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Based on the classical probability, the stability criteria for stochastic differential delay equations (SDDEs) where their coefficients are either linear or nonlinear but bounded by linear functions have been investigated intensively. Moreover, the dependent stability of the highly nonlinear hybrid stochastic differential equations is recently studied. In this paper, by using the nonlinear expectation theory, we explore the dependent stability of a class of highly nonlinear hybrid stochastic differential delay equations driven by $G$-Brownian motion ($G$-SDDEs). Firstly, we give preliminaries of sublinear expectation. Then, the delay-dependent criteria of the stability and boundedness of solutions to $G$-SDDEs is provided. Finally, an illustrative example is analyzed by the $\varphi$-max-mean algorithm., Comment: 24 pages; 3 figures

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-10$, we consider the generalized quadratic covariation $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ defined by $$ \bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\lim_{\varepsilon\downarrow 0}\frac{1+a+b}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} \left\{f(B^{a,b}_{s+\varepsilon}) -f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds, $$ provided the limit exists uniformly in probability. We construct a Banach space ${\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2(\Omega)$ and the generalized Bouleau-Yor identity $$ [f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\frac1{(1+b){\mathbb B}(a+1,b+1)} \int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t) $$ holds for all $f\in {\mathscr H}$, where ${\mathscr L}^{a,b}(x,t)=\int_0^t\delta(B^{a,b}_s-x)ds^{1+a+b}$ is the weighted local time of $B^{a,b}$ and ${\mathbb B}(\cdot,\cdot)$ is the Beta function., Comment: 31 pages

Let $B^H$ be a fractional Brownian motion with Hurst index $0\varepsilon\}}\frac1{B^H_s-a}ds^{2H}\equiv \frac1{\pi}{\mathscr H}{\mathscr L}^H(\cdot,t)(a) $$ in $L^2(\Omega)$ with $ a\in {\mathbb R}, t\geq 0$ and ${\mathscr H}$ denoting the Hilbert transform. We show that $$ {\mathcal C}^H_t(a)=2\left((B^H_t-a)\log|B^H_t-a|-B^H_t+a\log|a| -\int_0^t\log|B^H_s-a|\delta B^H_s\right) $$ for all $a\in {\mathbb R}, t\geq 0$ which is the fractional version of Yamada's formula, where the integral is the Skorohod integral. Moreover, we introduce the following {\it occupation type formula}: $$ \int_{\mathbb R}{\mathcal C}^H_t(a)g(a)da=2H\pi\int_0^t({\mathscr H}g)(B^H_s)s^{2H-1}ds $$ for all continuous functions $g$ with compact support., Comment: 30 pages

Let $u(t,x)$ be the solution to a stochastic heat equation $$ \frac{\partial}{\partial t}u=\frac12\frac{\partial^2}{\partial x^2}u+\frac{\partial^2}{\partial t\partial x}X(t,x),\quad t\geq 0, x\in {\mathbb R} $$ with initial condition $u(0,x)\equiv 0$, where $X$ is a time-space white noise. This paper is an attempt to study stochastic analysis questions of the solution $u(t,x)$. In fact, the solution is a Gaussian process such that the process $t\mapsto u(t,\cdot)$ is a bi-fractional Brownian motion seemed a fractional Brownian motion with Hurst index $H=\frac14$ for every real number $x$. However, the properties of the process $x\mapsto u(\cdot,x)$ are unknown. In this paper we consider the quadratic covariations of the two processes $x\mapsto u(\cdot,x),t\mapsto u(t,\cdot)$. We show that $x\mapsto u(\cdot,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with "It\^o's integral", but it is not a semimartingale. Moreover, some generalized It\^o's formulas and Bouleau-Yor identities are introduced., Comment: 35 pages

Let $B^{H_1}$ and $\tilde{B}^{H_2}$ be two independent fractional Brownian motions on ${\mathbb R}$ with respective indices $H_i\in (0,1)$ and $H_1\leq H_2$. In this paper, we consider their intersection local time $\ell_t(a)$. We show that $\ell_t(a)$ is differentiable in the spatial variable if $\frac1{H_1}+\frac1{H_2}>3$, and we introduce the so-called {\it hybrid quadratic covariation} $[f(B^{H_1}-\tilde{B}^{H_2}),B^{H_1}]^{(HC)}$. When $H_1<\frac12$, we construct a Banach space ${\mathscr H}$ of measurable functions such that the quadratic covariation exists in $L^2(\Omega)$ for all $f\in {\mathscr H}$, and the Bouleau-Yor type identity $$ [f(B^{H_1}-\tilde{B}^{H_2}),B^{H_1}]^{(HC)}_t=-\int_{\mathbb R}f(a)\ell_t(da) $$ holds. When $H_1\geq \frac12$, we show that the quadratic covariation exists also in $L^2(\Omega)$ and the above Bouleau-Yor type identity holds also for all H\"older functions $f$ of order $\nu>\frac{2H_1-1}{H_1}$., Comment: 34 pages

Let ${\mathscr L}$ be the local time of $G$-Brownian motion $B$. In this paper, we prove the existence of the quadratic covariation $_{t}$ and the integral $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$. Moreover, a sublinear version of the Bouleau-Yor identity $$ \int_{\mathbb R}f(x){\mathscr L}(dx,t)=-_{t} $$ is showed to hold under some suitable conditions. These allow us to write the It\^o's formula for $C^1$-functions., Comment: 24 pages

Let $B$ be a bi-fractional Brownian motion with indices $H\in (0,1),K\in (0,1]$, $2HK=1$ and let ${\mathscr L}(x,t)$ be its local time process. We construct a Banach space ${\mathscr H}$ of measurable functions such that the quadratic covariation $[f(B),B]$ and the integral $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$ exist provided $f\in {\mathscr H}$. Moreover, the Bouleau-Yor identity $$ [f(B),B]_t=-2^{1-K}\int_{\mathbb R}f(x){\mathscr L}(dx,t),\qquad t\geq 0, $$ holds for all $f\in {\mathscr H}$., Comment: 32 pages

Let $B^H$ be a fractional Brownian motion with Hurst index $0

Let ${\mathscr L}^H(x,t)=2H\int_0^t\delta(B^H_s-x)s^{2H-1}ds$ be the weighted local time of fractional Brownian motion $B^H$ with Hurst index $1/2

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2