The theory of stochastic integrals in appendix C and of stochastic differential equations dis- cussed in section 4.3 is suited to WIENER processes driving the SDE. A more general class of driving processes is the one of LEVY´ processes, for which the theory may be extended accordingly (see, e.g., [Protter2004; Bichteler2002]).
4.5.1 A short introduction to LEVY´ processes 4.32 Definition (LEVY´ process)
Let Xt be a stochastic process with X0 = 0 a.s. Then Xt is called a LEVY´ process, if the
following conditions hold:
i) the sample paths of Xt are c`adl`ag a.s.,
ii) Xt has independent increments,
iii) Xt has stationary increments.
4.33 Remark
The requirement in definition 4.32 of a LEVY´ processes being c`adl`ag can be substituted by requiring continuity in probability, i.e. for > 0 and t≥ 0, it holds lim
h→0IP |Xt+h− Xt| >
= 0 (or, equivalently, lim
s→tXs= Xt in probability).
One can then show that for every LEVY´ process there exists a c`adl`ag version of it. For a
proof, see e.g. [Protter2004].
4.34 Definition and Lemma (Counting process, (compound) POISSON process)
1. A stochastic process{Nt}t≥0is called a counting process, if the following conditions hold:
a) Nt∈ IN0 for all t≥ 0
b) N (t)≥ N(s) for t ≥ s
A counting process, as the name suggests, counts certain “events” – whatever they may be – that have been occurred up to and including time t. Thus, the increment
∆N (s, t]:= N (t) − N(s) is the number of events occurring in the interval (s, t].
2. A counting process {Nt}t≥0 is called a POISSON process with rate (or intensity) λ > 0,
if the following conditions hold: a) N0= 0 a.s.
b) the process has independent increments,
i.e. ∆N (s, s + t] has the same distribution as ∆N (0, t] for all t > s≥ 0. c) the increments are POISSON distributed
3. Let Nt be a POISSON process with rate λ > 0, and D(i) (i∈ IN) independent and iden-
tically distributed random variables with common cumulative distribution function FD.
Further, D(i) shall be independent of Ntfor all i∈ IN and t ≥ 0. Then,
Xt:= Nt
X
i=1
D(i) (4.10)
is called a POISSON process with rate λ and jump size distribution FD.
4. Every POISSON process and every compound POISSON process is a LEVY´ process. Proof: Let Nt be a POISSON process as above. Then, ∆N (s, t] ∼ Poi λ(t − s) and also
∆N (0, t− s]∼ Poi λ(t − s), i.e. the increments are stationary. Further, the increments are also independent by definition and Nt is c`adl`ag by construction, thus Ntis a LEVY´ process.
Let Xt be a compound POISSON process as above. Then, by construction, it is c`adl`ag and
has independent increments. For 0 ≤ s < t, we have (a) Xt− Xs = PNi=1t Di−PNi=1s Di =
PNt
i=Ns+1Di and (b) Xt−s=
PNt−s
i=1 Di. The number of jumps in (a) is Nt− Ns which has the
same distribution as Nt−sin (b), since the POISSONprocess Nthas stationary increments, and thus also Xt− Xs and Xt−shave the same distribution, as the Di are i.i.d. Thus, Xtis a LEVY´
process.
Figure 4.3 shows two realizations of LEVY´ processes: A POISSONprocess with rate λ = 5.0 and a compound POISSONprocess with rate λ = 5.0 and standard-normally distributed jumps. 4.35 Remark (LEVY´ -KHINTCHINE representation of LEVY´ processes)
We note that every LEVY´ process Ztmay be decomposed Zt= b(t) + aWt+ Mt
into its deterministic drift b(t), a scaled WIENER process Wt, and a jump process Mt being a superposition of independent POISSON processes, with Mt independent of Wt, W0 = 0, and
M0 = 0.
4.5.2 Numerical simulation of LEVY´ -driven SDE
Some numerical integration schemes for SDE with WIENER-driven diffusion are described in section 4.4. There, we deal with IT ˆO integrals w.r.t. WIENER processes, for which TAYLOR expansions are available, giving rise to several integration schemes with (in principle) arbitrary convergence rate [KloedenPlaten1995], with the EULER-MARUYAMA and MILSTEIN methods being the most prominently used ones.
In this section, we give a short introduction to a closely related numerical integration method for LEVY´ -driven SDE that follows ideas analogous to the explicit EULER-MARUYAMA method for WIENER-driven SDE.
