In this paper, motivated by mathematic finance we introduce the multiple G-Itô integral in the G-expectation space, then investigate how to calculate. We get the the relationship between Hermite polynomials and multiple G-Itô integrals which is a natural extension of the classical result obtained by Itô in 1951.
On some expectation and derivative operators related to integral representations of random variables with respect to a PII process
给定一个独立增量过程$X$（不一定是鞅）和一大类平方可积〔的〕r. v. $H=f(X_T)$，$f$是有限测度$μ$的傅里叶变换，我们提供明确的Kunita-Watanabe和Föllmer-Schweizer分解。该表示是通过两个显着的地图表示：期望和导数运营商的特点$X$。我们还提供了一个显式的表达时，对冲索赔$H$与底层过程$X$的方差最优误差。这些问题是由金融数学中著名的全局和局部二次风险最小化问题的求解引起的。
In this paper, we study the optimal multiple stopping problem under the filtration consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions rather than an RCLL process. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optim…
In this paper, we study the optimal multiple stopping problem under Knightian uncertainty both under discrete-time case and continuous-time case. The Knightian uncertainty is modeled by a single real-valued function g, which is the generator of a kind of backward stochastic differential equations (BSDEs). We show that the value function of the multiple stopping problem coincides with the one corresponding to a new reward sequence or process. For the discrete-time case, this problem can be solved by an induction method which is a straightforward generalization of the single stopping theory. For…
Given $\{W^{(m)}(t), t \in [0,T]\}_{m \ge 1}$ a sequence of approximations to a standard Brownian motion $W$ in $[0,T]$ such that $W^{(m)}(t)$ converges almost surely to $W(t)$ we show that, under regular conditions on the approximations, the multiple ordinary integrals with respect to $dW^{(m)}$ converge to the multiple Stratonovich integral. We are integrating functions of the type $$f(x_1,\ldots,x_n)=f_1(x_1)\ldots f_n(x_n) I_{\{x_1\le \ldots \le x_n\}},$$ where for each $i \in \{1,\ldots,n\}$, $f_i$ has continuous derivatives in $[0,T].$ We apply this result to approximations obtained from…
We derive a product formula for the multiple stochastic integrals with respect to Levy process. The idea is to use exponential vectors and the polarization technique which greatly simplify the argument.
Stochastic wave equations driven by space-time $G$-white noise under sublinear expectation
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性预期前沿科学中心（教育部））
Integral representations for expectations of functions of a stable Lévy process $X$ and its supremum $\bar X$ are derived. As examples, cumulative probability distribution functions (cpdf) of $X_T, \barX_T$, the joint cpdf of $X_T$ and $\barX_T$, and the expectation of $(\be X_T-\barX_T)_+$, $\be>1$, are considered, and efficient numerical procedures for cpdfs are developed. The most efficient numerical methods use the conformal acceleration technique and simplified trapezoid rule.
In this paper, we derive a representation for the value process associated to the solutions of FBSDEs in a jump-diffusion setting under multiple probability measures. Motivated by concrete financial problems, the latter representations are then applied to devise a generalization of the change of numéraire technique allowing to obtain recursive pricing formulas in the presence of multiple interest rates and collateralization.
