undefined
Matthew Varble mvarble@math.ucsb.edu Department of Mathematics
University of California, Santa Barbara
Overview Large deviations
Affine processes
Large deviation principle for affine processes
Large deviation rate functions ... require exponential asymptotics f ( ϵ ) ≈ r ϵ f(\epsilon) \approx r \epsilon f ( ϵ ) ≈ rϵ ⇝ f ( ϵ ) = r ϵ + o ( ϵ ) \leadsto\quad f(\epsilon) = r\epsilon + o(\epsilon) ⇝ f ( ϵ ) = rϵ + o ( ϵ ) ⟺ lim ϵ → 0 f ( ϵ ) ϵ = r \Longleftrightarrow\quad \lim_{\epsilon\rightarrow0} \frac{f(\epsilon)}{\epsilon} = r ⟺ ϵ → 0 lim ϵ f ( ϵ ) = r f ( ϵ ) ≈ r ϵ k f(\epsilon) \approx r \epsilon^k f ( ϵ ) ≈ r ϵ k ⇝ f ( ϵ ) = r ϵ k + o ( ϵ k ) \leadsto\quad f(\epsilon) = r\epsilon^k + o(\epsilon^k) ⇝ f ( ϵ ) = r ϵ k + o ( ϵ k ) ⟺ lim ϵ → 0 f ( ϵ ) ϵ k = r \Longleftrightarrow\quad \lim_{\epsilon\rightarrow0} \frac{f(\epsilon)}{\epsilon^k} = r ⟺ ϵ → 0 lim ϵ k f ( ϵ ) = r f ( ϵ ) ≈ exp ( − r / ϵ ) f(\epsilon) \approx \exp(-r/\epsilon) f ( ϵ ) ≈ exp ( − r / ϵ ) ⇝ f ( ϵ ) = exp ( − r / ϵ + o ( 1 / ϵ ) ) \leadsto\quad f(\epsilon) = \exp\big(-r/\epsilon + o(1/\epsilon) \big) ⇝ f ( ϵ ) = exp ( − r / ϵ + o ( 1/ ϵ ) ) ⟺ lim ϵ → 0 ϵ log f ( ϵ ) = − r \Longleftrightarrow\quad \lim_{\epsilon\rightarrow0} \epsilon \log f(\epsilon) = -r ⟺ ϵ → 0 lim ϵ log f ( ϵ ) = − r Large deviation principle Family of measures ( p ϵ ) ϵ > 0 (p^\epsilon)_{\epsilon>0} ( p ϵ ) ϵ > 0 satisfies large deviation principle on space X \bbX X if there exists lower semicontinuous I : X → [ 0 , ∞ ] I: \bbX \rightarrow [0,\infty] I : X → [ 0 , ∞ ] such that:
− inf ξ ∈ Γ ∘ I ( ξ ) ≤ lim inf ϵ → 0 ϵ log p ϵ ( Γ ) ≤ lim sup ϵ → 0 ϵ log p ϵ ( Γ ) ≤ − inf ξ ∈ Γ ‾ I ( ξ ) \begin{aligned}
-\inf_{\xi \in \Gamma^\circ} I(\xi)
&\leq \liminf_{\epsilon\rightarrow0} \epsilon \log p^\epsilon(\Gamma) \\
&\leq \limsup_{\epsilon\rightarrow0} \epsilon \log p^\epsilon(\Gamma)
&\leq -\inf_{\xi \in \overline\Gamma} I(\xi)
\end{aligned} − ξ ∈ Γ ∘ inf I ( ξ ) ≤ ϵ → 0 lim inf ϵ log p ϵ ( Γ ) ≤ ϵ → 0 lim sup ϵ log p ϵ ( Γ ) ≤ − ξ ∈ Γ inf I ( ξ ) lim δ → 0 lim ϵ → 0 ϵ log P ϵ ( Y ϵ ∈ B ( ξ , δ ) ) = − I ( ξ ) \lim_{\delta\rightarrow0}\lim_{\epsilon\rightarrow0} \epsilon\log\Prb^\epsilon\big(Y^\epsilon \in B(\xi,\delta)\big) = -I(\xi) δ → 0 lim ϵ → 0 lim ϵ log P ϵ ( Y ϵ ∈ B ( ξ , δ ) ) = − I ( ξ ) P ϵ ( Y ϵ ∈ B ( ξ , δ ) ) ≈ exp ( − I ( ξ ) / ϵ ) \Prb^\epsilon\big(Y^\epsilon \in B(\xi,\delta)\big) \approx \exp\big(-I(\xi)/\epsilon \big) P ϵ ( Y ϵ ∈ B ( ξ , δ ) ) ≈ exp ( − I ( ξ ) / ϵ ) Measure-change argument Q ϵ , θ ( d ω ) : = Z ϵ , θ ( ω ) ⋅ P ϵ ( d ω ) , Z ϵ , θ : = exp ( 1 ϵ ( ⟨ Y ϵ , θ ⟩ − Λ ϵ ( Y ϵ , θ ) ) ) \Qrb^{\epsilon,\theta}(\rmd\omega) \defeq Z^{\epsilon,\theta}(\omega) \cdot \Prb^\epsilon(\rmd\omega) , \quad
Z^{\epsilon,\theta} \defeq \exp\bigg( \frac1\epsilon \Big( \langle Y^\epsilon, \theta \rangle - \Lambda_\epsilon(Y^\epsilon, \theta) \Big) \bigg) Q ϵ , θ ( d ω ) := Z ϵ , θ ( ω ) ⋅ P ϵ ( d ω ) , Z ϵ , θ := exp ( ϵ 1 ( ⟨ Y ϵ , θ ⟩ − Λ ϵ ( Y ϵ , θ ) ) ) ϵ log P ϵ ( Y ϵ ∈ U ( ξ ) ) \epsilon\log\Prb^\epsilon\big(Y^\epsilon \in U(\xi)\big) ϵ log P ϵ ( Y ϵ ∈ U ( ξ ) ) = ϵ log E Q ϵ , θ ( Z ϵ , θ ⋅ 1 U ( ξ ) ( Y ϵ ) ) = \epsilon\log\Exp_{\Qrb^{\epsilon,\theta}}\Big( Z^{\epsilon,\theta} \cdot 1_{U(\xi)}(Y^\epsilon) \Big) = ϵ log E Q ϵ , θ ( Z ϵ , θ ⋅ 1 U ( ξ ) ( Y ϵ ) )
= ϵ log E Q ϵ , θ ( exp ( 1 ϵ ( ⟨ Y ϵ , θ ⟩ − Λ ϵ ( Y ϵ , θ ) ) ) 1 U ( ξ ) ( Y ϵ ) ) = \epsilon\log\Exp_{\Qrb^{\epsilon,\theta}}\Big( \exp\Big( \frac1\epsilon \big( \langle Y^\epsilon , \theta \rangle - \Lambda_\epsilon(Y^\epsilon, \theta) \big) \Big) 1_{U(\xi)}(Y^\epsilon) \Big) = ϵ log E Q ϵ , θ ( exp ( ϵ 1 ( ⟨ Y ϵ , θ ⟩ − Λ ϵ ( Y ϵ , θ ) ) ) 1 U ( ξ ) ( Y ϵ ) )
