In our paper [Bernoulli 26(2), 2020, 1381--1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^{\alpha}\circ dB$, $\alpha\in (0,1)$. These solutions have the form $X_t^\theta=F(B^\theta_t)$, where $F(x)=\frac{1}{1-\alpha}|x|^{1/(1-\alpha)}\text{sign}\, x$ and $B^\theta$ is the skew Brownian motion with skewness parameter $\theta\in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\varepsilon W$ restores uniqueness. In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $\theta=0$.

The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.

For a continuous-time lattice random walk $X^\Lambda=\set{X^\Lambda_t,t\ge 0}$ in a random environment $\Lambda$, we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X^\Lambda_s)ds$, $t\ge 0$. We establish a limit theorem for it, which is similar to that in the non-lattice case, under less restrictive assumptions.

We study Volterra processes X_t = ∫ _0^t K(t,s)dW_s , where W is a standard Wiener process, and the kernel has the form K(t,s) = a(s) ∫ _s^t b(u) c(u-s) du . This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index H>1/2 . We establish smoothness properties of X, including continuity and Hölder property. It happens that its Hölder smoothness is close to well-known Hölder smoothness of fBm but is a bit worse. We give a comparison with fBm for any smoothness theorem. Then we investigate the problem of inverse representation of W via X in the case where c∈ L^1[0,T] creates a Sonine pair, i.e. there exists h∈ L^1[0,T] such that c * h ≡ 1 . It is a natural extension of the respective property of fBm that generates the same filtration with the underlying Wiener process. Since the inverse representations of the Gaussian processes under consideration are based on the properties of Sonine pairs, we provide several examples of Sonine pairs, both well-known and new.

Motivated by the classical harmonic mean formula, estabished by Aldous in 1989, we investigate the relation between the sojourn time and supremum of a random process X(t), t is an element of R-d and extend the harmonic mean formula for general stochastically continuous X. We discuss two applications concerning the continuity of distribution of supremum of X and representations of classical Pickands constants.

A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed RV with respect to some properly localised boundedness $\mathcal{B}$ imposing weak assumptions on the structure of Polish space. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on general Polish metric spaces. Tail measures introduced in [2] appear naturally as limiting measures of regularly varying time series. We define tail measures on a measurable space indexed by $\mathcal{H}(D)$, a countable family of homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $\mathcal{B}$ is determined by $\mathcal{H}(D)$. This allows us to study the regular variation of cadlag processes on $D(R^l, R^d)$ retrieving in particular results obtained in [1] for RV of stationary cadlag processes on the real line removing $l=1$ therein. Further, we discuss potential applications and open questions.

We study the standard utility maximization problem for a non-decreasing upper-semicontinuous utility function satisfying mild growth assumption. In contrast to the classical setting, we do not impose the assumption that the utility function is concave. By considering the concave envelope, or concavification, of the utility function, we identify the optimal solution for the optimization problem. We also construct the optimal solution for the constrained optimization problem, where the final endowment is bounded from above by a discrete random variable. We present several examples illustrating that our assumptions cannot be totally avoided.

For a continuous-time random walk X = {X-t, t >= 0} (in general non-Markov), we study the asymptotic behaviour, as t -> infinity, of the normalized additive functional c(t) integral(t)(0) f(X.)ds, t >= 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to alpha-stable law with alpha > 1, we establish the convergence to the local time at zero of an alpha-stable Levy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: Lambda(x,gamma) = e(-Sigma y is an element of gamma phi(x-y)), where phi: R -> [0, infinity) is a bounded function decaying sufficiently fast, and gamma is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both 'quenched' component depending on Lambda, and a component, where Lambda is 'averaged'.

We consider a fractionally integrated Bessel process defined by Y s δ , H = ∫ 0 ∞ ( u H − ( 1 / 2 ) − ( u − s ) + H − ( 1 / 2 ) ) d X u δ , where X δ is the Bessel process of dimension δ > 2. We discuss the relation of this process to the fractional Brownian motion at its maximum, study the basic properties of the process and prove its Hölder continuity.

We consider a version of the secretary problem where elements may vanish during the selection and become unchoosable. We construct a selection strategy and identify the probability to select the best element, which turns out to be asymptotically maximal as number of elements increases indefinitely. As an auxiliary result of independent interest we establish large deviation probability estimates for sums of independent variables with distinct geometric distribution.

