Related papers: Mimicking an It\^{o} process by a solution of a st…

A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative…

Using the theory of stochastic integration developed recently by the authors, in this paper we prove an It\^{o} formula for Hilbert space-valued It\^{o} processes defined with respect to a cylindrical-martingale valued measure. As part of…

Stochastic optimization problems often involve data distributions that change in reaction to the decision variables. This is the case for example when members of the population respond to a deployed classifier by manipulating their features…

Stochastic processes are often used to model complex scientific problems in fields ranging from biology and finance to engineering and physical science. This paper investigates rate-optimal estimation of the volatility matrix of a…

We derive an It\^o-type formula for a measure-valued process that has a decomposition analogous to a classical semimartingale. The derivation begins with a time partitioning approach similar to the classical proof of It\^o's formula. To…

For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear…

In this paper, we study the diffusion approximation for slow-fast stochastic differential equations with state-dependent switching, where the slow component $X^{\varepsilon}$ is the solution of a stochastic differential equation with…

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct…

We shall study backward stochastic differential equations and we will present a new approach for the existence of the solution. This type of equation appears very often in the valuation of financial derivatives in complete markets.…

We construct a sequence of functions that uniformly converge (on compact sets) to the price of Asian option, which is written on a stock whose dynamics follows a jump diffusion, exponentially fast. Each of the element in this sequence…

We propose a model of fractal point process driven by the nonlinear stochastic differential equation. The model is adjusted to the empirical data of trading activity in financial markets. This reproduces the probability distribution…

We consider the stochastic continuity equation associated to an It\^{o} diffusion with irregular drift and diffusion coefficients. We give regularity conditions under which weak solutions are renormalized in the sense of DiPerna/Lions, and…

Stochastic differential equations (SDE) are widely used in modeling stochastic dynamics in literature. However, SDE alone is not enough to determine a unique process. A specified interpretation for stochastic integration is needed.…

We extend the It\=o formula \cite{MR1837298}*{Theorem 2.3} for semimartingales with rcll paths. We also comment on Local time process of such semimartingales. We apply the It\=o formula to L\'evy processes to obtain existence of solutions…

The mathematical model of a linear system with the short memory about own stochastic behavior is proposed. It is assumed that the system is under a continual influence of independent stochastic impulses. In a short memory approximation the…

This paper presents a framework of imitating the principal investor's behavior for optimal pricing and hedging options. We construct a non-deterministic Markov decision process for modeling stock price change driven by the principal…

To define oscillatory movements of securities market, we put in the non-local extension of Ito- equation for wavelet-images of random processes. It is proposed an algorithm of creation of evolutionary equation and a model of prediction of…

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model extending the decomposition obtained by E. Al\`os in [2] for the Heston model. We realize that a new term arises when the stock…

In this article we establish a new formula for the difference of a test function of the solution of a stochastic differential equation and of the test function of an It\^o process. The introduced formula essentially generalizes both the…

This article considers a model for alternative processes for securities prices and compares this model with actual return data of several securities. The distributions of returns that appear in the model can be Gaussian as well as…