Convergence of solution of stochastic equations in the space of formal series (Q2769709)

Let \(Y\) be a Hilbert space and \(Y^{\otimes k}\) its \(k\)-fold tensor product. Denote by \(Y^{\times\infty}\) the space of formal series \(y=\{y_k\}_{k\geq 1}\subset Y\). If \(a_k\) are linear continuous operators from \(Y^{\otimes k}\) into a Hilbert space \(Z\), then \(a=\{a_k\}_{k\geq 1}\) is a formal mapping from \(Y\) to \(Z\). Two formal mappings can be composed, and the action \(a(y)\) of a formal mapping \(a\) on a formal series \(y\) is the element of \(Z^{\times\infty}\) defined by NEWLINE\[NEWLINE \{a(y)\}_n:= \sum_{k=1}^n \sum_{j_1+\cdots+j_k=n} a_k(y_{j_1}\otimes\cdots\otimes y_{j_k}).NEWLINE\]NEWLINE If \(y_n\) are stochastic processes in \(Y\), then \(y(t)=\{y_n(t)\}_{n\geq 1}\) is called a stochastic process in \(Y^{\times\infty}\). Let \(H_0\) be a Hilbert space, \(H_+\subset H_0\subset H_{-}\) the canonical Hilbert-Schmidt triple, and \(w\) a standard Wiener process on \(H_{-}\). Consider the equation between formal series NEWLINE\[NEWLINE y(t)= y^s + \int_s^t a(\tau)(y(\tau)) d\tau + \int_s^t b(\tau)(y(\tau)) dw(\tau) \;,\quad 0\leq s\leq t\leq T,\tag{1} NEWLINE\]NEWLINE where \(y^0\) is a non-random initial condition, with \(y^0_k=0\) for \(k\geq 2\), \(a(\tau)\) and \(b(\tau)\) are formal mappings with \(a_k\) and \(b_k\) measurable bounded functions from \([0,T]\) into the space of linear continuous operators from \(Y^{\otimes k}\) to \(Y\) and to \(\mathcal L_2(H_0,Y)\), respectively, and the integrals are interpreted as formal series in \(Y^{\times\infty}\), whose components are NEWLINE\[NEWLINE \begin{aligned} \Big( \int_s^t a(\tau)(y(\tau)) d\tau \Big)_n &:= \int_s^t (a(\tau)(y(\tau)))_n d\tau_,\\ \Big( \int_s^t b(\tau)(y(\tau)) dw(\tau) \Big)_n &:= \int_s^t (b(\tau)(y(\tau)))_n dw(\tau).\end{aligned} NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEA process \(y(t)=\{y_n(t)\}_{k\geq 1} \in Y^{\times \infty}\) is called a solution if its components \(y_n\) satisfy the componentwise equations NEWLINE\[NEWLINE \begin{aligned} y_1(t)&=y^s_1 + \int_s^t a_1(\tau)(y_1(\tau)) d\tau + \int_s^t b_1(\tau)(y_1(\tau)) dw(\tau),\quad \\ y_n(t)&= \int_s^t \sum_{k=1}^n \sum_{j_1+\cdots+j_k=n} a_k(\tau)(y_{j_1}(\tau)\otimes\cdots\otimes y_{j_k}(\tau)) d\tau \\ &\phantom{=} + \int_s^t \sum_{k=1}^n \sum_{j_1+\cdots+j_k=n} b_k(\tau)(y_{j_1}(\tau)\otimes\cdots\otimes y_{j_k}(\tau)) dw(\tau),\quad n\geq 2.\end{aligned} NEWLINE\]NEWLINE Equation (1) has a unique adapted solution in \(Y^{\times \infty}\). In the case of linear diffusion and analytical drift, sufficient conditions are given for Equation (1) to have a solution in \(Y\), which coincides, until a random explosion time, with the formal series solution.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].