≈ ϵ log E Q ϵ , θ ( exp ( 1 ϵ ( ⟨ ξ , θ ⟩ − Λ ϵ ( ξ , θ ) ) ) 1 U ( ξ ) ( Y ϵ ) ) \approx \epsilon\log\Exp_{\Qrb^{\epsilon,\theta}}\Big( \exp\Big( \frac1\epsilon \big( \langle \xi, \theta \rangle - \Lambda_\epsilon(\xi, \theta) \big) \Big) 1_{U(\xi)}(Y^\epsilon) \Big) ≈ ϵ log E Q ϵ , θ ( exp ( ϵ 1 ( ⟨ ξ , θ ⟩ − Λ ϵ ( ξ , θ ) ) ) 1 U ( ξ ) ( Y ϵ ) )
= ⟨ ξ , θ ⟩ − Λ ϵ ( ξ , θ ) + ϵ log Q ϵ , θ ( Y ϵ ∈ U ( ξ ) ) = \langle \xi, \theta \rangle - \Lambda_\epsilon(\xi, \theta) + \epsilon\log\Qrb^{\epsilon,\theta}\Big( Y^\epsilon \in U(\xi) \Big) = ⟨ ξ , θ ⟩ − Λ ϵ ( ξ , θ ) + ϵ log Q ϵ , θ ( Y ϵ ∈ U ( ξ ) )
need Λ ϵ → Λ 0 ↑ \Lambda_\epsilon \rightarrow \Lambda_0 \quad \uparrow Λ ϵ → Λ 0 ↑
need θ \theta θ ↑ \uparrow ↑
what about sample-path deviations? (1966) Schilder gives LDP of scaled Brownian motion using Girsanov's theorem.
(1970-1979) Freidlin and Wentzell give LDP of small-noise diffusions (uniformly Lipschitz) with measure-changes
(1976) Donsker and Varadhan develop contraction principles
(1976) Mogulskii gives LDP of processes with independent increments
(1987) Dawson and Gärtner describe LDP for projective limits
(1994) Puhalskii develops weak-convergence analogies Puhalskii's analogy convergence in distribution
Y t ‾ ϵ → Y t ‾ Y^\epsilon_{\underline t} \rightarrow Y_{\underline t} Y t ϵ → Y t +
tightness
→ Prokhorov \xrightarrow{\text{Prokhorov}} Prokhorov convergence in distribution
Y ϵ → Y Y^\epsilon \rightarrow Y Y ϵ → Y LDP of
( Y t ‾ ϵ ) ϵ > 0 (Y^\epsilon_{\underline t})_{\epsilon>0} ( Y t ϵ ) ϵ > 0 with rate functions
I t ‾ I_{\underline t} I t +
exponential tightness
→ Puhalskii \xrightarrow{\text{Puhalskii}} Puhalskii LDP of
( Y ϵ ) ϵ > 0 (Y^\epsilon)_{\epsilon>0} ( Y ϵ ) ϵ > 0 with rate function
I = sup t ‾ I I = \sup_{\underline t} I I = sup t I Dawson-Gärtner Z ϵ , t ‾ , u ‾ = exp ( 1 ϵ ( ⟨ u ‾ , Y t ‾ ϵ ⟩ − Ψ ϵ ( t ‾ , u ‾ ) ) ) Z^{\epsilon, \underline t, \underline u} = \exp\Big( \frac1\epsilon \big( \langle \underline u, Y^\epsilon_{\underline t} \rangle - \Psi_\epsilon(\underline t, \underline u) \big) \Big) Z ϵ , t , u = exp ( ϵ 1 ( ⟨ u , Y t ϵ ⟩ − Ψ ϵ ( t , u ) ) ) If everything is nice... these produce a large deviation principle, and the rate function is:
I ( ξ ) = sup t ‾ , u ‾ ( ⟨ u ‾ , ξ ( t ‾ ) ⟩ − Ψ 0 ( t ‾ , u ‾ ) ) = sup t ‾ Ψ 0 ∗ ( t ‾ , ξ ( t ‾ ) ) I(\xi) = \sup_{\underline t, \underline u} \Big( \big\langle \underline u, \xi(\underline t) \big\rangle - \Psi_0(\underline t, \underline u) \Big) = \sup_{\underline t} \Psi^*_0\big(\underline t, \xi(\underline t)\big) I ( ξ ) = t , u sup ( ⟨ u , ξ ( t ) ⟩ − Ψ 0 ( t , u ) ) = t sup Ψ 0 ∗ ( t , ξ ( t ) ) Exponential martingale method Z ϵ , h = exp ( 1 ϵ ( ∫ 0 τ h ( t ) d Y t ϵ − ∫ 0 τ Λ ϵ ( h ( t ) , Y t ϵ ) d t ) ) Z^{\epsilon, h} = \exp\bigg( \frac1\epsilon \Big( \int_0^\tau h(t) \rmd Y^\epsilon_t - \int_0^\tau \Lambda_\epsilon\big(h(t), Y^\epsilon_t\big) \rmd t \Big) \bigg) Z ϵ , h = exp ( ϵ 1 ( ∫ 0 τ h ( t ) d Y t ϵ − ∫ 0 τ Λ ϵ ( h ( t ) , Y t ϵ ) d t ) ) If everything is nice... these produce a large deviation principle, and the rate function is:
I ( ξ ) = ∫ 0 τ sup θ ( ⟨ θ , ξ ˙ ( t ) ⟩ − Λ ( θ , ξ ( t ) ) ) d t = ∫ 0 τ Λ ∗ ( ξ ˙ ( t ) , ξ ( t ) ) d t I(\xi)
= \int_0^\tau \sup_\theta \Big( \big\langle \theta , \dot\xi(t) \big\rangle - \Lambda\big(\theta, \xi(t)\big) \Big) \rmd t
= \int_0^\tau \Lambda^*\big(\dot\xi(t), \xi(t)\big) \rmd t I ( ξ ) = ∫ 0 τ θ sup ( ⟨ θ , ξ ˙ ( t ) ⟩ − Λ ( θ , ξ ( t ) ) ) d t = ∫ 0 τ Λ ∗ ( ξ ˙ ( t ) , ξ ( t ) ) d t for absolutely continuous ξ \xi ξ I ( ξ ) = ∞ I(\xi) = \infty I ( ξ ) = ∞
Pros/cons Dawson-Gärtner
Exponential martingales
Definition Fix finite-dimensional real inner-product space V \bbV V , convex state space X ⊆ V \bbX \subseteq \bbV X ⊆ V .
