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Yuliya Mishura

Publications and source records attributed to Yuliya Mishura.

At least 91 records · Page 5Linked to original sources
arXiv2015

Utility maximization in Wiener-transformable markets

We consider a utility maximization problem in a broad class of markets. Apart from traditional semimartingale markets, our class of markets includes processes with long memory, fractional Brownian motion and related processes, and, in general, Gaussian processes satisfying certain regularity conditions on their covariance functions. Our choice of markets is motivated by the well-known phenomena of the so-called "constant" and "variable depth" memory observed in real world price processes, for which fractional and multifractional models are the most adequate descriptions. We introduce the notion of a Wiener-transformable Gaussian process, and give examples of such processes, and their representations. The representation for the solution of the utility maximization problem in our specific setting is presented for various utility functions.

math.PR↗
arXiv2015

Pricing the European call option in the model with stochastic volatility driven by Ornstein--Uhlenbeck process. Exact formulas

We consider the Black--Scholes model of financial market modified to capture the stochastic nature of volatility observed at real financial markets. For volatility driven by the Ornstein--Uhlenbeck process, we establish the existence of equivalent martingale measure in the market model. The option is priced with respect to the minimal martingale measure for the case of uncorrelated processes of volatility and asset price, and an analytic expression for the price of European call option is derived. We use the inverse Fourier transform of a characteristic function and the Gaussian property of the Ornstein--Uhlenbeck process.

q-fin.PR↗
arXiv2015

Small ball properties and representation results

We show that small ball estimates together with Holder continuity assumption allow to obtain new representation results in models with long memory. In order to apply these results, we establish small ball probability estimates for Gaussian processes whose incremental variance admits two-sided estimates and the incremental covariance preserves sign. As a result, we obtain small ball estimates for integral transforms of Wiener processes and of fractional Brownian motion with Volterra kernels.

math.PR↗
arXiv2015

The rate of convergence to the normal law in terms of pseudomoments

We establish the rate of convergence of distributions of sums of independent identically distributed random variables to the Gaussian distribution in terms of truncated pseudomoments by implementing the idea of Yu. Studnyev for getting estimates of the rate of convergence of the order higher than $n^{-1/2}$.

math.PR↗
arXiv2015

Construction of maximum likelihood estimator in the mixed fractional--fractional Brownian motion model with double long-range dependence

We construct an estimator of the unknown drift parameter $θ\in {\mathbb{R}}$ in the linear model \[X_t=θt+σ_1B^{H_1}(t)+σ_2B^{H_2}(t),\;t\in[0,T],\] where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $H_1$ and $H_2$ satisfying the condition $\frac{1}{2}\leq H_1<H_2<1.$ Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence--uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.

math.PR↗
arXiv2015

Bounds for expected maxima of Gaussian processes and their discrete approximations

The paper deals with the expected maxima of continuous Gaussian processes $X = (X_t)_{t\ge 0}$ 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 $E \max_{0\le t\le 1}X_t$ and investigate the rate of convergence to that quantity of its discrete approximation $E \max_{0\le i\le n}X_{i/n}$. Some further properties of these two maxima are established in the special case of the fractional Brownian motion.

math.PR↗
arXiv2015

Optimal stopping for Levy processes with polynomial rewards

Explicit solution of an infinite horizon optimal stopping problem for a Levy processes with a polynomial reward function is given, in terms of the overall supremum of the process, when the solution of the problem is one-sided. The results are obtained via the generalization of known results about the averaging function associated with the problem. This averaging function can be directly computed in case of polynomial rewards. To illustrate this result, examples for general quadratic and cubic polynomials are discussed in case the process is Brownian motion, and the optimal stopping problem for a quartic polynomial and a Kou's process is solved.

math.PR↗
arXiv2015

Maximum likelihood drift estimation for the mixing of two fractional Brownian motions

We construct the maximum likelihood estimator (MLE) of the unknown drift parameter $θ\in \mathbb{R}$ in the linear model $X_t=θt+σB^{H_1}(t)+B^{H_2}(t),\;t\in[0,T],$ where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $\frac12<H_1<H_2<1.$ The formula for MLE is based on the solution of the integral equation with weak polar kernel.

math.PR↗
arXiv2015

Practical approaches to the estimation of the ruin probability in a risk model with additional funds

We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.

math.PR↗
arXiv2015

Consistency of the drift parameter estimator for the discretized fractional Ornstein-Uhlenbeck process with Hurst index $H\in(0,\frac12)$

We consider Langevin equation involving fractional Brownian motion with Hurst index $H\in(0,\frac12)$. 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 $m>\frac{1}{2H}$.

math.PR↗
arXiv2014

Convergence of solutions of mixed stochastic delay differential equations with applications

The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a $γ$-Hölder continuous process with $γ>1/2$ (e.g. a fractional Brownian motion with Hurst parameter greater than $1/2$). It is shown that its solution depends continuously on the coefficients and the initial data. Two applications of this result are given: the convergence of solutions to equations with vanishing delay to the solution of equation without delay and the convergence of Euler approximations for mixed stochastic differential equations. As a side result of independent interest, the integrability of solution to mixed stochastic delay differential equations is established.

math.PR↗
arXiv2014

Boundary Non-Crossings of Additive Wiener Fields

Let $W_i=\{W_i(t), t\in \mathbb{R}_+\}, i=1,2$ be two Wiener processes and $W_3=\{W_3(\mathbf{t}), \mathbf{t}\in \mathbb{R}_+^2\}$ be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary non-crossing probability $$P_f=P\{W_1(t_1)+W_2(t_2)+W_3(\mathbf{t})+h(\mathbf{t})\leq u(\mathbf{t}), \mathbf{t}\in\mathbb{R}_+^2\},$$ where $h, u: \mathbb{R}_+^2\rightarrow \mathbb{R}_+$ are two measurable functions. We show further that for large trend functions $γf>0$ asymptotically when $γ\to \infty$ we have that $\ln P_{γf}$ is the same as $\ln P_{γ\underline{f}}$ where $\underline{f}$ is the projection of $f$ on some closed convex set of the reproducing kernel Hilbert Space of $W$. It turns out that our approach is applicable also for the additive Brownian pillow.

math.PR↗
arXiv2014

Ruin probability in a risk model with a variable premium intensity and risky investments

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.

math.PR↗
arXiv2014

Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet. This Gaussian field is an extension of fractional Brownian motion. We prove some properties of covariance function for self-similar fields with rectangular increments. Using Lamperti transformation we obtain properties of covariance function for the corresponding stationary fields. We present an example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet.

math.PR↗
arXiv2013

Boundary non-crossing probabilities for fractional Brownian motion with trend

In this paper we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product we solve a minimization problem related to the norm of the trend function.

math.PR↗
arXiv2013

Asymptotic Properties of Drift Parameter Estimator Based on Discrete Observations of Stochastic Differential Equation Driven by Fractional Brownian Motion

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.

math.PR↗