Parameters Estimation in Stochastic Process Model

Transcription

1 Parameters Estimation in Stochastic Process Model A Quasi-Likelihood Approach Ziliang Li University of Maryland, College Park GEE RIT, Spring 28

2 Outline 1 Model Review The Big Model in Mind: Signal + Noise The Small Model in Business 2 Constructing MLE for Jump-Diffusion Process Examples of MLE s

3 The Big Model in Mind: Signal + Noise In many physical phenomenon, the Response/Feedback can be decomposed as: Response = Signal + Noise One way to represent this model is through the form of a SDE related to a semimartingale X t as: dx t (θ) = da t (θ) + dm t (θ) t > X = x where A t (θ), a predictable finite variation(fv) process, and M t (θ), a Càdlàg local martingale, are defined on a complete probability space (Ω, F, P) where F is a right-continuous filtration.

4 The Big Model in Mind: Signal + Noise OR, written in the form of a stochastic integral as: X t = X + f s (θ)dλ s + M t (θ), t where {λ s } is a real nondecreasing right-continuous predictable process with λ =, {f t } is a predictable process, and {M t (θ)} is an Càdlàg local martingale with M (θ) =. Note: we say a process X t is adapted if X t is F t measurable; we say a process X t is predictable if X t is a left-continuous adapted process.

5 Model of Interest: Jump-Diffusion Process General form of Jump-Diffusion Process The one-dimensional Jump-Diffusion process can be defined as: dx t = d t (θ; X t )dt + γ t (θ; X t )dw t + δ t (θ; X t, Z θ t )dz θ t, t > (1) For some θ Θ R p, let d t (θ; ), γ t (θ; ) and δ t (θ; ) be predictable functionals. The process {W t } is the Wiener process and {Z θ t } is a Compound Poison process Z θ t = N t i=1 Y i where {N t } is a counting process with intensity {α t (θ)}, and {Y i } are i.i.d random variables with distribution F θ independent of {N t }.

6 Examples of Jump-Diffusion Processes 1 A Neurophysiological Model dv t = ( ρv t + λ)dt + dm t (e.g. Kallianpur(1983)), where M t is a discontinuous martingale with a (centered) generalized Poisson distribution and V(t) is the membrane potential. 2 A CIR Interest Rate Model dx t = α(β X t )dt + σ X t dw t where X >, α >, β > and σ >.

7 Examples of Jump-Diffusion Processes 1 A Neurophysiological Model dv t = ( ρv t + λ)dt + dm t (e.g. Kallianpur(1983)), where M t is a discontinuous martingale with a (centered) generalized Poisson distribution and V(t) is the membrane potential. 2 A CIR Interest Rate Model dx t = α(β X t )dt + σ X t dw t where X >, α >, β > and σ >.

8 Examples of Jump-Diffusion Processes 3 A Counting Process Model dx t = θj t dt + dm t with multiplicative intensity Λ t = θj t, J t > a.s. being predictable and M t a square integrable martingale.

9 Formal Steps in Deriving Likelihood and MLE: Find out the Drift, Diffusion and Jump characteristics of X t. Verify there exist a θ Θ s.t. the measure P θ induced by X t will be dominated by P θ. Verify certain integrability conditions under the measure P θ. Derive the Radon-Nikodym Derivative, so the Likelihood function. Derive the MLE.

10 Formal Steps Applied on a Simplified Jump-Diffusion Process We consider a sub-model of SDE (1): dx t = d t (θ; X t )dt + γ t (X t )dw t + δ t (X t, Z t )dz t, t >, X = x deriving from SDE (1) by assuming the follow: d t (θ; X t ) = d t (X t ) + γ t (X t )π t θ where π t is a known nonrandom quantity. The jump intensity λ t and jump size distribution F(y) do NOT depend on θ.

11 Formal Steps Applied on a Simplified Jump-Diffusion Process Then it can be checked that the Log-Likelihood function of θ given the continuously observed data X t for any fixed θ is given by: l t (θ) = θ π s γ t (X t ) 1 dx (c) s (θ ) 1 2 θ2 π 2 sds with = π s γ t (X t ) 1 dx (c) s (θ ) π s γ t (X t ) 1 dx s (θ ) π s γ t (X t ) 1 d s (θ, X s )ds s t π s γ t (X t ) 1 X s where X (c) t (θ) = X t X s t X s d s (θ, X s )ds

12 Formal Steps Applied on a Simplified Jump-Diffusion Process Thus the MLE is given by: Under P θ [ ] 1 { } ^θ t = π 2 sds π s γ s (X s ) 1 dx (c) s (θ ) Or under P θ [ ^θ t = +θ ] 1 { π 2 sds π s γ s (X s ) 1 dx (c) s (θ) } π s γ s (X s ) 1 γ s (X s )π s ds [ ] 1 = π 2 sds π s dw s + θ

13 Formal Steps Applied on a Simplified Jump-Diffusion Process So one sees, under P θ, ( [ ^θ t N θ, ] 1 ) π 2 sds

14 Examples 1 dx t = θdt + dw t + dn t, t, X = x W t : standard Wiener process N t : Poisson process with intensity λ Then the MLE s of (θ, λ) are given by: ^θ t = X(c) t t, ^λ t = N t t 2 dx t = θx t dt + σdw t + dn t, t, X = x Then the MLE s of (θ, λ) are given by: ^θ t = X s dx (c) s X2 sds and ^λ t = N t t

15 3 Model for Security Prices: Examples N t dx t = θx t dt + X t dw t + X t dz t, Z t = N t is a Poisson process with parameter λ ε i are i.i.d. rv s bounded below by 1. The MLE for θ is given by i=1 ( ) ^θ t = t 1 Xt log + 1 x 2 ( ) Xs t 1 X s s t ε i and ^θ t N(θ, 1 t )