Professor Alex Novikov

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.
View/Download from: Publisher's site
View description>>

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.
View/Download from: UTSePress | Publisher's site
View description>>

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.
View/Download from: UTSePress

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.
View/Download from: UTSePress
View description>>

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. & Shiryaev, A. 2013, 'Remarks on moment inequalities and identities for martingales', Statistics and Probability Letters, vol. 83, pp. 1260-1261.

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.
View/Download from: UTSePress | Publisher's site
View description>>

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: UTSePress
View description>>

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.
View/Download from: UTSePress
View description>>

NA

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: UTSePress | Publisher's site
View description>>

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.
View/Download from: UTSePress | Publisher's site
View description>>

The paper deals with the problem of finding an optimal one-time rebalancing strategy assuming that in the BlackÔ++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.
View/Download from: UTSePress | Publisher's site
View description>>

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: UTSePress | Publisher's site
View description>>

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: UTSePress | Publisher's site
View description>>

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.
View/Download from: UTSePress | Publisher's site

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.
View/Download from: UTSePress
View description>>

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: UTSePress | Publisher's site
View description>>

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., 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: UTSePress | Publisher's site
View description>>

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. & 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.
View/Download from: UTSePress | Publisher's site
View description>>

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.
View/Download from: UTSePress

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.
View/Download from: UTSePress

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.
View/Download from: UTSePress | Publisher's site

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

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.

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.
View/Download from: UTSePress
View description>>

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.