radiotekhnika), vol. 32-33, no. 8, pp. 40-44.
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.
Novikov, A.A. 1978, 'Self-oscillations in a segment of a line with discrete active elements', Radiophysics and Quantum Electronics, vol. 21, no. 6, pp. 643-645.
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Novikov, A.A. 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.
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Novikov, A.A. 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.
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For the parameter of a diffusion process(t), satisfying the stochastic differential equation d(t)=f (t,)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 (t) having a linear stochastic differential. &copy; 1973 Consultants Bureau.

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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.