Representing Gaussian Processes with Martingales

Transcription

1 Representing Gaussian Processes with Martingales with Application to MLE of Langevin Equation Tommi Sottinen University of Vaasa Based on ongoing joint work with Lauri Viitasaari, University of Saarland Stochastic Visit Workshop of the FDPSS, Tartu, Estonia September 11 12, / 23

2 Outline 1 Gaussian Processes and Martingales 2 Invertible Gaussian Volterra Processes and 3 MLE for Langevin Equation with Gaussian Noise 4 Invertible Gaussian Fredholm Processes and 2 / 23

3 Outline 1 Gaussian Processes and Martingales 2 Invertible Gaussian Volterra Processes and 3 MLE for Langevin Equation with Gaussian Noise 4 Invertible Gaussian Fredholm Processes and 3 / 23

4 Gaussian Processes and Martingales A centered Gaussian process X on [, T ] is completely defined by its covariance function R X (t, s) = E [X t X s ]. This seems simple enough. But for Gaussian martingales M on [, T ] the situation is even simpler. Indeed, R M (t, s) = E [M t M s ] = M t s is a function of one variable only. Moreover, for Gaussian martingales we have the powerful stochastic calculi of Ito and Malliavin at our disposal. 4 / 23

5 Gaussian Processes and Martingales So it seems desirable to connect Gaussian processes with Gaussian martingales to invoke the tools of Ito Malliavin calculus. This connection is called the. As an example we show how to construct an MLE for the drift parameter of the Langevin equation with relatively general Gaussian noise. 5 / 23

6 Outline 1 Gaussian Processes and Martingales 2 Invertible Gaussian Volterra Processes and 3 MLE for Langevin Equation with Gaussian Noise 4 Invertible Gaussian Fredholm Processes and 6 / 23

7 Invertible Gaussian Volterra Processes and A Gaussian process X on [, T ] is Invertible Gaussian Volterra process (IGV) if the exist a Gaussian martingale M with bracket M t = E[M 2 t ] such that X t = M t = k(t, s) dm s, k 1 (t, s) dx s. For convenience we assume that both X and M are continuous. The Wiener integral k 1 (t, s)dx s is defined via a Transfer Principle explained later. 7 / 23

8 Invertible Gaussian Volterra Processes and Remark: A good guess for M is the Prediction Martingale [ ] M t = E X T Ft X. Alas, now M depend on T. This is not too bad, or at least unavoidable for the transfer principle in general. Define the adjoint operators K T and (K T ) 1 by linearly extending the relations K T 1 [,t] = k(t, s), (K T ) 1 1 [,t] = k 1 (t, s). 8 / 23

9 Invertible Gaussian Volterra Processes and Then we have the (TP) for Wiener and Skorohod integrals f (t) dx t = g(t) dm t = K T f (t) dm t, (K T ) 1 g(t) dx t. Remark: If k is smooth enough and vanishes on the diagonal, then Ditto for (K ) 1. KT f (t) = f (s) k(ds, t). t 9 / 23

10 Outline 1 Gaussian Processes and Martingales 2 Invertible Gaussian Volterra Processes and 3 MLE for Langevin Equation with Gaussian Noise 4 Invertible Gaussian Fredholm Processes and 1 / 23

11 MLE for Langevin Equation with Gaussian Noise We show how to apply the for MLE s. Consider the Langevin equation on t [, T ]: dg θ t = θg θ t dt + dg t, where θ is the parameter to be estimated and G is a continuous process with mean zero. The solution is ) Gt θ = e (G θt + e θs dg s. The integral can be defined via integration by parts as e θs dg s = e θt G t G s de θs = e θt G t θ G s e θs ds. We have unique pathwise solution if G is continuous. No additional assumptions are needed. 11 / 23

12 MLE for Langevin Equation with Gaussian Noise Now assume that G in the Langevin equation is IGV with representations G t = M t = k(t, s) dm s, k 1 (t, s) dg s. Here M is a Gaussian martingale with bracket M and k L 2 ([, T ] 2, d M ). The Wiener integral k 1 (t, s)dg s is defined via the transfer principle and the kernel k 1 (t, s) is assumed to be a true function (as opposed to a generalized function). 12 / 23

