Lecture on Parameter Estimation for Stochastic Differential Equations. Erik Lindström

Transcription

1 Lecture on Parameter Estimation for Stochastic Differential Equations Erik Lindström

2 Recap We are interested in the parameters θ in the Stochastic Integral Equations X(t) = X(0) + t 0 µ θ (s, X(s))ds + t 0 σ θ (s, X(s))dW(s) (1)

3 Recap We are interested in the parameters θ in the Stochastic Integral Equations X(t) = X(0) + t Why? Model validation 0 µ θ (s, X(s))ds + t 0 σ θ (s, X(s))dW(s) (1)

4 Recap We are interested in the parameters θ in the Stochastic Integral Equations X(t) = X(0) + t Why? Model validation Risk management 0 µ θ (s, X(s))ds + t 0 σ θ (s, X(s))dW(s) (1)

5 Recap We are interested in the parameters θ in the Stochastic Integral Equations X(t) = X(0) + t 0 µ θ (s, X(s))ds + t 0 σ θ (s, X(s))dW(s) Why? Model validation Risk management Advanced hedging (Greeks and quadratic hedging (P/Q)) (1)

6 Some asymptotics Consider the arithmetic Brownian motion dx(t) = µdt + σdw(t) (2)

7 Some asymptotics Consider the arithmetic Brownian motion dx(t) = µdt + σdw(t) (2) The drift is estimated by computing the mean, and compensating for the sampling δ = t n+1 t n ˆµ = 1 N 1 X(t n+1 ) X(t n ). (3) δn n=0

8 Some asymptotics Consider the arithmetic Brownian motion dx(t) = µdt + σdw(t) (2) The drift is estimated by computing the mean, and compensating for the sampling δ = t n+1 t n ˆµ = 1 N 1 X(t n+1 ) X(t n ). (3) δn n=0 Expanding this expression reveals that the MLE is given by ˆµ = X(t N) X(t 0 ) t N t 0 = µ + σ W(t N) W(t 0 ) t N t 0. (4)

9 Some asymptotics Consider the arithmetic Brownian motion dx(t) = µdt + σdw(t) (2) The drift is estimated by computing the mean, and compensating for the sampling δ = t n+1 t n ˆµ = 1 N 1 X(t n+1 ) X(t n ). (3) δn n=0 Expanding this expression reveals that the MLE is given by ˆµ = X(t N) X(t 0 ) t N t 0 = µ + σ W(t N) W(t 0 ) t N t 0. (4) The MLE for the diffusion (σ) parameter is given by ˆσ 2 = N 1 1 (X(t n+1 ) X(t n ) ˆµδ) 2 d σ 2 χ2 (N 1) δ(n 1) N 1 n=0

10 A simple method Many data sets are sampled at high frequency, making the bias due to discretization of the SDEs some of the schemes in Chapter 12 acceptable.

11 A simple method Many data sets are sampled at high frequency, making the bias due to discretization of the SDEs some of the schemes in Chapter 12 acceptable. The simplest discretization, the Explicit Euler method, would for the stochastic differential equation dx(t) = µ(t, X(t))dt + σ(t, X(t))dW(t) (6) correspond the Discretized Maximum Likelihood (DML) estimator given by N 1 ˆθ DML = arg max log ϕ (X(t n+1 ), X(t n ) + µ(t n, X(t n )), Σ(t n, X(t θ Θ n=1 (7) where ϕ(x, m, P) is the density for a multivariate Normal distribution with argument x, mean m and covariance P and Σ(t, X(t)) = σ(t, X(t))σ(t, X(t)) T. (8)

12 Consistency The DMLE is generally NOT consistent.

13 Consistency The DMLE is generally NOT consistent. Approximate ML estimators (13.5) are, provided enough computational resources are allocated Simulation based estimators Fokker-Planck based estimators Series expansions.

14 Consistency The DMLE is generally NOT consistent. Approximate ML estimators (13.5) are, provided enough computational resources are allocated Simulation based estimators Fokker-Planck based estimators Series expansions. GMM-type estimators (13.6) are consistent if the moments are correctly specified (which is a non-trivial problem!)