A stochastically continuous time-homogeneous Markov process X X X on X \bbX X affine if we have the following.
affine transform formula E P x exp ⟨ u , X t ⟩ = exp Ψ ( t , u , x ) Ψ ( t , u , x ) = ψ 0 ( t , u ) + ⟨ ψ ( t , u ) , x ⟩ t ≥ 0 , u ∈ i V , x ∈ X \begin{aligned}
\Exp_{\Prb_x}\exp\langle u, X_t \rangle &= \exp\Psi(t, u, x) \\
\Psi(t, u, x) &= \psi_0(t, u) + \big\langle \psi(t, u), x \big\rangle
\end{aligned} \qquad t \geq 0, ~ u \in \rmi \bbV, ~ x \in \bbX E P x exp ⟨ u , X t ⟩ Ψ ( t , u , x ) = exp Ψ ( t , u , x ) = ψ 0 ( t , u ) + ⟨ ψ ( t , u ) , x ⟩ t ≥ 0 , u ∈ i V , x ∈ X Semimartingales Cuchiero (2011).
Any affine process X X X χ \chi χ ( ( ( B χ B^\chi B χ , , , A A A , , , q ^ X \hat q^X q ^ X ) ) )
B t χ = ∫ 0 t β χ ( X s ) d s , \displaystyle B^\chi_t = \int_0^t \beta^\chi(X_s) \rmd s, B t χ = ∫ 0 t β χ ( X s ) d s , A t = ∫ 0 t α ( X s ) d s , \displaystyle A_t = \int_0^t \alpha(X_s) \rmd s, A t = ∫ 0 t α ( X s ) d s , q ^ X ( d s , d v ) = μ ( X s , d v ) d s \hat q^X(\rmd s, \rmd v) = \mu(X_s, \rmd v) \rmd s q ^ X ( d s , d v ) = μ ( X s , d v ) d s Differentiability of B χ B^\chi B χ , , , A A A , , , q ^ X \hat q^X q ^ X X X X L \calL L
L f ( x ) = \calL f(x) = L f ( x ) = D f ( x ) \Der f(x) D f ( x ) β χ ( x ) \beta^\chi(x) β χ ( x ) + 1 2 tr ( D 2 f ( x ) \displaystyle + \frac12 \tr \big( \Hess f(x) + 2 1 tr ( D 2 f ( x ) α ( x ) \alpha(x) α ( x ) ) \big) ) + ∫ V ( f ( x + v ) − f ( x ) − D f ( x ) \displaystyle + \int_\bbV \big( f(x + v) - f(x) - \Der f(x) + ∫ V ( f ( x + v ) − f ( x ) − D f ( x ) χ ( v ) \chi(v) χ ( v ) ) \big) ) μ ( x , d v ) \mu(x, \rmd v) μ ( x , d v ) Real-moments Keller-Ressel and Mayerhofer (2015).
An equivalence for u ∈ V u \in \bbV u ∈ V
Λ ( u , x ) = ⟨ u , \Lambda(u, x) = \langle u, Λ ( u , x ) = ⟨ u , β χ ( x ) \beta^\chi(x) β χ ( x ) ⟩ + 1 2 ⟨ u , \rangle + \displaystyle \frac12\langle u, ⟩ + 2 1 ⟨ u , α ( x ) \alpha(x) α ( x ) u ⟩ + u \rangle + u ⟩ + ∫ V ( e ⟨ u , v ⟩ − 1 − ⟨ u , \displaystyle \int_\bbV \Big( e^{\langle u, v \rangle} - 1 - \langle u, ∫ V ( e ⟨ u , v ⟩ − 1 − ⟨ u , χ ( v ) \chi(v) χ ( v ) ⟩ ) \rangle \Big) ⟩ ) μ ( x , d v ) \mu(x, \rmd v) μ ( x , d v ) generalized Riccati system. ∀ x ∈ X , { Ψ ˙ ( t , u , x ) = Λ ( ψ ( t , u ) , x ) t ∈ [ 0 , τ ] Ψ ( 0 , u , x ) = ⟨ u , x ⟩ \forall x \in \bbX, \quad \left\{\begin{array}{ll}
\dot\Psi(t, u, x) = \Lambda\big(\psi(t, u), x\big) & t \in [0,\tau] \\
\Psi(0, u, x) = \langle u, x \rangle
\end{array}\right. ∀ x ∈ X , { Ψ ˙ ( t , u , x ) = Λ ( ψ ( t , u ) , x ) Ψ ( 0 , u , x ) = ⟨ u , x ⟩ t ∈ [ 0 , τ ] affine transform formula. ∀ t ∈ [ 0 , τ ] , x ∈ X , E P x exp ⟨ u , X t ⟩ = exp Ψ ( t , u , x ) < ∞ \forall t \in [0,\tau],~x \in \bbX, \quad \Exp_{\Prb_x}\exp\langle u, X_t \rangle = \exp\Psi(t, u, x) < \infty ∀ t ∈ [ 0 , τ ] , x ∈ X , E P x exp ⟨ u , X t ⟩ = exp Ψ ( t , u , x ) < ∞
3. Large deviation principle for affine proceses Previous results Kang and Kang (2014).
Large deviation of families ( Y ϵ ) ϵ > 0 (Y^\epsilon)_{\epsilon>0} ( Y ϵ ) ϵ > 0
Y t ϵ Y^\epsilon_t Y t ϵ = = = y + ∫ 0 t y + \displaystyle \int_0^t y + ∫ 0 t β ( Y t ϵ ) \beta(Y^\epsilon_t) β ( Y t ϵ ) d t \rmd t d t + ∫ 0 t \displaystyle + \int_0^t + ∫ 0 t ϵ σ ( Y t ϵ ) \sqrt\epsilon\sigma(Y^\epsilon_t) ϵ σ ( Y t ϵ ) d W t \rmd W_t d W t i.e. β \beta β α = σ σ ∗ \alpha = \sigma\sigma^* α = σ σ ∗ μ ( ⋅ , d v ) = 0 \mu(\cdot, \rmd v) = 0 μ ( ⋅ , d v ) = 0
Established LDP using and Dawson-Gärtner exponential martingale methods.
