Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes

This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure Ito and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite quadratic variation processes, Dirichlet and weak Dirichlet processes.

Clark–Ocone type formula for non-semimartingales with finite quadratic variation

We consider the stochastic differential equation (SDE) f(s,w, X.(w))dZs((w), where V and Z are vector valued process indexed by . The assumptions we make on Z and on the increasing process controlling Z are satisfied by certain classes of square integrable martingales, by processes of finite variation and by mixtures of these types of processes. Existence, uniqueness and the possibility of explosions of a strong solution X are investigated under Lipschitz conditions on f. A well-known sufficient condition for non-explosion is shown to work also in the multiparameter case and stability of X under perturbation of V, f and Z is proved. Finally more special SDE without Lipschitz conditions are considered, including a class of SDE of the Tsirel'son type.

Solution of the fourth and fifth boundary value problems for a one-dimensional second-order parabolic equation in a curvilinear region

This paper deals with the pathwise uniqueness of solutions to one-dimensional stochastic differential equations driven by Brownian motion and involving local times of unknown processes. The existence and uniqueness results of Legall are extended to cover, in particular, the case of degenerate diffusion coefficients. In addition, the class of time homogeneous stochastic differential equations with a fairly general diffusion coefficient and measurable drift term is briefly studied. In this case pathwise uniqueness of solutions is established under some integrability condition on the diffusion and drift coefficients

This article discusses an alternative method for estimating marginal probability densities of the solution to stochastic differential equations (SDEs). Two algorithms for calculating the numerical–statistical projection estimate for distributions of solutions to SDEs using Legendre polynomials are proposed. The root-mean-square error of this estimate is studied as a function of the projection expansion length, while the step of a numerical method for solving SDE and the sample size for expansion coefficients are fixed. The proposed technique is successfully verified on three one-dimensional SDEs that have stationary solutions with given one-dimensional distributions and exponential correlation functions. A comparative analysis of the proposed method for calculating the numerical–statistical projection estimate and the method for constructing the histogram is carried out.

We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy to handled. Similar equations without jumps were studied in the same context by [10, 15] and others authors.

Stochastic averaging for estimating a ship roll in random longitudinal or oblique waves

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at $\varepsilon \to 0$ were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.

Let Wt be a one‐dimensional Brownian motion on the probability space (Ω,F,P), and let dxt = a(xt)dt + b(xt)dwt, b2(x) > 0, be a one‐dimensional Ito stochastic differential equation. For a(x) = a0 + a1x + … + anxn on a bounded interval we obtain a lower bound for p(t,x,y), the transition density function of the homogeneous Markov process xt, depending directly on the coefficients a0,a1, …, an, and b(x).

In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable ( | z | ⁢ | ln ⁡ | z | | ) (\lvert z\rvert\sqrt{\lvert\ln\lvert z\rvert\rvert}) . We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.

Nous resolvons le probleme de plongement de Skorokhod pour une classe de processus stochastiques satisfaisant une equation differentielle stochastique (EDS) non homogene de la forme $\mathrm{d}A_{t}=\mu(t,A_{t})\,\mathrm{d}t+\sigma(t,A_{t})\,\mathrm{d}W_{t}$. Nous fournissons des conditions suffisantes garantissant que, pour une mesure de probabilite $\nu$ sur $\mathbb{R}$, il existe un temps d’arret borne $\tau$ et un reel $a$ tels que la solution $(A_{t})$ de l’EDS avec condition initiale $a$ satisfait $A_{\tau}\sim\nu$. Nous distinguons ici les cas ou $(A_{t})$ est une solution de l’EDS dans un sens faible ou fort. Notre construction des temps d’arret de plongement est basee sur une solution d’une EDS progressive retrograde totalement couplee. Nous utilisons la methode des decoupling fields pour verifier que l’EDSPR a une solution unique. Enfin, nous esquissons un algorithme pour mettre en pratique notre construction theorique et l’illustrons par une simulation numerique.

In this paper, we study the strong rate of convergence for the Euler-Maruyama approximation of a class of one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process, corresponding to divergence form operator with a discontinuous coefficients at zero. We use a space transform in order to remove the local time $L_{t}^{0}$ from the stochastic differential equation of type, \begin{equation*} dX_{t}=\sigma(X_{t})dB_{t}+\psi(X_{t})dt+\beta dL_{t}^{0}(X). \end{equation*} Here $B$ is a standard one-dimensional Brownian motion. $\sigma$ and $\psi$ are a bounded measurable functions, and $\beta\in(-1,1)$ . We provide the approximation of Euler-Maruyama for the stochastic differential equation without local time. After that the approximation can be transformed back, giving an approximation of Euler-Maruyama $X_{t}^{n}$ to the solution of the original SDELT, and we provide the rate of strong convergence Error $=\mathbb{E}[\vert X_{T}^{\mathrm{n}}-X_{T}\vert]$ .

In this paper we prove the existence and uniqueness of a solution for a one-dimensional quasilinear stochastic differential equation of the Skorohod type, with a non-linear boundary condition of the form . The method of proof uses the Girsanov transformation and some basic exponential estimates. When the coefficients of the equation are deterministic we show that the solution is weakly differentiable on the Wiener space and possesses a continuous modification.

ABSTRACTConsider a class of mixed stochastic differential equation (SDE) involving both a Brownian motion and a fractional Brownian motion with Hurst parameter We get the mean square rate of convergence by using a modified Euler method, here δ is the diameter of partition. As we know, the classical Euler method has the rate of convergence for mixed SDE and δ2H − 1 (resp. δH) for pathwise (resp. Skorokhod) SDE driven only by fBm, which were proved by Mishura and Shevchenko Mishura and Shevchenko (2011) and Mishura and Shevchenko (2008), respectively. Therefore, we obtain a better result than those of them.