Periodic averaging method for impulsive stochastic differential equations with Lévy noise
This paper deals with a stochastic differential equation of following type \[ dx(t)=f(t,x(t))dt+g(t,x(t))dW(t)+ \int_Z h(t,x(t),z)\tilde{N}(dt,dz),\, t\neq t_{i}, \] \(\Delta x(t_i)=x(t_i^+)- x(t_i^-)=J_i (x(t_i^-))\), \(t=t_i\), \(i\in \mathbb{N}\), \(x(0)=x_0\), where \(W(t)\) is a \(n\)-dimensional standard Brownian motion, \(\tilde N(dt,dz)=N(dt,dz)-\lambda(dz)dt\), \(N(dt,dz)\) is a Poisson measure independent on \(W(t)\). It is supposed that there exists a concave nondecreasing function \(\psi(\cdot)\), such that \(\psi(0)=0\), \(\int_{0^+}\psi^{-1}(u)du=\infty\) and \[ \vert f(t,y)-f(t,x)\vert^2 +\vert g(t,y)-g(t,x)\vert^2+\int_Z \vert h(t,y,z)-h(t,x,z)\vert^2\lambda(dz)\leq\psi(\vert y-x\vert^2), \] and there exist \(L_1>0, L_2>0\) such that \[ \vert J_i (x)\vert^2\leq L_1,\, \vert J_i(x)-J_i(y)\vert^2\leq L_2\vert x-y\vert^2\text{ for every }x,\, y\in R^n. \] The main result of the paper is following. Let the functions \(f\), \(g\), \(h\) be \(T\)-periodic in the first argument and the impulses are periodic in the sense that there exists a \(k\in\mathbf{ N}\) such that \(0\leq t_1<\dots<t_k<T\) and for every integer \(i>k\), we have \(t_i =t_{i-k}+T\), \(J_i=J_{i-k}\). Let \(\bar f(x)\), \(\bar g(x)\), \(\bar h(x,z)\) be the corresponding averaged functions, and \(\bar J(x)=T^{-1}\sum_{i=1}^k J_i(x)\). Suppose that for every \(\varepsilon\in(0,\varepsilon_0]\), the impulsive stochastic differential equation \[ dx_\varepsilon (t)=\varepsilon f(t,x_\varepsilon (t))dt+ \sqrt{\varepsilon}g(t,x_\varepsilon (t))dW(t)+\sqrt{\varepsilon} \int_Z h(t,x_\varepsilon (t),z)\tilde N(dt,dz),\,t\neq t_i, \] \[ \Delta x_\varepsilon (t_i)=\varepsilon J_i (x_\varepsilon (t_i^-)),\, t=t_i,\, i\in \mathbf{ N},\, x_\varepsilon(0)=x_0 \] and the stochastic differential equation \(dy_\varepsilon(t)=\varepsilon \bar{f}(y_\varepsilon(t))dt +\sqrt{\varepsilon}\bar{g} (y_\varepsilon (t))dW(t)+\sqrt{\varepsilon} \int_Z \bar h(y_\varepsilon (t),z)\tilde N(dt,dz)\), \(y_\varepsilon (0)=y_0\), have solutions \(x_\varepsilon\) and \(y_ \varepsilon\), respectively. Assume that the functions \(f,g,h\) are bounded in the first argument, and there is a constant \(C>0\) such that \(\vert x_0-y_0\vert^2\leq C\varepsilon\), then there exist constants \(\eta_1>0\) and \(\eta_2>0\) such that \(E\vert x_\varepsilon (t)- y_\varepsilon (t) \vert^2\leq \eta_1\varepsilon \forall t\in [0,\eta_2\varepsilon^{-1}]\) for all \(\varepsilon\in(0,\varepsilon_1]\) and for some \(\varepsilon_1\in (0,\varepsilon_0]\).
impulsive stochastic differential equations
scientific article; zbMATH DE number 7048167
scientific article