Chester Nov08 Terry Lynch

1. Introduction Bounded Noise Extensions Future work On the Growth of the Extreme Fluctuations of SDEs with Markovian Switching Terry Lynch (joint work with Dr. John Appleby) Dublin City University, Ireland. Leverhulme International Network University of Chester, UK. November 7th 2008 Supported by the Irish Research Council for Science, Engineering and Technology

2. Introduction Bounded Noise Extensions Future work Outline 1 Introduction Motivation Regular Variation 2 Bounded Noise Main Results Outline of Proofs 3 Extensions 4 Future work

4. Introduction Bounded Noise Extensions Future work Recap - one year ago Theorem 1 Let X be the unique adapted continuous solution satisfying dX (t) = f (X (t), t) dt + g (X (t), t) dB(t). If there exist ρ > 0 and real numbers K1 and K2 such that ∀ (x, t) ∈ R × [0, ∞) xf (x, t) ≤ ρ, 0 < K2 ≤ g 2 (x, t) ≤ K1 then X satisﬁes |X (t)| lim sup √ ≤ K1 , a.s. t→∞ 2t log log t Question: Why the hell is everyone using that iterated logarithm?!

8. Introduction Bounded Noise Extensions Future work Introduction We consider the rate of growth of the partial maxima of the solution {X (t)}t≥0 of a nonlinear finite-dimensional stochastic differential equation (SDE) with Markovian Switching. We find upper and lower estimates on this rate of growth by finding constants α and β and an increasing function ρ s.t. X (t) 0 < α ≤ lim sup ≤ β, a.s. t→∞ ρ(t) We look at processes which have mean-reverting drift terms and which have either: (a) bounded noise intensity, or (b) unbounded noise intensity.

13. Introduction Bounded Noise Extensions Future work Motivation The ability to model a self-regulating economic system which is subjected to persistent stochastic shocks is facilitated by the use of mean-reverting drift terms. The type of ﬁnite-dimensional system studied allows for the analysis of heavy tailed returns distributions in stochastic volatility market models in which many assets are traded. Equations with Markovian switching are motivated by econometric evidence which suggests that security prices often move from conﬁdent to nervous (or other) regimes.

16. Introduction Bounded Noise Extensions Future work Motivational case study To demonstrate the potential impact of swift changes in investor sentiment, we look at a case study of Elan Corporation.

21. Introduction Bounded Noise Extensions Future work Regular Variation Our analysis is aided by the use of regularly varying functions In its basic form, may be viewed as relations such as f (λx) lim = λζ ∈ (0, ∞) ∀ λ > 0 x→∞ f (x) We say that f is regularly varying at inﬁnity with index ζ, or f ∈ RV∞ (ζ) Regularly varying functions have useful properties such as: x f ∈ RV∞ (ζ) ⇒ F (x) := f (t) dt ∈ RV∞ (ζ + 1) 1 −1 f ∈ RV∞ (ζ) ⇒ f (x) ∈ RV∞ (1/ζ).

26. Introduction Bounded Noise Extensions Future work S.D.E with Markovian switching We study the finite-dimensional SDE with Markovian switching dX (t) = f (X (t), Y (t)) dt + g (X (t), Y (t)) dB(t) (1) where Y is a Markov chain with finite irreducible state space S, f : Rd × S → Rd , and g : Rd × S → Rd × Rr are locally Lipschitz continuous functions, B is an r -dimensional Brownian motion, independent of Y . Under these conditions, there is a unique continuous and adapted process which satisfies (1).

31. Introduction Bounded Noise Extensions Future work Bounded Noise Consider the i th component of our SDE r dXi (t) = fi (X (t), Y (t)) dt + gij (X (t), Y (t)) dBj (t). j=1 We suppose that the noise intensity g is bounded in the sense that there exists K2 > 0 s.t. g (x, y ) F ≤ K2 for all y ∈ S, 2 r d j=1 i=1 xi gij (x,y ) 2 there exists K1 > 0 s.t. inf x ∈Rd x 2 ≥ K1 . y ∈S By Cauchy-Schwarz we get the following upper and lower estimate 2 2 2 K1 ≤ g (x, y ) F ≤ K2 , for all y ∈ S.