13 MLE for Langevin Equation with Gaussian Noise Let us then consider MLE for θ. It makes sense to consider the case G = G θ =. So, we have the equation dgt θ = θgt θ dt + dg t with the solution Gt θ = e θt e θs dg s = e θt G t θ G s e θs ds. 13 / 23

14 MLE for Langevin Equation with Gaussian Noise To make MLE non-trivial we must assume the G θ and G are equivalent in law. Indeed, in the Gaussian case the only alternative would be singularity. Girsanov Hitsuda theorem suggest that the bracket d M is consistent with the drift dt. Indeed, we have the following necessary and sufficient condition for the existence of non-trivial MLE: Assume that (A) Q t = d d M t k 1 (t, s)g θ s ds exists in L 2 (Ω [, T ]). 14 / 23

15 MLE for Langevin Equation with Gaussian Noise Let us find the MLE by using a kind of inverse transfer principle. Start with the Langevin equation dg θ s = θg θ s ds + dg s and integrate against the kernel k 1 (t, s) on both sides. This yields k 1 (t, s) dgs θ = θ k 1 (t, s)gs θ ds + k 1 (t, s) dg s Or, by denoting Mt θ = k 1 (t, s)dgs θ, Mt θ = θ k 1 (t, s)gs θ ds + M t. 15 / 23

16 MLE for Langevin Equation with Gaussian Noise Now we use the assumption (A) and obtain dm θ t = θq t d M t + dm t. So, we have written the linear-drift Langevin equation with noise G as a Langevin equation with non-linear drift and noise M. By the Girsanov theorem for martingales we can write the log-likelihood as l T (θ) = log dpθ T dp T = θ Q t dm θ t θ2 2 Q 2 t d M t. 16 / 23

17 MLE for Langevin Equation with Gaussian Noise Maximizing l T (θ) with respect to θ is trivial. Indeed, l T (θ) = Q t dmt θ θ Qt 2 d M t. Setting l T (θ) = yields immediately the MLE ˆθ T = Q t dmt θ. Q2 t d M t 17 / 23

18 MLE for Langevin Equation with Gaussian Noise Let us then consider the Strong Consistency of the MLE. Now Thus, ˆθ T = = Q t dm θ t Q2 t d M t Q t {θq t d M t dm t } Q2 t d M t Q2 t d M t = θ Q2 t d M t Q t dm t ˆθ T θ = Q t dm t. Q2 t d M t 18 / 23

19 MLE for Langevin Equation with Gaussian Noise Note that the downstairs of the difference is nothing but the quadratic variation of the martingale appearing in the upstairs of the difference. Thus, by the strong law of large numbers for martingales, we obtain that ˆθ T is strongly consistent if (B) lim Qt 2 d M t = + T a.s. Assumption (B), and assumption (A), and also the IGV assumption, can be written in terms of the covariance of G, but it is complicated. 19 / 23

20 Outline 1 Gaussian Processes and Martingales 2 Invertible Gaussian Volterra Processes and 3 MLE for Langevin Equation with Gaussian Noise 4 Invertible Gaussian Fredholm Processes and 2 / 23

21 Invertible Gaussian Fredholm Processes and It follows from the Mercer s theorem that any centered Gaussian process X with X = on [, T ] with continuous (say) covariance function admits the Fredholm representation X t = where W is a Brownian motion. k T (t, s) dw s, Obviously, in general there is no hope in having the inversion W t = k 1 T (t, s) dx s. Indeed, X can be, e.g., generated by a single Gaussian variable. 21 / 23

22 Invertible Gaussian Fredholm Processes and There seems to be hope, when one replaces the Brownian motion W with a Gaussian martingale M that generates dynamically the same amount of randomness as X, whatever that means. So, with minimal assumptions one hopes to get X t = M t = k T (t, s) dm s, k 1 T (t, s) dx s. The obvious problem here is that the kernels k T and k 1 T depend on T. This is, I think, a minor inconvenience. Indeed, the adjoint operators K T and (K T ) 1 depend on T even in the Volterra case. The works pretty much the same way in the Fredholm case as in the Volterra case. 22 / 23

23 Thank you for listening! Any questions? 23 / 23