The functional Itō formula under the family of continuous semimartingale measures

Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Itō’s formula that applies to functionals with a continuous dependence on the trajectories of the underlying process. In this paper, we study nonlinear functionals that do not have such continuity. By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.

Nonexplosion and Pathwise Uniqueness of Stochastic Differential Equation Driven by Continuous Semimartingale with Non-Lipschitz Coefficients

We study a class of stochastic differential equations driven by semimartingale with non-Lipschitz coefficients. New sufficient conditions on the strong uniqueness and the nonexplosion are derived ford-dimensional stochastic differential equations onRd(d>2)with non-Lipschitz coefficients, which extend and improve Fei’s results.

On Henstock Method to Stratonovich Integral with respect to Continuous Semimartingale

We will use the Henstock (or generalized Riemann) approach to define the Stratonovich integral with respect to continuous semimartingale in L2 space. Our definition of Stratonovich integral encompasses the classical definition of Stratonovich integral.

STATISTICAL CAUSALITY AND MARTINGALE REPRESENTATION PROPERTY WITH APPLICATION TO STOCHASTIC DIFFERENTIAL EQUATIONS

The paper considers a statistical concept of causality in continuous time between filtered probability spaces, based on Granger’s definition of causality. This causality concept is connected with the preservation of the martingale representation property when the filtration is getting smaller. We also give conditions, in terms of causality, for every martingale to be a continuous semimartingale, and we consider the equivalence between the concept of causality and the preservation of the martingale representation property under change of measure. In addition, we apply these results to weak solutions of stochastic differential equations. The results can be applied to the economics of securities trading.

A Note on Asymptotic Exponential Arbitrage with Exponentially Decaying Failure Probability

The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer (2007) in a slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer (2007), our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.

A Note on Asymptotic Exponential Arbitrage with Exponentially Decaying Failure Probability

The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer (2007) in a slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer (2007), our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.

Backward stochastic PDEs related to the utility maximization problem

We study utility maximization problem for general utility functions using the dynamic programming approach. An incomplete financial market model is considered, where the dynamics of asset prices is described by an -valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic partial differential equation related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward stochastic differential equation. The cases of power, exponential and logarithmic utilities are considered as examples.

A generalized occupation time formula for continuous semimartingales

We show that for a wide class of functions F we have lim \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{\varepsilon }\int\limits_0^t {\{ F(s,X_s ) - F(s,X_s - \varepsilon )\} d\left\langle {X,X} \right\rangle _s = - } \int\limits_0^t {\int\limits_\mathbb{R} {F(s,x)dL_s^x } }$$ \end{document} where Xt is a continuous semimartingale, ( Ltx , x ∈ ℝ, t ≧ 0) its local time process and (〈 X, X 〉 t , t ≧ 0) its quadratic variation process.

KUNITA-TYPE STOCHASTIC FLOWS OF HOMEOMORPHISMS IN EUCLIDEAN SPACE

In this paper we prove Kunita-type stochastic differential equation (SDE) [Formula: see text], t > s, driven by a spatial continuous semimartingale F (x, t) =(F1 (x, t), …, Fm (x, t)), x ∈ ℜm, with local characteristic (a, b), in the sense of Kunita (Chap. 3 of Ref. 10), can produce a stochastic flow of homeomorphisms of ℜm into itself almost surely under the (a, b) satisfies non-Lipschitz conditions. This result bases on recent works12 by the author.