Large Deviations for processes on half-line

We consider a sequence of processes defined on half-line for all non negative t. We give sufficient conditions for Large Deviation Principle (LDP) to hold in the space of continuous functions with a new metric that is more sensitive to behaviour at infinity than the uniform metric. LDP is established for Random Walks, Diffusions, and CEV model of ruin, all defined on the half-line. LDP in this space is"more precise"than that with the usual metric of uniform convergence on compacts.


§1. Introduction
In this work we derive sufficient conditions for a sequence {X n } ∞ n=1 of stochastic processes X n (t); 0 ≤ t < ∞, to satisfy the Large Deviation Principle (LDP) in the space of continuous on [0, ∞) functions, which we denote by C.
In the recent literature [1], [2], [3] the space C is considered with the metric (1.1) [4] (Theorem 2.6) gives sufficient conditions for X n to satisfy LDP in the space (C, ρ (P ) ). As noted in [4], convergence f n → f in metric ρ (P ) is equivalent to convergence in C[0, T ] with uniform metric for any T ≥ 0. A considerable drawback of metric ρ (P ) is that it is " not sensitive" to behaviour of functions as t → ∞.
We consider the space C with metric for a fixed κ ≥ 0.
As we shall see in §2, the LDP in the space (C, ρ) is " more precise" than the LDP in (C, ρ (P ) ).
In this work we treat continuous processes on infinite interval. As we envisage, a treatment of discontinuous processes on infinite interval will need essentially different to ρ metric, (see [5], for the LDP for Compound Poisson processes on infinite interval).
The paper is organised as follows: Sufficient conditions for LDP in the space (C, ρ) are given in §2, Theorem 2.1. We also compare Theorem 2.1 and Theorem 2.6 of [4], and show that Theorem 2.1 is more precise. Next we demonstrate Theorem 2.1 on different kind of processes. In §3 LDP and Moderate Deviation Principle are obtained for Random Walks on half-line. We also give LDP for Winner process in the space (C, ρ) and an example showing that the metric ρ = ρ κ is preferable to ρ (P ) . §4 is devoted to LDP for diffusion processes on half-line. We obtain LDP in space (C, ρ) for solutions of stochastic differential equations with various conditions on the coefficients. In §5 we give LDP for a diffusion model of ruin (a CEV model in finance). §2. Main Result Note that lower semi-continuity (2.1) can be written as: It is obvious that (2.1) and (2.2) are equivalent. We shall keep the same notations as defined above also for an arbitrary space M with metric ρ = ρ M .
For a function f ∈ C, f (T ) denotes its projection on C[0, T ], Let now X n (t); t ∈ [0, ∞), be a sequence of processes in space C 0 . We shall assume in the remainder of §2 that the following conditons hold: Moreover, for any f ∈ C[0, T ] there is g = g f ∈ C 0 , such that g (T ) = f , and for any U ≥ T it holds Remark that condition (2.3) means that one can extend any f ∈ C[0, T ] for t > T such that the rate function will stay the same. It is natural to call the function g = g f most likely extension of f beyond [0, T ].
II. For any r ≥ 0 It is known (see e.g. [6] or [7]), that LDP implies local LDP: for any f ∈ C[0, T ] For U ≥ T , with obvious notations, we have for f ∈ C Thus we established that for U ≥ T , I T 0 (f (T ) ) ≤ I U 0 (f (U ) ), i.e. the rate function I T 0 (f (T ) ) of argument T is non-decreasing in T . Therefore for any f ∈ C there exists limit In what follows it will be shown (see Lemma 2.1), that I(f ) is a good rate function in the space (C, ρ). For a non-empty set B ⊂ C Let we show that the following equality holds: Indeed, for any ε > 0 let f ∈ B be such that Then due to (2.3) there is g ∈ C 0 such that g (T ) = f (consequently g ∈ B (T ) ) with I(g) = I T 0 (f ). Therefore Since ε > 0 is arbitrary, Let now g ∈ B (T ) such that Then g (T ) ∈ B with I T 0 (g (T ) ) ≤ I(g). Therefore and Inequalities (2.6), (2.7) now prove equality (2.5). For a ε > 0 (f ) ε , (B) ε denote ε-neighborhood of f ∈ C, and set B ⊂ C, respectively. For any N < ∞, ε > 0 there is T = T N,ε < ∞ such that Since Since N < ∞ and ε > 0 are arbitrary, the latter implies (2.9). As noted earlier, (2.9) implies lower semi-continuity (2.8).
We show next that the set B r is completely bounded. For any ε > 0 due to condition II there is T = T r < ∞ such that for any f ∈ B r sup t≥T |f (t)| 1 + t 1+κ < ε. (2.10) Denote Since by I the set B T,r is a compact in C[0, T ], it is possible to find finite ε-net: We have for this i due to (2.10) therefore the collection {f 1 , · · · , f M } represents a 3ε-net in the set B r . Thus we have shown that the set B r is completely bounded in C 0 . From lower semi-continuity of I(f ), established earlier, it follows that B r is closed in C 0 . Since a closed completely bounded subset of a Polish space is a compact (see [8], Theorem 3, p. 109), we have shown that B r is a compact in C 0 , thus completing the proof of Lemma 2.1. describes precision of LDP: the smaller the difference the more precise is the theorem. In [4] (Theorem 2.6) sufficient conditions are given for a sequence X n to satisfy LDP in the space (C, ρ (P ) ). Assume, that conditions of both theorems 2.1 and 2.6 in [4] are satisfied, and compare their statements. It follows that the rate functions in both theorems are the same. This is because projections X As noted earlier, (see also [4]), The opposite is not true, as the following example demonstrates. Thus In §3 we give an example of a measurable set B satisfying simultaneously Hence in this example Theorem 2.1 allows to give "precise" logarithmic asymptotic for P(X n ∈ B), whereas Theorem 2.6 in [4] does not. Hence we arrive at conclusion: LDP in the space (C, ρ) is more precise than in the space (C, ρ (P ) ).
For the proof of Theorem 2.1 we need two Lemmas (Lemma 2.2 and Lemma 2.3). (2.14) P r o o f (i). First we prove the bound from below (2.14) (as it is also used in the proof of upper bound (2.13)). If I(f ) = ∞, then (2.14) is trivially satisfied. Let now For a large T the event D(T ) is a certainty (due to I(f ) < ∞). Therefore there exists T 0 < ∞, such that for all T ≥ T 0 it holds that and for this T due to I we have (2.14) now follows from (2.15) by using (2.16), (2.17).
(ii). Now we prove the upper bound (2.13). It is obvious that for any T ∈ (0, ∞) Due to condition I for any δ > 0 . For any T ∈ (0, ∞) and chosen ε and δ, in this way we have the inequality Choose now T < ∞ so large, that simultaneously the following holds: where N < ∞ is arbitrary, and Due to property (2.5) (which was obtained from (2.3) in condition I) we have Take an arbitrary g ∈ (f (T ) ) T,ε+δ . Then either and then which contradicts (2.23). We have proved (see (2.22) and (2.25)), that From the latter we obtain (2.26) Further, due to (2.20) where the last inequality for an open set R(T, ε) follows from the established lower bound (2.14). Therefore and, in view of (2.26), Going back to (2.21), we obtain the inequality in which δ > 0 and N < ∞ are arbitrary. Taking 2δ = ε and sending N to ∞, we obtain the required upper bound L(ε) ≤ −I((f ) 2ε ).