Unable to derive nice rate function for Dawson-Gärtner.
No indication on how to regularize jumps. Previous results β ϵ = β \beta_\epsilon = \beta β ϵ = β α ϵ = ϵ α \alpha_\epsilon = \epsilon\alpha α ϵ = ϵ α μ ϵ ( x , d v ) = 1 ϵ τ # ϵ μ ( x , d v ) \mu_\epsilon(x, \rmd v) = \frac1\epsilon \tau^\epsilon_\#\mu(x, \rmd v) μ ϵ ( x , d v ) = ϵ 1 τ # ϵ μ ( x , d v )
What we do Intuitively formulate asymptotics
Establish an equivalence between Dawson-Gärtner and exponential martingale methods
Establish a large deviation principle for our asymptotic family Intuitively formulate asymptotics
Establish an equivalence between Dawson-Gärtner and exponential martingale methods
Establish a large deviation principle for our asymptotic family Addressing big jumps Proposition.
If 0 ∈ V 0 \in \bbV 0 ∈ V u u u
∫ ∣ v ∣ > 1 e ⟨ u , v ⟩ μ ( x , d v ) < ∞ , x ∈ X \int_{|v| > 1} e^{\langle u, v \rangle} \mu(x, \rmd v) < \infty, \quad x \in \bbX ∫ ∣ v ∣ > 1 e ⟨ u , v ⟩ μ ( x , d v ) < ∞ , x ∈ X then X X X
L f ( x ) = \calL f(x) = L f ( x ) = D f ( x ) \Der f(x) D f ( x ) β ( x ) \beta(x) β ( x ) + 1 2 tr ( D 2 f ( x ) \displaystyle + \frac12 \tr \big( \Hess f(x) + 2 1 tr ( D 2 f ( x ) α ( x ) \alpha(x) α ( x ) ) \big) ) + ∫ V ( f ( x + v ) − f ( x ) − D f ( x ) v ) \displaystyle + \int_\bbV \big( f(x + v) - f(x) - \Der f(x) v \big) + ∫ V ( f ( x + v ) − f ( x ) − D f ( x ) v ) μ ( x , d v ) \mu(x, \rmd v) μ ( x , d v ) β ( x ) \beta(x) β ( x ) = = = β χ ( x ) \beta^\chi(x) β χ ( x ) + ∫ V ⟨ v − χ ( v ) ⟩ \displaystyle+ \int_\bbV \langle v - \chi(v) \rangle + ∫ V ⟨ v − χ ( v )⟩ μ ( x , d v ) \mu(x, \rmd v) μ ( x , d v ) equivalent to Riccati/AT
u u u 's!
Asymptotic family Process X ϵ X^\epsilon X ϵ drift β ϵ \beta_\epsilon β ϵ , diffusion α ϵ \alpha_\epsilon α ϵ , jump-kernel μ ϵ \mu_\epsilon μ ϵ .
β ϵ ( x ) = 1 ϵ β ( ϵ x ) \displaystyle \beta_\epsilon(x) = \frac1\epsilon \beta(\epsilon x) β ϵ ( x ) = ϵ 1 β ( ϵ x ) ,
α ϵ ( x ) = 1 ϵ α ( ϵ x ) \displaystyle \alpha_\epsilon(x) = \frac1\epsilon \alpha(\epsilon x) α ϵ ( x ) = ϵ 1 α ( ϵ x ) ,
μ ϵ ( x , d v ) = 1 ϵ μ ( ϵ x , d v ) \displaystyle \mu_\epsilon(x, \rmd v) = \frac1\epsilon \mu(\epsilon x, \rmd v) μ ϵ ( x , d v ) = ϵ 1 μ ( ϵ x , d v ) Note.
This is the same as selecting familiar expressions for ϵ X ϵ \epsilon X^\epsilon ϵ X ϵ
β ( x ) \beta(x) β ( x ) ,
ϵ α ( x ) \epsilon \alpha(x) ϵ α ( x ) ,
1 ϵ τ # ϵ μ ( x , d v ) \displaystyle \frac1\epsilon \tau^\epsilon_\#\mu(x, \rmd v) ϵ 1 τ # ϵ μ ( x , d v ) Stochastic differential equations Proposition.
