We consider a generalization of the classical risk model when the premium intensity depends on th... more We consider a generalization of the classical risk model when the premium intensity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals.
Journal of Applied Mathematics and Stochastic Analysis, 1999
The criterion and sufficient condition for the existence of moments of one-parameter increasing p... more The criterion and sufficient condition for the existence of moments of one-parameter increasing predictable processes is presented in terms of an associated potential. The estimates of moments of special functional connected with two-parameter increasing predictable processes are given in the case when the associated potential is bounded. The application of these estimates to the local time for purely discontinuous strong martingales in the plane is also presented.
Assume that X is a continuous square integrable process with zero mean defined on some probabilit... more Assume that X is a continuous square integrable process with zero mean defined on some probability space (Ω, F, P). The classical characterization due to P. Lévy says that X is a Brownian motion if and only if X and X 2 t − t, t ≥ 0 are martingales with respect to the intrinsic filtration IF X. We extend this result to fractional Brownian motion.
Theory of Probability and Mathematical Statistics, 2012
The convergence in probability of the sets of maximum probability of success is studied in the pr... more The convergence in probability of the sets of maximum probability of success is studied in the problem of quantile hedging for a model of an asset price process involving Brownian and fractional Brownian motions.
We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a know... more We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a known nonrandom function, and B is a centered Gaussian process. We construct the maximum likelihood estimators of the drift parameter θ based on discrete and continuous observations of the process X and prove their strong consistency. The results obtained generalize the paper [13] in two directions: the drift may be nonlinear, and the noise may have nonstationary increments. As an example, the model with subfractional Brownian motion is considered.
We find the best approximation of the fractional Brownian motion with the Hurst index H ∈ (0, 1/2... more We find the best approximation of the fractional Brownian motion with the Hurst index H ∈ (0, 1/2) by Gaussian martingales of the form t 0 s γ dWs, where W is a Wiener process, γ > 0.
Theory of Probability and Mathematical Statistics, 2014
We study the problem of approximation of a fractional Brownian motion with the help of Gaussian m... more We study the problem of approximation of a fractional Brownian motion with the help of Gaussian martingales that can be represented as the integrals with respect to a Wiener process and with nonrandom integrands being "similar" to the kernel of the fractional Brownian motion. The "similarity" is understood in the sense that an integrand is the value of the kernel at some point. We establish analytically and evaluate numerically the upper and lower bounds for the distance between the fractional Brownian motion and the space of Gaussian martingales.
The paper deals with the expected maxima of continuous Gaussian processes X = (X t) t≥0 that are ... more The paper deals with the expected maxima of continuous Gaussian processes X = (X t) t≥0 that are Hölder continuous in L 2-norm and/or satisfy the opposite inequality for the L 2-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 E max 0≤t≤1 X t and investigate the rate of convergence to that quantity of its discrete approximation E max 0≤i≤n X i/n. Some further properties of these two maxima are established in the special case of the fractional Brownian motion.
If a random function is stochastically continuous (continuous with probability one, continuous in... more If a random function is stochastically continuous (continuous with probability one, continuous in the L p sense) at every point of the parametric set Note that sometimes, while dealing with continuity of the function X in the L p sense
In this chapter, we consider a problem of statistical estimation of an unknown drift parameter fo... more In this chapter, we consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value, as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates for fractional derivative of fractional Brownian motion.
Proceedings of the American Mathematical Society, 1998
We prove the existence of entire solutions to some abstract higher order Cauchy problem for a den... more We prove the existence of entire solutions to some abstract higher order Cauchy problem for a dense subset of initial values.
The article is devoted to models of financial markets with stochastic volatility, which is define... more The article is devoted to models of financial markets with stochastic volatility, which is defined by a functional of Ornstein-Uhlenbeck process or Cox-Ingersoll-Ross process. We study the question of exact price of European option. The form of the density function of the random variable, which expresses the average of the volatility over time to maturity is established using Malliavin calculus.The result allows calculate the price of the option with respect to minimum martingale measure when the Wiener process driving the evolution of asset price and the Wiener process, which defines volatility, are uncorrelated.