We employ appropriate smoothness conditions on f and g (e.g. standard LIPSCHITZcondi- tions) as well as a sufficient integrability condition on the driving m-dimensional LEVY´ process Zt (e.g. square integrability, IE|Zt|2<∞) for strong solutions to exist.
(a) Sample path of a POISSONprocess, rateλ = 5.0 (b) Sample path of a compound POISSONprocess, rateλ = 5.0, jump size distribution FD∼N 0, 1
Figure 4.3: Sample paths of (compound) POISSON processes.
Let f : [0, T ]× IRn→ IRn be the deterministic drift function, and g : [0, T ]× IRn→ IRm×n be
a coefficient function for the driving LEVY´ process Zt. Then ˜
Xk+1 := ˜Xk+ f (τk, ˜Xk)∆τ + g(τk, ˜Xk)∆Zk, X˜0:= x0 (4.11)
delivers an EULERapproximation ˜Xk of Xτk for the S-IVP
dXt= f (t, Xt) dt + g(t, Xt−) dZt, X0= x0 (4.12)
with initial value x0, constant time increments ∆τ := NT (N ∈ IN), and τk := k · ∆τ for
k = 0, ..., N . Note that since Zt is c`adl`ag, the evaluation of the coefficient function g in the
S-IVP occurs at the left limits, “right before” a possible jump. 4.36 Remark
It would be mathematically sufficient to formulate a LEVY´ -driven SDE as dXt = g(X
t−)dZt
due to the LEVY´ -KHINTCHINErepresentation of Z
t (see remark 4.35).
4.37 Remark
If we are able to compute (simulate) the increments of the LEVY´ process ∆Z
k:= Zτk+1− Zτk
exactly, the method is called a genuine EULER method.
However, for many LEVY´ processes, simulation of exact increments is computationally hard. Often, approximations ∆ ˜Zk on the increments are used, introducing a second source of error
apart from the unavoidable discretization error. In this case, the method is frequently called
an approximative EULER method.
Under mild assumptions on the first moment of Zt, and still requiring a finite second
moment of Zt, in addition to sufficient smoothness of f and g, it can be shown that the genuine
EULERmethod has an error ofO(∆τ), see [ProtterTalay1997] and [DereichHeidenreich2011]. JACOD et al. [Jacod2005] give precise error bounds on both the genuine and approximate EULERmethods.
Differential Equation Models
In this chapter, we present and analyse a new method for parameter estimation in stochastic differential equations, based on a piecewise deterministic approach.
In the first section, we give an introduction into the topic of parameter estimation in SDE, describe our new approach in words, and formulate some technical assumptions.
Section 2 outlines existing estimation techniques for parameter estimation in continuous- time SDE with discretely sampled observations. References are given within the discussion.
The third section presents some results (with proofs) on the distance of solutions of sto- chastic initial value problems, S-IVP, to be defined in eq. (5.10), to the corresponding solutions of deterministic initial value problems, D-IVP, defined in eq. (5.11). For both, distance at the interval end as well the maximum distance throughout the interval, upper bounds in expecta- tion and mean-square are given.
In section 4, we present our new approach, the piecewise deterministic parameter esti- mation method for SDE in detail. After introducing notation and basic assumptions, mainly to ensure existence of strong solutions to the SDE, the new method is derived on the basis of the multiple shooting technique for parameter estimation in ODE presented in chapter 2. The continuity condition is replaced by a carefully weighted jump regularization term in the objective, allowing for a discontinuous trajectory and thus mimicking the stochasticity of an SDE formulation. It is shown that this regularization is necessary in order to get correct pa- rameter estimates. Further, results from the third section are used to prove that the jumps asymptotically converge to zero if the number of equidistantly chosen shooting intervals goes to infinity.
Section 5 gives an numerical analysis of the proposed method. The sparsity pattern of the Jacobian of the combined residual vector , composed of measurement and jump residuals, is investigated, and it is shown that the number of nonzero elements grows only linearly in the number of shooting intervals. Further, we prove that the sparsity is maintained under
HOUSEHOLDERbased decomposition techniques. The section concludes with the proposition
and analysis of a lifting technique based on interval-wise decoupling of parameters.
In section 6, we propose two extensions: an homotopy ansatz for pathological problems and a grid refinement strategy that is elaborated in more detail in the numerical examples chapter 6.
We also refer to appendix S for a discussion of the software package :sfit that implements the presented parameter estimation method.