Each ϵ X ϵ \epsilon X^\epsilon ϵ X ϵ ( W , p ) (W, p) ( W , p )
fluid-limit
W t ϵ = W t / ϵ , p ϵ ( [ 0 , t ] × Γ ) = p ( [ 0 , t / ϵ ] × Γ ) W^\epsilon_t = W_{t/\epsilon}, \quad p^\epsilon([0,t] \times \Gamma) = p([0,t/\epsilon] \times \Gamma) W t ϵ = W t / ϵ , p ϵ ([ 0 , t ] × Γ ) = p ([ 0 , t / ϵ ] × Γ ) ϵ X t ϵ = x + ∫ 0 t β ( ϵ X s ϵ ) d s + ∫ 0 t ϵ σ ( ϵ X s ϵ ) d W s ϵ + ∫ [ 0 , t ] × V ϵ c ( ϵ X s − ϵ , v ) p ~ ϵ ( d s , d v ) \begin{aligned}
\epsilon X^\epsilon_t
&=
x
+ \int_0^t \beta(\epsilon X^\epsilon_s) \rmd s
+ \int_0^t \epsilon\sigma(\epsilon X^\epsilon_s) \rmd W^\epsilon_s \\
&\hspace{40mm}+ \int_{[0,t] \times \bbV} \epsilon c\big(\epsilon X^\epsilon_{s-}, v) \tilde p^\epsilon(\rmd s, \rmd v)
\end{aligned} ϵ X t ϵ = x + ∫ 0 t β ( ϵ X s ϵ ) d s + ∫ 0 t ϵ σ ( ϵ X s ϵ ) d W s ϵ + ∫ [ 0 , t ] × V ϵc ( ϵ X s − ϵ , v ) p ~ ϵ ( d s , d v ) small-noise ϵ X t ϵ = x + ∫ 0 t β ( ϵ X s ϵ ) d s + ∫ 0 t ϵ σ ( ϵ X s ϵ ) d W s + ∫ [ 0 , t ] × V ϵ c ( ϵ X s − ϵ , ϵ d ⋅ v ) p ~ ( d s , d v ) \begin{aligned}
\epsilon X^\epsilon_t
&=
x
+ \int_0^t \beta(\epsilon X^\epsilon_s) \rmd s
+ \int_0^t \sqrt{\epsilon}\sigma(\epsilon X^\epsilon_s) \rmd W_s \\
&\hspace{40mm}+ \int_{[0,t] \times \bbV} \epsilon c\big(\epsilon X^\epsilon_{s-}, \sqrt[d]{\epsilon} \cdot v) \tilde p(\rmd s, \rmd v)
\end{aligned} ϵ X t ϵ = x + ∫ 0 t β ( ϵ X s ϵ ) d s + ∫ 0 t ϵ σ ( ϵ X s ϵ ) d W s + ∫ [ 0 , t ] × V ϵc ( ϵ X s − ϵ , d ϵ ⋅ v ) p ~ ( d s , d v ) What we do Intuitively formulate asymptotics
Establish an equivalence between Dawson-Gärtner and exponential martingale methods
Establish a large deviation principle for our asymptotic family Dawson-Gärtner, for us Z ‾ ϵ , t ‾ , u ‾ = exp ( ⟨ u ‾ , X t ‾ ϵ ⟩ − Ψ ϵ ( t ‾ , u ‾ , x ) ) , Ψ ϵ ( t ‾ , u ‾ , x ) = log E P x exp ⟨ u ‾ , X t ‾ ϵ ⟩ \underline Z^{\epsilon, \underline t, \underline u} = \exp\Big( \langle \underline u, X^\epsilon_{\underline t} \rangle - \Psi_\epsilon(\underline t, \underline u, x) \Big), \quad
\Psi_\epsilon(\underline t, \underline u, x) = \log\Exp_{\Prb_x}\exp\langle \underline u, X^\epsilon_{\underline t} \rangle Z ϵ , t , u = exp ( ⟨ u , X t ϵ ⟩ − Ψ ϵ ( t , u , x ) ) , Ψ ϵ ( t , u , x ) = log E P x exp ⟨ u , X t ϵ ⟩ Ψ ϵ ( t ‾ , u ‾ , x ) = 1 ϵ Ψ ( t ‾ , u ‾ , ϵ x ) , \displaystyle\Psi_\epsilon(\underline t, \underline u, x) = \frac1\epsilon\Psi(\underline t, \underline u, \epsilon x), Ψ ϵ ( t , u , x ) = ϵ 1 Ψ ( t , u , ϵ x ) , φ ( t , θ , x ) = log E P x ( exp ⟨ θ , X τ + t − X τ ⟩ ∣ X τ = x ) \varphi(t, \theta, x) = \log\Exp_{\Prb_x}\big( \exp\langle \theta, X_{\tau+t} - X_\tau \rangle | X_\tau = x \big) φ ( t , θ , x ) = log E P x ( exp ⟨ θ , X τ + t − X τ ⟩ ∣ X τ = x ) ⟨ u ‾ , x ‾ ⟩ − Ψ ( t ‾ , u ‾ , x ‾ ) = ∑ k = 1 ∣ t ‾ ∣ ( ⟨ θ k , x k − x k − 1 ⟩ − φ ( Δ t k , θ k , x k − 1 ) ) \big\langle \underline u, \underline x \rangle - \Psi(\underline t, \underline u, \underline x) = \sum_{k=1}^{|\underline t|} \Big( \langle \theta_k, x_k-x_{k-1} \rangle - \varphi(\Delta t_k, \theta_k, x_{k-1}) \Big) ⟨ u , x ⟩ − Ψ ( t , u , x ) = k = 1 ∑ ∣ t ∣ ( ⟨ θ k , x k − x k − 1 ⟩ − φ ( Δ t k , θ k , x k − 1 ) ) I ( ξ ) = sup t ‾ ∑ k = 1 ∣ t ‾ ∣ sup θ k ( ⟨ θ k , ξ ( t k ) − ξ ( t k − 1 ) ⟩ − φ ( Δ t k , θ k , ξ ( t k − 1 ) ) ) I(\xi) = \sup_{\underline t} \sum_{k=1}^{|\underline t|} \sup_{\theta_k} \Big( \big\langle \theta_k, \xi(t_k) - \xi(t_{k-1}) \big\rangle - \varphi\big(\Delta t_k, \theta_k, \xi(t_{k-1}) \big) \Big) I ( ξ ) = t sup k = 1 ∑ ∣ t ∣ θ k sup ( ⟨ θ k , ξ ( t k ) − ξ ( t k − 1 ) ⟩ − φ ( Δ t k , θ k , ξ ( t k − 1 ) ) ) Calibrating twists.
solve θ k : ξ ( t k ) − ξ ( t k − 1 ) = ∇ θ φ ( Δ t k , θ , ξ ( t k − 1 ) ) ∣ θ = θ k \text{solve } \theta_k: \quad \xi(t_k)-\xi(t_{k-1})
= \nabla_\theta \varphi\big(\Delta t_k, \theta, \xi(t_{k-1}) \big) \Big|_{\theta = \theta_k} solve θ k : ξ ( t k ) − ξ ( t k − 1 ) = ∇ θ φ ( Δ t k , θ , ξ ( t k − 1 ) ) ∣ ∣ θ = θ k
Exponential martingales, for us Λ ( u , x ) = ⟨ u , β ( x ) ⟩ + 1 2 ⟨ u , α ( x ) u ⟩ + ∫ V ( e ⟨ u , v ⟩ − 1 − ⟨ u , v ⟩ ) μ ( x , d v ) \Lambda(u, x)
=
\langle u, \beta(x) \rangle
+ \frac12 \langle u, \alpha(x) u \rangle
+ \int_\bbV \Big( e^{\langle u, v \rangle} - 1 - \big\langle u, v \big\rangle \Big) \mu(x, \rmd v) Λ ( u , x ) = ⟨ u , β ( x )⟩ + 2 1 ⟨ u , α ( x ) u ⟩ + ∫ V ( e ⟨ u , v ⟩ − 1 − ⟨ u , v ⟩ ) μ ( x , d v ) Λ ϵ ( u , x ) = 1 ϵ Λ ( u , ϵ x ) , \Lambda^\epsilon(u, x) = \frac1\epsilon\Lambda(u, \epsilon x), Λ ϵ ( u , x ) = ϵ 1 Λ ( u , ϵ x ) , ϵ X t ϵ = x + ∫ 0 t ( β ( ϵ X s ϵ ) + ∇ θ Λ ( θ , ϵ X s ϵ ) ∣ θ = h ( s ) ) d s + M t \epsilon X^\epsilon_t = x + \int_0^t \Big( \beta\big(\epsilon X^\epsilon_s\big) + \nabla_\theta\Lambda\big(\theta, \epsilon X^\epsilon_s) \big|_{\theta = h(s)} \Big) \rmd s + M_t ϵ X t ϵ = x + ∫ 0 t ( β ( ϵ X s ϵ ) + ∇ θ Λ ( θ , ϵ X s ϵ ) ∣ ∣ θ = h ( s ) ) d s + M t I ( ξ ) = ∫ 0 τ sup θ ( ⟨ θ , ξ ˙ ( t ) ⟩ − Λ ( θ , ξ ( t ) ) ) d t = ∫ 0 τ Λ ∗ ( ξ ˙ ( t ) , ξ ( t ) ) d t I(\xi) = \int_0^\tau \sup_\theta \Big( \big\langle \theta, \dot\xi(t) \big\rangle - \Lambda\big( \theta, \xi(t) \big) \Big) \rmd t = \int_0^\tau \Lambda^*\big(\dot\xi(t), \xi(t)\big) \rmd t I ( ξ ) = ∫ 0 τ θ sup ( ⟨ θ , ξ ˙ ( t ) ⟩ − Λ ( θ , ξ ( t ) ) ) d t = ∫ 0 τ Λ ∗ ( ξ ˙ ( t ) , ξ ( t ) ) d t Calibrating control.