36. Introduction Bounded Noise Extensions Future work Upper bound The strength of the restoring force is characterised by a locally Lipschitz continuous function φ : [0, ∞) → (0, ∞) with xφ(x) → ∞ as x → ∞, where x, f (x, y ) lim sup sup ≤ −c2 ∈ (−∞, 0). (2) x →∞ y ∈S x φ( x ) Theorem 1 Let X be the unique adapted continuous solution satisfying (1). Suppose there exists a function φ satisfying (2) and that g satisﬁes the bounded noise conditions. Then X satisﬁes X (t) lim sup 2 ≤ 1, a.s. on Ω , t→∞ Φ−1 K2 (1+ ) log t 2c2 x where Φ(x) = 1 φ(u) du and Ω is an almost sure event.

39. Introduction Bounded Noise Extensions Future work Lower bound Moreover, we ensure that the degree of non-linearity in f is characterised by φ also, by means of the assumption | x, f (x, y ) | lim sup sup ≤ c1 ∈ (0, ∞). (3) x →∞ y ∈S x φ( x ) Theorem 2 Let X be the unique adapted continuous solution satisfying (1). Suppose there exists a function φ satisfying (3) and that g satisﬁes the bounded noise conditions. Then X satisﬁes X (t) lim sup 2 ≥ 1, a.s. on Ω , t→∞ Φ−1 K1 (1− ) log t 2c1 x where Φ(x) = 1 φ(u) du and Ω is an almost sure event.

42. Introduction Bounded Noise Extensions Future work Growth rate of large fluctuations Taking both theorems together, in the special case where φ is a regularly varying function, we get the following: Theorem 3 Let X be the unique adapted continuous solution satisfying (1). Suppose there exists a function φ ∈ RV∞ (ζ) satisfying (2) and (3) and that g satisfies the bounded noise conditions. Then X satisfies 1 1 K12 ζ+1 X (t) K22 ζ+1 ≤ lim sup −1 ≤ a.s., 2c1 t→∞ Φ (log t) 2c2 x where Φ(x) = 1 φ(u) du ∈ RV∞ (ζ + 1).

45. Introduction Bounded Noise Extensions Future work Comments and Example Note that φ small ⇒ Φ small ⇒ Φ−1 large. Thus, as expected, weak mean-reversion results in large ﬂuctuations. Take a simple one-dimensional Ornstein Uhlenbeck process dX (t) = −cX (t)dt + K dB(t) where, in our notation, c1 = c2 = c and K1 = K2 = K , φ(x) = x ∈ RV∞ (1) ⇒ ζ = 1 2 √ Φ(x) = x2 and Φ−1 (x) = 2x Then applying theorem 3 recovers the well known result 1 1 1 K12 |X (t)| K 2 2 K2 ζ+1 2 lim sup √sup X (t) ζ+1 ≤ lim = ≤ , a.s. 2c1 t→∞ t→∞ Φ−1 (log2c 2 log t t) 2c2

54. Introduction Bounded Noise Extensions Future work Outline of proofs Apply time–change and Itˆ transformation to reduce o dimensionality and facilitate the use of one-dimensional theory, Use the bounding conditions on f and g to construct upper and lower comparison processes whose dynamics are not determined by the switching parameter Y , Use stochastic comparison theorem to prove that our transformed process is bounded above and below by the comparison processes for all t ≥ 0 a.s., The large deviations of the comparison processes are determined by means of a classical theorem of Motoo.

59. Introduction Bounded Noise Extensions Future work Extensions We can impose a rate of nonlinear growth of g with x as x → ∞ through an increasing scalar function γ. Then the growth rate of the deviations are of the order x Ψ−1 (log t) where Ψ(x) = 1 φ(u)/γ 2 (u) du. Using norm equivalence in Rd we can study the size of the largest component of the system rather than the norm, to get upper and lower bounds of the form 1 max1≤j≤d |Xj (t)| 0 < √ α ≤ lim sup ≤ β ≤ +∞, a.s. d t→∞ ρ(t) We can extend the state space of the Markov chain to a countable state space.

64. Introduction Bounded Noise Extensions Future work Future work To study ﬁnite-dimensional processes which are always positive, e.g. the Cox Ingersoll Ross (CIR) model. To investigate the growth, explosion and simulation of our developed results. This will involve stochastic numerical analysis and Monte Carlo simulation. This simulation will allow us to determine whether the qualitative features of our dynamical systems are preserved when analysed in discrete time. To investigate whether our results can be extended to SDEs with delay.

Chester Nov08 Terry Lynch

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