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Professor Alex Novikov

Biography

Alexander Novikov is Professor of Mathematics at the Department of Mathematical 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

Image of Alex Novikov
Professor of Mathematics, School of Mathematical Sciences
Core Member, Quantitative Finance Research Centre
MAppM (Steklov MI), DSc (Steklov MI)
 
Phone
+61 2 9514 2242
Room
CB01.15.42

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.

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 (35361)
Stochastic Calculus in Finance (35365)
Advanced Stochastic Processes (35466)

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

Book Chapters

Kordzakhia, N., Novikov, A. & Tsitsiashvili, G. 2012, 'On ruin probabilities in risk models with interest rate' in Sibillo, Marilena; Perna, Cira (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.

Conference Papers

Novikov, A. 2007, 'Pricing of Defaultable Securities under Stochastic Interest', Workshop on Mathematical Control Theory and Finance, Lisbon, Portugal, April 2007 in Workshop on Mathematical Control Theory and Finance, ed Guerra, A., Instituto Superior de Economia e Gest+o, Lisbon.
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 Mathematic Symposium, Perth, Australia, August 2005 in International Mathematica Symposium 2005, ed Abbott, P; McCarthy, S, Wolfram Research, 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', Sydney, Australia, November 2000 in Applications of Statistics and Probability, ed R.E. Melchers, M.G. Stewart, A.A. Balkeme,, Netherlands, pp. 35-43.

Journal Articles

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.
Skryabin, I., Maisano, J.P., 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.
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.
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.
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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.
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.
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++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.
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. 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. & 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.
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.
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
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. 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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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.

Other research activity

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.