15 Simultion based estimators Discretely observed SDEs are Markov processes

16 Simultion based estimators Discretely observed SDEs are Markov processes Then it follows that p θ (x t x s ) = E θ [p θ (x t x τ ) F(s)], t > τ > s (9) This is the Pedersen algorithm.

17 Simultion based estimators Discretely observed SDEs are Markov processes Then it follows that p θ (x t x s ) = E θ [p θ (x t x τ ) F(s)], t > τ > s (9) This is the Pedersen algorithm. Improved by Durham-Gallant (2002) and Lindström (2012)

18 Simultion based estimators Discretely observed SDEs are Markov processes Then it follows that p θ (x t x s ) = E θ [p θ (x t x τ ) F(s)], t > τ > s (9) This is the Pedersen algorithm. Improved by Durham-Gallant (2002) and Lindström (2012) Works very well for Multivariate models! and is easily (...) extended to Levy driven SDEs.

19 Some key points Naive implementation only provides a point wise estimate - use CRNs or importance sampling Variance reduction helps (antithetic variates, control variates) The near optimal importance sampler is a Bridge process, as it reduces variance AND improves the asymptotics. There is a version that is completely bias free, albeit somewhat restrictive in terms of the class of feasible models.

20 Fokker-Planck Consider the expectation E [h(x(t)) F(0)] = h(x(t))p(x(t) x(0))dx(t) (10) and then E [h(x(t)) F(0)] (11) t

21 Fokker-Planck Consider the expectation E [h(x(t)) F(0)] = h(x(t))p(x(t) x(0))dx(t) (10) and then E [h(x(t)) F(0)] (11) t Two possible ways to compute this, direct and using the Itō formula.

22 Fokker-Planck Consider the expectation E [h(x(t)) F(0)] = h(x(t))p(x(t) x(0))dx(t) (10) and then E [h(x(t)) F(0)] (11) t Two possible ways to compute this, direct and using the Itō formula. Equating these yields where p t (x(t) x(0)) = A p(x(t) x(0)) (12) A p(x(t)) = x(t) (µ( )p(x(t)))+1 2 x 2 (t) 2 ( ) σ 2 ( )p(x(t)). (13)

23 Example of the Fokker-Planck equation From (Lindström, 2007) p(s,x s ;t,x t ) Grid Time Figure: Fokker-Planck equation computed for the CKLS process

24 Comments on the PDE approach Generally better than the Monte Carlo method in low dimensional problems Durham Gallant Poulsen Order2 Pade(1,1) Order4 Pade(2,2) 10 2 MAE time Figure: Comparing Monte Carlo, 2nd order and 4th order numerical approximations of the Fokker-Planck equation

25 Discussion Fokker-Planck is the preferred method is the state space is non-trivial (see Pedersen et. al, 2011)

26 Discussion Fokker-Planck is the preferred method is the state space is non-trivial (see Pedersen et. al, 2011) Successfully used in 1-d and 2-d problems

27 Discussion Fokker-Planck is the preferred method is the state space is non-trivial (see Pedersen et. al, 2011) Successfully used in 1-d and 2-d problems but the ``curse of dimensionality'' will eventually make the method infeasable

28 Series expansion The solution to the Fokker-Planck equation when dx(t) = µdt + σdw(t) (14) is p(x(t) x(0)) = N(x(t); x(0) + µt, σ 2 t).

29 Series expansion The solution to the Fokker-Planck equation when dx(t) = µdt + σdw(t) (14) is p(x(t) x(0)) = N(x(t); x(0) + µt, σ 2 t). Hermite polynomials are the orthogonal polynomial basis when using a Gaussian as weight function.

30 Series expansion The solution to the Fokker-Planck equation when dx(t) = µdt + σdw(t) (14) is p(x(t) x(0)) = N(x(t); x(0) + µt, σ 2 t). Hermite polynomials are the orthogonal polynomial basis when using a Gaussian as weight function. This is used in the 'series expansion approach', see e.g. (Ait-Sahalia, 2002, 2008)

31 Key ideas Transform from X Y Z where Z is approximately standard Gaussian. We assume that dx(t) = µ(x(t))dt + σ(x(t))dw(t) (15)