We estimate the kernel function of a symmetric alpha stable ( $$S\alpha S$$ ) moving average random function which is observed on a regular grid of points. The proposed estimator relies on the empirical normalized (smoothed) periodogram. It is shown to be weakly consistent for positive definite kernel functions, when the grid mesh size tends to zero and at the same time the observation horizon tends to infinity (high-frequency observations). A simulation study shows that the estimator performs well at finite sample sizes, when the integrator measure of the moving average random function is $$S\alpha S$$ and for some other infinitely divisible integrators.

In this paper, we study the Stratonovich stochastic differential equation dX = vertical bar X vertical bar(alpha) circle dB, alpha is an element of (-1, 1), which has been introduced by Cherstvy et al. (New J. Phys. 15 (2013) 083039) in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes spending zero time at 0: for alpha is an element of (0, 1), these solutions have the form X-t(theta) = ((1 - alpha)B-t(theta))(1/(1-alpha)). where B-theta is the theta-skew Brownian motion driven by B and starting at 1/1-alpha(X-0)(1-alpha), theta is an element of [-1, 1], and (x)(gamma) = vertical bar x vertical bar(gamma) sign x; for alpha is an element of (-1, 0], only the case theta = 0 is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation [f (B-theta), B] for a locally square integrable function f and is based on the time-reversion technique for Markovian diffusions.

We study boundary non-crossing probabilities $$\begin{aligned} P_{f,u} := \mathrm {P}\big (\forall t\in {\mathbb {T}}\ X_t + f(t)\le u(t)\big ) \end{aligned}$$ for a continuous centered Gaussian process X indexed by some arbitrary compact separable metric space $${\mathbb {T}}$$ . We obtain both upper and lower bounds for $$P_{f,u}$$ . The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $$P_{{y}f,u}$$ , $${y}\rightarrow +\infty $$ , which are two-term approximations (up to $$o({y})$$ ). The asymptotics are formulated in terms of the solution $${\tilde{f}}$$ to the constrained optimization problem $$\begin{aligned} \left\Vert h\right\Vert _{{\mathbb {H}}_X}\rightarrow \min , \quad h\in {\mathbb {H}}_X, h\ge f \end{aligned}$$ in the reproducing kernel Hilbert space $${\mathbb {H}}_X$$ of X. Several applications of the results are further presented.

A wave equation with external forces is considered in this paper for a homogeneous string with fixed ends. The distribution of the right-hand side of the equation is symmetric $\alpha$-stable. It is proved that the function constructed by the Fourier method is a generalized solution of the equation. The regularity of the trajectories is also established.

For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset {\mathbb{R}^{d}}$ and driven by an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.

In this paper we study the Stratonovich stochastic differential equation d X=|X|^α∘d B, α∈(-1,1), which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes spending zero time at 0: for α∈ (0,1), these solutions have the form X_t^θ=((1-α)B_t^θ)^1/(1-α), where B^θ is the θ-skew Brownian motion driven by B and starting at 1/1-α(X_0)^1-α, θ∈ [-1,1], and (x)^γ=|x|^γsign x; for α∈(-1,0], only the case θ=0 is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation [f(B^θ),B] for a locally square integrable function f and is based on the time-reversion technique for Markovian diffusions.

This paper deals with stochastic differential heat equation which is the typical example of stochastic partial differential equations (SPDE). In particular, paper is devoted to the estimation of diffusion parameter $\sigma$ for the random field which is the solution of stochastic differential heat equation for R^d, d = 1, 2, 3. The estimtion of diffusion parameter was constructed in accordance with observations on the grid. It was shown that the constructed estimate is strictly consistent and asymptotically normal, the asymptotic variance was calculated.

We investigate Wiener-transformable markets, where the driving process is given by an adapted transformation of a Wiener process. This includes processes with long memory, like 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. Motivated by integral representation results in general Gaussian setting, we study the conditions under which random variables can be represented as pathwise integrals with respect to the driving process. From financial point of view, it means that we give the conditions of replication of contingent claims on such markets. As an application of our results, we consider the utility maximization problem in our specific setting. Note that the markets under consideration can be both arbitrage and arbitrage-free, and moreover, we give the representation results in terms of bounded strategies.

The Cramer–Lundberg model is considered as a model of insurance company. Since it is impossible to obtain an explicit solution for the non-ruin probability function of insurance company for an arbitrary distribution of the values of insurance claims, the authors consider the problem of estimating the convergence of the original non-ruin probability to one that would be obtained by approximating the values of claim distribution function.

We study the equation for forced vibrations of a homogeneous string with a random force having a symmetric alpha-stable distribution. We show that the function constructed by the Fourier method is a generalized solution to the equation, and establish its pathwise regularity.