Invariant measures for the Musiela equation with deterministic diffusion term
This paper deals with the Musiela forward rates equation in the Gaussian case, i.e. the stochastic partial differential equation \[ dr(t,x)=\left({{\partial}\over{\partial x}} r(t,x)+\sum_{n=1}^k \tau_n(t,x) \int_0^x \tau_n(t,u) du\right)dt+ \sum_{n=1}^k \tau_n(t,x) dW_t^n, \] where \(r(t,x)\) is an instantaneous forward rate at time \(t\), \(\tau_n(t,x)\) are \({\mathcal F_t}\)-measurable random fields, and \(W_t\) is a Brownian motion adapted to \({\mathcal F_t}.\) This equation is studied in the Sobolev spaces \(H_{\gamma}^1(\mathbb R^+)\), \(\gamma>0\), and \(H^1(\mathbb R^+)\). For each of the spaces explicit solutions are found. Conditions of existence and uniqueness of invariant measures are given.
stochastic partial differential equation
scientific article; zbMATH DE number 1452411