The paper is devoted to stochastic differential equations with a fractional Brownian component. T... more The paper is devoted to stochastic differential equations with a fractional Brownian component. The fractional Brownian motion is constructed on the white noise space with the help of "forward" and "backward" fractional integrals. The fractional white noise and Wick products are considered. A similar construction for the "complete" fractional integral is considered by Elliott and van der Hoek. We consider two possible approaches to the existence and uniqueness of solutions of stochastic differential equation with a fractional Brownian motion.
We obtain the weak convergence of measures generated by the price process and prove the continuit... more We obtain the weak convergence of measures generated by the price process and prove the continuity of the price of a barrier call option with respect to the parameter of a series for the Black-Scholes model of a complete market. The explicit form of the price of the barrier option is not required. The result obtained allows one to prove the continuity of a solution of the corresponding boundary-value problem for the parabolic partial differential equation with respect to the parameter of a series. Applying the Malliavin calculus, we establish the existence of a bounded continuous density of the distribution of a Wiener integral with shift restricted to an arbitrary "positive" ray and prove the differentiability of the fair price with respect to the barrier (the differentiability with respect to other parameters is a classical result).
Theory of Probability and Mathematical Statistics, 2004
We consider maximum likelihood statistical estimates for the number of individuals in a biologica... more We consider maximum likelihood statistical estimates for the number of individuals in a biological population modelled by a compound Poisson process. We prove the local asymptotic normality and asymptotic efficiency of the estimates.
We consider Langevin equation involving fractional Brownian motion with Hurst index H ∈ (0, 1 2).... more We consider Langevin equation involving fractional Brownian motion with Hurst index H ∈ (0, 1 2). Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter θ. We construct the estimator that is similar in form to maximum likelihood estimator for Langevin equation with standard Brownian motion. Observations are discrete in time. It is assumed that the interval between observations is n −1 , i.e. tends to zero (high frequency data) and the number of observations increases to infinity as n m with m > 1. It is proved that for positive θ the estimator is strongly consistent for any m > 1 and for negative θ it is consistent when m > 1 2H .
We consider stochastic differential equation involving pathwise integral with respect to fraction... more We consider stochastic differential equation involving pathwise integral with respect to fractional Brownian motion. The estimates for the Hurst parameter are constructed according to first-and second-order quadratic variations of observed values of the solution. The rate of convergence of these estimates to the true value of a parameter is established.
We consider a generalization of the classical risk model when the premium intensity depends on th... more We consider a generalization of the classical risk model when the premium intensity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals.
Journal of Applied Mathematics and Stochastic Analysis, 1999
The criterion and sufficient condition for the existence of moments of one-parameter increasing p... more The criterion and sufficient condition for the existence of moments of one-parameter increasing predictable processes is presented in terms of an associated potential. The estimates of moments of special functional connected with two-parameter increasing predictable processes are given in the case when the associated potential is bounded. The application of these estimates to the local time for purely discontinuous strong martingales in the plane is also presented.
Assume that X is a continuous square integrable process with zero mean defined on some probabilit... more Assume that X is a continuous square integrable process with zero mean defined on some probability space (Ω, F, P). The classical characterization due to P. Lévy says that X is a Brownian motion if and only if X and X 2 t − t, t ≥ 0 are martingales with respect to the intrinsic filtration IF X. We extend this result to fractional Brownian motion.
Theory of Probability and Mathematical Statistics, 2012
The convergence in probability of the sets of maximum probability of success is studied in the pr... more The convergence in probability of the sets of maximum probability of success is studied in the problem of quantile hedging for a model of an asset price process involving Brownian and fractional Brownian motions.
We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a know... more We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a known nonrandom function, and B is a centered Gaussian process. We construct the maximum likelihood estimators of the drift parameter θ based on discrete and continuous observations of the process X and prove their strong consistency. The results obtained generalize the paper [13] in two directions: the drift may be nonlinear, and the noise may have nonstationary increments. As an example, the model with subfractional Brownian motion is considered.
We find the best approximation of the fractional Brownian motion with the Hurst index H ∈ (0, 1/2... more We find the best approximation of the fractional Brownian motion with the Hurst index H ∈ (0, 1/2) by Gaussian martingales of the form t 0 s γ dWs, where W is a Wiener process, γ > 0.
Theory of Probability and Mathematical Statistics, 2014
We study the problem of approximation of a fractional Brownian motion with the help of Gaussian m... more We study the problem of approximation of a fractional Brownian motion with the help of Gaussian martingales that can be represented as the integrals with respect to a Wiener process and with nonrandom integrands being "similar" to the kernel of the fractional Brownian motion. The "similarity" is understood in the sense that an integrand is the value of the kernel at some point. We establish analytically and evaluate numerically the upper and lower bounds for the distance between the fractional Brownian motion and the space of Gaussian martingales.