32 Key ideas Transform from X Y Z where Z is approximately standard Gaussian. We assume that dx(t) = µ(x(t))dt + σ(x(t))dw(t) (15) First step. Y(t) = du σ(u) (16)

33 Key ideas Transform from X Y Z where Z is approximately standard Gaussian. We assume that dx(t) = µ(x(t))dt + σ(x(t))dw(t) (15) First step. Y(t) = It then follows that du σ(u) (16) dy(t) = µ Y (Y(t))dt + dw(t) (17)

34 Key ideas Transform from X Y Z where Z is approximately standard Gaussian. We assume that dx(t) = µ(x(t))dt + σ(x(t))dw(t) (15) First step. Y(t) = It then follows that du σ(u) (16) Second step: Transform dy(t) = µ Y (Y(t))dt + dw(t) (17) Z(t k ) = Y(t k) Y(t k 1 ) t k t k 1. (18)

35 Expansion A Hermite expansion for the density p Z at order J is given by p J Z (z y(0), t k t k 1 ) = ϕ(z) J η j (t k t k 1, y 0 )H j (z) (19) j=0 where H j (z) = e z2 /2 dj dz j e z2 /2. (20)

36 Expansion A Hermite expansion for the density p Z at order J is given by p J Z (z y(0), t k t k 1 ) = ϕ(z) where J η j (t k t k 1, y 0 )H j (z) (19) j=0 H j (z) = e z2 /2 dj dz j e z2 /2. (20) The coefficients are computed by projecting the density onto the basis functions H j (z) (recall Hilbert space theory) η j (t, y 0 ) = 1 H j! j (z)p J Z (z y(0), t k t k 1 )dz (21)

37 Practical concerns The series expansion can be extremely accurate. The standard approach is to compute η j by Taylor expansion up to order (t k t k 1 ) K = ( t) K Some restrictions 'so-called reducible diffusion' when using the method for multivariate diffusions.

38 Other alternatives - GMM/EF What about non-likelihood methods? The model is governed by some p-dimensional parameter.

39 Other alternatives - GMM/EF What about non-likelihood methods? The model is governed by some p-dimensional parameter. Suppose some set of features are important, h l (x), l = 1,..., q p

40 Other alternatives - GMM/EF What about non-likelihood methods? The model is governed by some p-dimensional parameter. Suppose some set of features are important, h l (x), l = 1,..., q p Compute h 1 (x(t)) E θ [h 1 (X t )] f(x(t); θ) =... (22) h q (x t ) E θ [h q (X t )] and form J N (θ) = ( 1 N T ( N 1 f(x(n); θ)) W N n=1 ) N f(x(n); θ) n=1 (23)

41 GMM The Generalized Methods of Moments (GMM) estimators is then given by ˆθ = arg min J N (θ) (24)

42 GMM The Generalized Methods of Moments (GMM) estimators is then given by ˆθ = arg min J N (θ) (24) It can be shown that N (ˆθN θ 0 ) N(0, Σ) (25) where Σ = ( Γ T N Ω 1 N Γ N) 1. (26) and Γ N and Ω N are estimates of [( )] f(x, θ) Γ = E θ T, Ω = Var [f(x, θ)]. (27)

43 Quasi Likelihood What if we treat the distribution as Gaussian? This is called quasi likelihood. Asymptotics can be derived from the GMM asymptotics (Höök & Lindström, 2016) showed that this can be extremely efficient from a computational point of view, essentially O(1) for N observations.

44 References Pedersen, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scandinavian journal of statistics, AïtSahalia, Y. (2002). Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closedform Approximation Approach. Econometrica, 70(1), Lindström, E. (2007). Estimating parameters in diffusion processes using an approximate maximum likelihood approach. Annals of Operations Research, 151(1),

45 References Durham, G. B. & Gallant, A. R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. Journal of Business & Economic Statistics, 20(3), Lindström, E. (2012). A regularized bridge sampler for sparsely sampled diffusions. Statistics and Computing, 22(2), Whitaker, G. A., Golightly, A., Boys, R. J., & Sherlock, C. (2016). Improved bridge constructs for stochastic differential equations. Statistics and Computing, Höök, L. J., & Lindström, E. (2016). Efficient computation of the quasi likelihood function for discretely observed diffusion processes. Computational Statistics & Data Analysis.

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