Parameter Estimation in Stochastic Differential Equations

Drift estimation for a periodic mean reversion process

“…We will see in the proof of Proposition 4.5 that the limit of 1 T Q −1 T is well defined since we show that the limit of γ T denoted by γ is greater than zero. Consequently, 1 T Q −1 T exists almost surely if T is large enough.…”

“…Thus we want to consider the more general process satisfying the stochastic differential equation (1) dX t = (L(t) − αX t )dt + σdB t , t ≥ 0, where L(t) is a time-dependent mean reversion level and where α, σ are positive constants. Note that model (1) differs from the original Ornstein-Uhlenbeck process in the position of α within the drift term.…”

“…Due to the linear form of S(θ, t, ·) the global Lipschitz condition is satisfied in our setting such that there exists at most one solution of (1).…”

“…The analysis of asymptotic properties of drift estimates for time-inhomogeneous diffusion models has been paid much less attention. Among some other authors, Bishwal [1] studies the maximum likelihood estimator for a timeinhomogeneous diffusion provided with drift function f (θ, t, X) defined on Θ×[0, T ]×C[0, T ], where Θ is the parameter space and C[0, T ] is the space of all continuous, real-valued functions on [0, T ]. Note that our model presented above belongs to this class of diffusion processes and that our results comply with Bishwal's findings.…”

“…where m T is an increasing, non-random sequence, see Bishwal [1] condition (A6) on p. 65. The verification of this condition does not go without saying in our model and requires some auxiliary ideas and results.…”