In many settings, e.g. in cellular biology, occurring processes happen in a “mostly” determin- istic way, so to speak. For example, in enzyme kinetics, the reactions follow clear principles: reaction partners are known and if there are enough molecules of each occurring species, mod-
(a) deterministic (ODE) interpretation of the BISTABAER model
(b) stochastic (SDE) interpretation of the BISTABAER model
Figure 5.1: Stochasticity in the BISTABAER model completely alters its behaviour. The ODE system (a) quickly approaches the steady state, whereas in the SDE interpretation (b), ongoing transitions between two steady states are observed. See section 6.3 for details.
However, this is not always the case. As an illustrative example, we have a look at the allosteric regulation(1) of an enzyme, which might switch between its inactive form X and active form X∗ due to allosterically binding by its activator or inhibitor, respectively.
Figure 5.1 shows the time course of a simplified allosteric regulation model. In a deter- ministic interpretation, the system quickly reaches its (quasi-)steady state, as can be seen on section 5.1, whereas the stochastic version reveals a (random) switching between two stable steady states.
In both settings, a hidden control (the bistable system ˙L = L(1−L4)) is implemented that influences the amount of activator (effector) concentration (details are given in section 6.3). In the ODE interpretation, this control system quickly reaches and remains on a (locally) stable steady state, allowing the enzyme model to run towards its equilibrium. In the stochastic setting, the bistable control system gets permanent input from a WIENER process. While the control system tries to maintain its stability, a displacement by the driving WIENER process that is high enough might bring the control system into the contraction area of the second steady state, resulting in a different concentration of the activator. As a consequence, the equilibrium point of the enzyme model is changed.
However, the underlying dynamics of this system are – in a way – deterministic: If we knew the state of the (stochastically driven) control system L, the resulting kinetics can be calculated using an ODE integrator to arbitrary precision.
In other cases, e.g. in mathematical finance, some general principles of interest rates are known or assumed. Properties like (exponential) mean reversion to a level µ with reversion rate θ can be described by a simple deterministic model
˙x(t) = θ(µ− x(t)),
however, the actual sample paths are stochastic, possibly not even continuous (see section 6.4). As soon as there are stochastic parts in a system, modelling with ODE is, in general, inap- propriate, as the system is likely to show behaviour that one is unable to describe using pure ODE formulations, albeit the main occurring processes possess deterministic properties.
(1) The activity of an enzyme might be regulated by one or more effector proteins, that bind to the enzyme at
its allosteric site (not the active site). An activator might stabilize an enzyme in its active form, whereas the binding of an inhibitor might change the enzyme’s conformation in a way that the active site becomes
On notation: Xt vs. x(t)
Throughout this and the following chapter, capital letters with indices, e.g. Xt, denote sto-
chastic processes, especially solutions of an S-IVP
dXt= f (t, Xt) dt + g(t, Xt) dWt, Xt0 = x0
whereas the corresponding ODE solution, i.e. the solution of the D-IVP
˙x(t) = f (t, x(t)) , x(t0) = x0
is denoted by the respective small letter with arguments set in parenthesis, x(t).
5.1.1 The idea: ODE solutions resemble SDE solution paths on short time scales
When the influence of a driving stochastic process Wtis not “too strong”, we intuitively expect
a solution Xt of an SDE with constant diffusionD, i.e. with diffusion function g≡ D,
dXt= f (Xt) dt + D dWt Xt0 = x0
to be close to the solution x(t) of the corresponding ODEwith same initial value
˙x(t) = f (x(t)) x(t0) = x0
at least for a small period of time. Indeed, this can be shown to true; estimates of the maximum distance and of the distance at the interval and are given in section 5.3.
We can also make this observation in the introductory example of an allosteric enzyme regulation model. Though the deterministic and stochastic interpretation differ qualitatively over the whole time domain, the ODE and SDE trajectories stay closely together over a short time span as depicted in figure 5.2.
If we know the state of the stochastically influenced system at a specific time point t, say Xt, and simulate the corresponding ODE in the interval [t, t + ∆t] with this state as
initial condition, we may expect to stay “close” to the SDE solution, where the admissible length ∆t obviously depends on the driving process’s activity in that interval. Thus, we may try to approximate the solution process Xt over the whole time domain by a number of
discontinuously concatenated ODE solutions.
The finer we choose the grid of ODE approximations, the smaller the gaps eventually become, approaching zero in the limit, as we will show in section 5.4.6.
