A Generalized It$\hat {\rm o}$'s Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals

Pathwise integrals and Ito-Tanaka Formula for Gaussian processes

We prove the Ito-Tanaka formula and the existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue-Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of Ito-Tanaka formula for pricing and hedging of financial derivatives.

Pathwise Construction of Stochastic Integrals

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. This method applies to any predictable integrand.

On pathwise uniform approximation of processes with càdlàg trajectories by processes with finite total variation

For any real-valued stochastic process X with càdlàg paths we define non-empty family of processes, which have finite total variation, have jumps of the same order as the process X and uniformly approximate its paths: This allows to decompose any real-valued stochastic process with càdlàg paths and infinite total variation into a sum of uniformly close, finite variation process and an adapted process, with arbitrary small amplitude but infinite total variation. Another application of the defined class is the definition of the stochastic integral with respect to the process X as a limit of path…

Two-parameter $p, q$-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter $p, q$-variation path integrals. Our condition of locally bounded $p,q$-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time $\int_{-\infty}^\infty\int_0^t g(s,x)d_{s,x}L_s(x)$ pathwise and then give generalized It$\hat {\rm o}$'s formula when $\nabla^-f(s,x)$ is only of bounded $p,q$-variation in $(s,x)$. In the case that $g(s,x)=\nabla^-f(s,x)$…

Pathwise stochastic integration with finite variation processes uniformly approximating càdlàg processes

For any real-valued stochastic process $X$ with cŕdlŕg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on compacts. The application of the defined class is the definition of stochastic integral with semimartingale integrand and integrator as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term (different from the Stratonovich integral). We compare the obtained result with classical results of Wong-…

区间型可料集上的随机积分及其金融应用

Stochastic Integrals on Predictable Sets of Interval Type with Financial Applications

本文推广经典随机积分，定义并刻画区间型可料集上的三类随机积分，建立相应半鞅 Itô 公式，并展示其在数理金融中的应用。