solve h ( t ) : ξ ˙ ( t ) = ∇ θ Λ ( θ , ξ ( t ) ) ∣ θ = h ( t ) \text{solve } h(t): \quad \dot\xi(t)
= \nabla_\theta \Lambda\big(\theta, \xi(t) \big) \Big|_{\theta = h(t)} solve h ( t ) : ξ ˙ ( t ) = ∇ θ Λ ( θ , ξ ( t ) ) ∣ ∣ θ = h ( t )
Equivalence Theorem.
For each Dawson-Gärtner measure-change P ϵ , t ‾ , θ ‾ \Prb^{\epsilon,\underline t, \underline\theta} P ϵ , t , θ P ϵ , h \Prb^{\epsilon,h} P ϵ , h
P ϵ , t ‾ , θ ‾ = P ϵ , h \Prb^{\epsilon, \underline t, \underline\theta} = \Prb^{\epsilon, h} P ϵ , t , θ = P ϵ , h ∑ k = 1 ∣ t ‾ ∣ ( ⟨ θ k , X t k − X t k − 1 ⟩ − φ ( Δ t k , θ k , X t k − 1 ) ) = ∫ 0 τ h ( t ) d X t − ∫ 0 τ Λ ( h ( t ) , X t ) d t \sum_{k=1}^{|\underline t|} \Big( \big\langle \theta_k, X_{t_k} - X_{t_{k-1}} \rangle - \varphi\big(\Delta t_k, \theta_k, X_{t_{k-1}} \big) \Big) = \int_0^\tau h(t) \rmd X_t - \int_0^\tau \Lambda\big(h(t), X_t\big) \rmd t k = 1 ∑ ∣ t ∣ ( ⟨ θ k , X t k − X t k − 1 ⟩ − φ ( Δ t k , θ k , X t k − 1 ) ) = ∫ 0 τ h ( t ) d X t − ∫ 0 τ Λ ( h ( t ) , X t ) d t h ( t , t ‾ , θ ‾ ) = ∑ k = 1 ∣ t ‾ ∣ ψ ( t k − t , θ k ) 1 [ t k − 1 , t k ) ( t ) h(t, \underline t, \underline\theta) = \sum_{k=1}^{|\underline t|} \psi(t_k - t, \theta_k) 1_{[t_{k-1},t_k)}(t) h ( t , t , θ ) = k = 1 ∑ ∣ t ∣ ψ ( t k − t , θ k ) 1 [ t k − 1 , t k ) ( t ) What we do Intuitively formulate asymptotics
Establish an equivalence between Dawson-Gärtner and exponential martingale methods
Establish a large deviation principle for our asymptotic family Simple summary
Establish LDP with Dawson-Gärtner
Tighten to Skorokhod topology by showing exponential tightness.
Use equivalence theorem to establish rate function.
h ( t , t ‾ , θ ‾ ) → refine t ‾ , choose θ ‾ h h(t, \underline t, \underline\theta) \xrightarrow{\text{refine } \underline t, \text{ choose } \underline\theta} h h ( t , t , θ ) refine t , choose θ h Main result Theorem.