(a) ODE interpretation (b) SDE interpretation
Figure 5.2: Short term similarity of ODE and SDE interpretation of the BISTABAER model. This picture is a detail enlargement of figure 5.1. For a short time scale, the ODE and SDE trajectories are very similar. The impact of the driving WIENERprocess manifests as a small jitter in the SDE interpre- tation.
Clearly, if we already have the stochastic solution process Xt (i.e. a certain realization of it),
there is no need to approximate it any more. In the context of parameter estimation, we (usually) do not have a continuous observation of the system, but rather some measurement values of its state or functions thereof.
Now, the above observation gives rise to the following idea: Dissect the time horizon into small intervals, and determine an ODE solution on each interval, such that their (discon- tinuous) concatenation is close to the observations. Further, since the (unknown) stochastic solution process is continuous, choose the initial conditions on each interval in a way such that the discontinuities at the interval borders become small.
The above description resembles the method of multiple shooting for parameter estima- tion in ODE (section 2.2.3). What is most important, is that this ansatz gives us access to derivatives and allows the application of gradient-based optimization methods, although the approximated stochastic process Xt is nowhere differentiable in general.
We will see, that this idea is not just a pious hope, as section 5.3 gives some general results on the convergence of WIENER-driven SDE to ODE solutions. Further, section 5.4.6 gives a convergence result for the proposed parameter estimation method.
The precise formulation of the parameter estimation method with jump regularization is done in section 5.4.
5.1.2 SDE with constant diffusion by LAMPERTI transform
In this thesis, the focus lies on SDE with constant diffusion. Under the conditions of theo- rem 4.28 (existence and uniqueness of SDE solutions), a 1-dimensional SDE
dXt= f (t, Xt) dt + g(t, Xt) dWt (t∈ [0, T ])
with state-dependent diffusion may be transformed into an SDE with constant diffusion D = 1, using the LAMPERTI transform L(t, Xt) [Iacus2008; MollerMadsen2010; LuschgyPages2006], which is based on the IT ˆOformula (see, e.g. [Oksendal1998], Theorems 4.1.2 and 4.2.1):
Zt:= L(t, Xt) := Xt Z ξ 1 g(t, x) dx (5.1)
with an arbitrary value ξ from the state space of Xt. We note that the LAMPERTI transfor-
mation in eq. (5.1) is bijective if g(t, Xt) > 0 ∀ (t, Xt), as for every t∈ [0, T ], x 7→ L(t, x) is
continuous and strictly increasing [LuschgyPages2006]. The transformed process Zt solves the SDE
dZt=h d dtL t, L −1(t, Z t)+ f L−1(t, Zt) g L−1(t, Zt) − 1 2 d dxg t, L −1(t, Z t)idt + 1 dWt
from which, after simulation, the original process may be reconstructed as Xt = L−1(t, Zt).
For time-independent diffusion g(Xt), the SDE for Zt simplifies to
dZt= hf L−1(Z t) g L−1(Zt) − 1 2 dg dx L −1(Z t)idt + 1 dWt.
The application of the LAMPERTI transform is limited by the fact that we need to able to compute its inverse; however, this is possible for quite general classes of diffusion processes [MollerMadsen2010].
A general multivariate version of the LAMPERTI transform is currently not available [MollerMadsen2010], but MØLLERand MADSEN give a multivariate LAMPERTI transform for the class of SDEs that are of the form
dXt= f (t, Xt) dt + g(t, Xt)M (t) dWt with g(t, Xt) =
g1(t,X1,t)
...
gn(t,Xn,t)
!
with state variable Xt ∈ IRn, an n-dimensional WIENER process Wt ∈ IRn, a matrix func-
tion M (t) ∈ IRn×n of the time variable t, and a diagonal diffusion matrix g(t, Xt) ∈ IRn×n,
g(t, Xt) = diag{g1(t, X1,t), ..., gn(t, Xn,t)}, such that the state-dependent parts of the diffu-
sion are not influencing across components. Then, the 1-dimensional LAMPERTI transform, eq. (5.1), can be applied component-wise.
We refer to [MollerMadsen2010] for details.
5.1.3 Restricting w.l.o.g. to time-homogeneous SDE
Using the “standard trick” of adding time as an extra dimension to the state vector, augmenting its dimension by 1, we can always transform time-inhomogeneous SDE into time-homogeneous, so it is sufficient to study time-homogeneous SDE (and the corresponding autonomous ODE).