The paper deals with the expected maxima of continuous Gaussian processes X = (X t) t≥0 that are ... more The paper deals with the expected maxima of continuous Gaussian processes X = (X t) t≥0 that are Hölder continuous in L 2-norm and/or satisfy the opposite inequality for the L 2-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 E max 0≤t≤1 X t and investigate the rate of convergence to that quantity of its discrete approximation E max 0≤i≤n X i/n. Some further properties of these two maxima are established in the special case of the fractional Brownian motion.
If a random function is stochastically continuous (continuous with probability one, continuous in... more If a random function is stochastically continuous (continuous with probability one, continuous in the L p sense) at every point of the parametric set Note that sometimes, while dealing with continuity of the function X in the L p sense
In this chapter, we consider a problem of statistical estimation of an unknown drift parameter fo... more In this chapter, we consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value, as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates for fractional derivative of fractional Brownian motion.
Proceedings of the American Mathematical Society, 1998
We prove the existence of entire solutions to some abstract higher order Cauchy problem for a den... more We prove the existence of entire solutions to some abstract higher order Cauchy problem for a dense subset of initial values.
The article is devoted to models of financial markets with stochastic volatility, which is define... more The article is devoted to models of financial markets with stochastic volatility, which is defined by a functional of Ornstein-Uhlenbeck process or Cox-Ingersoll-Ross process. We study the question of exact price of European option. The form of the density function of the random variable, which expresses the average of the volatility over time to maturity is established using Malliavin calculus.The result allows calculate the price of the option with respect to minimum martingale measure when the Wiener process driving the evolution of asset price and the Wiener process, which defines volatility, are uncorrelated.
The paper is devoted to stochastic differential equations with a fractional Brownian component. T... more The paper is devoted to stochastic differential equations with a fractional Brownian component. The fractional Brownian motion is constructed on the white noise space with the help of "forward" and "backward" fractional integrals. The fractional white noise and Wick products are considered. A similar construction for the "complete" fractional integral is considered by Elliott and van der Hoek. We consider two possible approaches to the existence and uniqueness of solutions of stochastic differential equation with a fractional Brownian motion.
We obtain the weak convergence of measures generated by the price process and prove the continuit... more We obtain the weak convergence of measures generated by the price process and prove the continuity of the price of a barrier call option with respect to the parameter of a series for the Black-Scholes model of a complete market. The explicit form of the price of the barrier option is not required. The result obtained allows one to prove the continuity of a solution of the corresponding boundary-value problem for the parabolic partial differential equation with respect to the parameter of a series. Applying the Malliavin calculus, we establish the existence of a bounded continuous density of the distribution of a Wiener integral with shift restricted to an arbitrary "positive" ray and prove the differentiability of the fair price with respect to the barrier (the differentiability with respect to other parameters is a classical result).
Theory of Probability and Mathematical Statistics, 2004
We consider maximum likelihood statistical estimates for the number of individuals in a biologica... more We consider maximum likelihood statistical estimates for the number of individuals in a biological population modelled by a compound Poisson process. We prove the local asymptotic normality and asymptotic efficiency of the estimates.
We consider Langevin equation involving fractional Brownian motion with Hurst index H ∈ (0, 1 2).... more We consider Langevin equation involving fractional Brownian motion with Hurst index H ∈ (0, 1 2). Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter θ. We construct the estimator that is similar in form to maximum likelihood estimator for Langevin equation with standard Brownian motion. Observations are discrete in time. It is assumed that the interval between observations is n −1 , i.e. tends to zero (high frequency data) and the number of observations increases to infinity as n m with m > 1. It is proved that for positive θ the estimator is strongly consistent for any m > 1 and for negative θ it is consistent when m > 1 2H .
We consider stochastic differential equation involving pathwise integral with respect to fraction... more We consider stochastic differential equation involving pathwise integral with respect to fractional Brownian motion. The estimates for the Hurst parameter are constructed according to first-and second-order quadratic variations of observed values of the solution. The rate of convergence of these estimates to the true value of a parameter is established.
Uploads
Papers by Y. Mishura