For each x ∈ X ∘ x \in \bbX^\circ x ∈ X ∘ ( P x ϵ ) ϵ > 0 (\Prb^\epsilon_x)_{\epsilon>0} ( P x ϵ ) ϵ > 0 I x : D ( [ 0 , τ ] , X ) → [ 0 , ∞ ] I_x: \bbD([0,\tau], \bbX) \rightarrow [0,\infty] I x : D ([ 0 , τ ] , X ) → [ 0 , ∞ ]
I x ( ξ ) = { ∫ 0 τ Λ ∗ ( ξ ˙ ( t ) , ξ ( t ) ) d t , ξ ( 0 ) = x , ξ absolutely continuous, ∞ , otherwise I_x(\xi) = \left\{\begin{array}{ll}
\displaystyle \int_0^\tau \Lambda^*\big(\dot\xi(t), \xi(t)\big) \rmd t, & \xi(0) = x, ~ \xi \text{ absolutely continuous,} \\[1em]
\infty, &\text{otherwise}
\end{array}\right. I x ( ξ ) = ⎩ ⎨ ⎧ ∫ 0 τ Λ ∗ ( ξ ˙ ( t ) , ξ ( t ) ) d t , ∞ , ξ ( 0 ) = x , ξ absolutely continuous, otherwise P x ϵ ( ϵ X ϵ ∈ B ( ξ , δ ) ) ≈ exp ( − I x ( ξ ) / ϵ ) \Prb^\epsilon_x\Big( \epsilon X^\epsilon \in B(\xi, \delta) \Big) \approx \exp\Big(-I_x(\xi)/\epsilon\Big) P x ϵ ( ϵ X ϵ ∈ B ( ξ , δ ) ) ≈ exp ( − I x ( ξ ) / ϵ )
4. Representation of rate function Example: Brownian motion Brownian motions ( ϵ W ) ϵ > 0 (\sqrt\epsilon W)_{\epsilon>0} ( ϵ W ) ϵ > 0
Our principle:
β ( x ) = 0 \beta(x) = 0 β ( x ) = 0 α ( x ) = id V \alpha(x) = \operatorname{id}_\bbV α ( x ) = id V μ ( x , d v ) = 0 \mu(x, \rmd v) = 0 μ ( x , d v ) = 0
Λ ∗ ( x ˙ ) = sup u ∈ V ( ⟨ u , x ˙ ⟩ − 1 2 ⟨ u , u ⟩ ) = 1 2 ∣ x ˙ ∣ 2 \Lambda^*(\dot x) = \sup_{u \in \bbV} \Big( \langle u, \dot x \rangle - \frac12 \langle u, u \rangle \Big) = \frac12 |\dot x|^2 Λ ∗ ( x ˙ ) = u ∈ V sup ( ⟨ u , x ˙ ⟩ − 2 1 ⟨ u , u ⟩ ) = 2 1 ∣ x ˙ ∣ 2 ∫ 0 τ 1 2 ∣ ξ ˙ ( t ) ∣ 2 d t \int_0^\tau \frac12 \big|\dot\xi(t)\big|^2 \rmd t ∫ 0 τ 2 1 ∣ ∣ ξ ˙ ( t ) ∣ ∣ 2 d t Example: Poisson process For Poisson process N N N ( ϵ N ⋅ / ϵ ) ϵ > 0 (\epsilon N_{\cdot/\epsilon})_{\epsilon>0} ( ϵ N ⋅ / ϵ ) ϵ > 0
Our principle:
β ( x ) = 1 \beta(x) = 1 β ( x ) = 1 α ( x ) = 0 \alpha(x) = 0 α ( x ) = 0 μ ( x , d v ) = δ 1 \mu(x, \rmd v) = \delta_1 μ ( x , d v ) = δ 1
Λ ∗ ( x ˙ ) = sup u ∈ V ( ⟨ u , x ˙ ⟩ − ( e u − 1 ) ) = x ˙ log x ˙ − x ˙ + 1 \Lambda^*(\dot x) = \sup_{u \in \bbV} \Big( \langle u, \dot x \rangle - (e^u-1) \Big) = \dot x \log \dot x - \dot x + 1 Λ ∗ ( x ˙ ) = u ∈ V sup ( ⟨ u , x ˙ ⟩ − ( e u − 1 ) ) = x ˙ log x ˙ − x ˙ + 1 ∫ 0 τ ( ξ ˙ ( t ) log ( ξ ˙ ( t ) ) − ξ ˙ ( t ) + 1 ) d t \int_0^\tau \Big( \dot\xi(t) \log\big(\dot\xi(t)\big) - \dot\xi(t) + 1 \Big) \rmd t ∫ 0 τ ( ξ ˙ ( t ) log ( ξ ˙ ( t ) ) − ξ ˙ ( t ) + 1 ) d t Example: Diffusion ϵ X t ϵ = x + ∫ 0 t β ( ϵ X s ϵ ) d s + ∫ 0 t ϵ σ ( ϵ X ϵ ) d W s \epsilon X^\epsilon_t = x + \int_0^t \beta(\epsilon X^\epsilon_s) \rmd s + \int_0^t \sqrt\epsilon\sigma(\epsilon X^\epsilon) \rmd W_s ϵ X t ϵ = x + ∫ 0 t β ( ϵ X s ϵ ) d s + ∫ 0 t ϵ σ ( ϵ X ϵ ) d W s Our principle:
β ( x ) \beta(x) β ( x ) α ( x ) = σ σ ∗ ( x ) \alpha(x) = \sigma\sigma^*(x) α ( x ) = σ σ ∗ ( x ) μ ( x , d v ) = 0 \mu(x, \rmd v) = 0 μ ( x , d v ) = 0
Λ ∗ ( x ˙ , x ) = sup u ∈ V ( ⟨ u , x ˙ ⟩ − ⟨ u , β ( x ) ⟩ − 1 2 ⟨ u , α ( x ) u ⟩ ) = 1 2 ⟨ ( x ˙ − β ( x ) ) , α ( x ) † ( x ˙ − β ( x ) ) ⟩ \begin{aligned}
\Lambda^*(\dot x, x)
&= \sup_{u\in\bbV} \Big( \langle u, \dot x \rangle - \langle u, \beta(x) \rangle - \frac12\langle u, \alpha(x)u \rangle \Big) \\
&= \frac12 \Big\langle \big(\dot x - \beta(x) \big), \alpha(x)^\dagger \big(\dot x - \beta(x) \big) \Big\rangle
