The Risk Measurement under the Variance-Gamma Process with Drift Switching

The paper discusses an extension of the variance-gamma process with stochastic linear drift coefficient. It is assumed that the linear drift coefficient may switch to a different value at the exponentially distributed time. The size of the drift jump is supposed to have a multinomial distribution. We have obtained the distribution function, the probability density function and the lower partial expectation for the considered process in closed forms. The results are applied to the calculation of the value at risk and the expected shortfall of the investment portfolio in the related multivariate stochastic model.

Drift Estimation of Multiscale Diffusions Based on Filtered Data

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

Controlled Switching Diffusions Under Ambiguity: The Average Criterion

This paper concerns controlled switching diffusions. In particular, we consider that the drift coefficient of the diffusion process depends on an unknown (and possibly nonobservable) parameter. For giving solution to our control problem, we formulate it as a game against nature, where the ambiguity is represented by nature that chooses values of the unknown parameter through actions so playing the role as an opposite player of the controller. Our objective is to give conditions to characterize the ergodic optimality and guarantee the existence of optiaml policies for the central controller. Finally, we provide two examples to illustrate our results.

ANALYSIS OF THE PARAMETERS OF DYNAMIC SUPERPLASTICITY THE INDUSTRIAL ALUMINUM ALLOYS

The problem of determining the features of the development of blurred phase transitions observed under conditions of dynamic superplasticity of aluminum alloys is solved using the specific heat capacity function. Within the framework of the developed model representations, the deformation mechanisms characteristic of superplasticity and boundary metastable states are analyzed using the Fokker-Planck equation. Using a macrokinetic model, an explicit expression is obtained for the function that characterizes the mechanism of grain boundary slippage (the “drift " coefficient) that prevails in superplasticity. By integrating the differential equations resulting from the model, the solution of which establishes the type of functions responsible for the implementation of the mechanisms of grain boundary slippage and diffusion processes. It is proposed that the diffusion coefficient is responsible for the accumulation of irreversible deformations outside the velocity range of superplasticity. The function responsible for the effects of grain boundary slippage (the "drift" coefficient) is particularly active towards the middle of the superplasticity velocity interval. It is confirmed that outside the velocity range of superplasticity, there is a redistribution of mass transfer forms, the responsibility for which is assigned to the diffusion coefficient. It is shown that the diffusion function shows a tendency to decrease when approaching the range of superplasticity rates. Metastable states are characterized by the competition of diffusion mechanisms and grain boundary slippage.

Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives

<p style='text-indent:20px;'>In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, <i>J. Math. Anal. Appl. 502</i>, 2 (2021)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunov-type condition (ensuring the existence of a unique solution to the SDE) depend in the strong sense in a logarithmically Hölder continuous way on their initial values. One might then wonder whether this result can be sharpened and whether in fact, SDEs from this class necessarily have solutions which depend in the strong sense locally Lipschitz continuously on their initial value. The key contribution of this article is to establish that this is not the case. More precisely, we supply a family of examples of additive noise driven SDEs, which have smooth drift coefficient functions with at most polynomially growing derivatives and whose solutions do not depend in the strong sense on their initial value in a locally Lipschitz continuous, nor even in a locally Hölder continuous way.</p>

Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control

<p style='text-indent:20px;'>In this work, the issue of stabilization for a class of continuous-time hybrid stochastic systems with Lévy noise (HLSDEs, in short) is explored by using periodic intermittent control. As for the unstable HLSDEs, we design a periodic intermittent controller. The main idea is to compare the controlled system with a stabilized one with a periodic intermittent drift coefficient, which enables us to use the existing stability results on the HLSDEs. An illustrative example is proposed to show the feasibility of the obtained result.</p>

A Bayesian sequential test for the drift of a fractional Brownian motion

We consider a fractional Brownian motion with linear drift such that its unknown drift coefficient has a prior normal distribution and construct a sequential test for the hypothesis that the drift is positive versus the alternative that it is negative. We show that the problem of constructing the test reduces to an optimal stopping problem for a standard Brownian motion obtained by a transformation of the fractional Brownian motion. The solution is described as the first exit time from some set, and it is shown that its boundaries satisfy a certain integral equation, which is solved numerically.