\end{aligned} Λ ∗ ( x ˙ , x ) = u ∈ V sup ( ⟨ u , x ˙ ⟩ − ⟨ u , β ( x )⟩ − 2 1 ⟨ u , α ( x ) u ⟩ ) = 2 1 ⟨ ( x ˙ − β ( x ) ) , α ( x ) † ( x ˙ − β ( x ) ) ⟩ I ( ξ ) = ∫ 0 τ 1 2 ⟨ ξ ˙ ( t ) − β ( ξ ( t ) ) , α ( ξ ( t ) ) † ( ξ ˙ ( t ) − β ( ξ ( t ) ) ) ⟩ d t I(\xi) = \int_0^\tau \frac12\Big\langle \dot\xi(t) - \beta\big(\xi(t)\big), \alpha\big(\xi(t)\big)^\dagger \Big( \dot\xi(t) - \beta\big(\xi(t)\big) \Big) \Big\rangle \rmd t I ( ξ ) = ∫ 0 τ 2 1 ⟨ ξ ˙ ( t ) − β ( ξ ( t ) ) , α ( ξ ( t ) ) † ( ξ ˙ ( t ) − β ( ξ ( t ) ) ) ⟩ d t Example: Compound-Poisson X t ϵ = ∑ k = 1 N t / ϵ V k , V k ∼ κ X^\epsilon_t = \sum_{k=1}^{N_{t/\epsilon}} V_k, \quad V_k \sim \kappa X t ϵ = k = 1 ∑ N t / ϵ V k , V k ∼ κ Regime for ( ϵ X ϵ , ϵ N ⋅ / ϵ ) ϵ > 0 (\epsilon X^\epsilon, \epsilon N_{\cdot/\epsilon})_{\epsilon>0} ( ϵ X ϵ , ϵ N ⋅ / ϵ ) ϵ > 0
Our principle:
β ( x ^ ) = ( κ ‾ , 1 ) \beta(\hat x) = (\overline\kappa, 1) β ( x ^ ) = ( κ , 1 ) α ( x ^ ) = 0 \alpha(\hat x) = 0 α ( x ^ ) = 0 μ ( x ^ , d v ^ ) = κ ( d v 1 ) δ 1 ( d v 2 ) \mu(\hat x, \rmd\hat v) = \kappa(\rmd v_1) \delta_1(\rmd v_2) μ ( x ^ , d v ^ ) = κ ( d v 1 ) δ 1 ( d v 2 )
Λ ∗ ( x ˙ , x ) = sup u ∈ V ( ⟨ u 1 , x ˙ 1 ⟩ + u 2 x ˙ 2 − exp ( Λ κ ( u 1 ) + u 2 ) + 1 ) = x ˙ 2 log x ˙ 2 − x ˙ 2 + 1 + x ˙ 2 Λ κ ∗ ( x ˙ 1 x ˙ 2 ) \begin{aligned}
\Lambda^*(\dot x, x)
&= \sup_{u\in\bbV} \Big( \langle u_1, \dot x_1 \rangle + u_2\dot x_2 - \exp\big(\Lambda_\kappa(u_1) + u_2\big) + 1 \Big) \\
&= \dot x_2 \log \dot x_2 - \dot x_2 + 1 + \dot x_2 \Lambda_\kappa^*\big(\frac{\dot x_1}{\dot x_2}\big)
\end{aligned} Λ ∗ ( x ˙ , x ) = u ∈ V sup ( ⟨ u 1 , x ˙ 1 ⟩ + u 2 x ˙ 2 − exp ( Λ κ ( u 1 ) + u 2 ) + 1 ) = x ˙ 2 log x ˙ 2 − x ˙ 2 + 1 + x ˙ 2 Λ κ ∗ ( x ˙ 2 x ˙ 1 ) I ( ξ , η ) = ∫ 0 τ ( η ˙ ( t ) log η ˙ ( t ) − η ˙ ( t ) + 1 ) d t + ∫ 0 τ Λ κ ∗ ( ξ ˙ ( t ) η ˙ ( t ) ) d t I(\xi, \eta) = \int_0^\tau \Big( \dot\eta(t) \log \dot\eta(t) - \dot\eta(t) + 1 \Big) \rmd t + \int_0^\tau \Lambda_\kappa^*\Big(\frac{\dot\xi(t)}{\dot\eta(t)} \Big) \rmd t I ( ξ , η ) = ∫ 0 τ ( η ˙ ( t ) log η ˙ ( t ) − η ˙ ( t ) + 1 ) d t + ∫ 0 τ Λ κ ∗ ( η ˙ ( t ) ξ ˙ ( t ) ) d t General closed form Theorem.
Suppose μ ( ⋅ , V ) \mu(\cdot, \bbV) μ ( ⋅ , V )
μ ( x , d v ) = λ ( x ) κ ( x , d v ) \mu(x, \rmd v) = \lambda(x) \kappa(x, \rmd v) μ ( x , d v ) = λ ( x ) κ ( x , d v ) Otherwise, as general as we want:
β ( x ) \beta(x) β ( x ) α ( x ) \alpha(x) α ( x ) μ ( x , d v ) = λ ( x ) κ ( x , d v ) \mu(x, \rmd v) = \lambda(x) \kappa(x, \rmd v) μ ( x , d v ) = λ ( x ) κ ( x , d v )
Can define coupling X ^ = ( X , X c , X d , N X ) \hat X = (X, X^\rmc, X^\rmd, N^X) X ^ = ( X , X c , X d , N X ) X ^ ϵ \hat X^\epsilon X ^ ϵ
...said closed form I ( ξ , ω , γ , η ) = ∫ 0 τ ( η ˙ ( t ) log ( η ˙ ( t ) λ ( ξ ( t ) ) ) − η ˙ ( t ) + λ ( ξ ( t ) ) ) d t + ∫ 0 τ 1 2 ⟨ ω ˙ ( t ) , α ( ξ ( t ) ) ω ˙ ( t ) ⟩ d t + ∫ 0 τ η ˙ ( t ) Λ κ ( ξ ( t ) , ⋅ ) ( ξ ˙ ( t ) − β ( ξ ( t ) ) − γ ˙ ( t ) + λ ( ξ ( t ) ) ν ( ξ ( t ) , ⋅ ) ‾ η ˙ ( t ) ) d t \begin{aligned}
I(\xi, \omega, \gamma, \eta)
&= \int_0^\tau \bigg( \dot\eta(t) \log\Big(\frac{\dot\eta(t)}{\lambda\big(\xi(t)\big)}\Big) - \dot\eta(t) + \lambda\big(\xi(t)\big) \bigg) \rmd t \\
&\quad + \int_0^\tau \frac12 \Big\langle \dot\omega(t), \alpha\big(\xi(t)\big) \dot\omega(t) \Big\rangle \rmd t \\
&\quad + \int_0^\tau \dot\eta(t) \Lambda_{\kappa(\xi(t),\cdot)}\Big(\frac{\dot\xi(t) - \beta\big(\xi(t)\big) - \dot\gamma(t) + \lambda\big(\xi(t)\big)\overline{\nu(\xi(t), \cdot)}}{\dot\eta(t)}\Big) \rmd t
\end{aligned} I ( ξ , ω , γ , η ) = ∫ 0 τ ( η ˙ ( t ) log ( λ ( ξ ( t ) ) η ˙ ( t ) ) − η ˙ ( t ) + λ ( ξ ( t ) ) ) d t + ∫ 0 τ 2 1 ⟨ ω ˙ ( t ) , α ( ξ ( t ) ) ω ˙ ( t ) ⟩ d t + ∫ 0 τ η ˙ ( t ) Λ κ ( ξ ( t ) , ⋅ ) ( η ˙ ( t ) ξ ˙ ( t ) − β ( ξ ( t ) ) − γ ˙ ( t ) + λ ( ξ ( t ) ) ν ( ξ ( t ) , ⋅ ) ) d t ξ ˙ ( t ) = β ( ξ ( t ) ) + ω ˙ ( t ) + γ ˙ ( t ) \dot\xi(t) = \beta\big(\xi(t)\big) + \dot\omega(t) + \dot\gamma(t) ξ ˙ ( t ) = β ( ξ ( t ) ) + ω ˙ ( t ) + γ ˙ ( t )