Lemma 2.2 is now proved.
Local LDP for X n in C 0 follows from Lemma 2.2: Then due to condition III there is T < ∞ such that lim n→∞ 1 n ln P(X n ∈ R T (ε)) ≤ −N.
(2.27) For this T due to condition I the process X (T ) n satisfies LDP in the space C[0, T ]. Therefore for a chosen N by a theorem of Puhalskii (see [4] For a given ε > 0 take a finite ε-net f 1 , · · · , f M ∈ C[0, T ] in K: Then lim . We bound P 3 as follows: Since P 1 ≤ P(X n ∈ R T (ε)), Using bounds (2.27) and (2.28) with (2.29), we obtain the required inequality As it is known (see e.g. [6] or [7]), that a good rate function I(f ) satisfies the deviation function of ξ. It is a convex non-negative lower-semiconscious function with a single zero at α = 0, (see e.g. [6] or [9]). Denote S 0 := 0, S k := ξ 1 + · · · + ξ k for k ≥ 1, where {ξ n } is a sequence of i.i.d. copies of ξ. Consider a random piece-wise linear function s n = s n (t) ∈ C, going through the nodes where x = x(n) is a fixed sequence of positive constants such that x ∼ n as n → ∞. Rate function corresponding to the procees s n , define as )dt, f (0) = 0, f is absolutely continuous, +∞, otherwise. In what follows superscript (0) denotes quantities for the centered random variable ξ (0) := ξ − a. The deviation function for ξ (0) is given by Therefore the rate function for s (0) n , is given by where e a = e a (t) := at; t ≥ 0. Clearly, s n = s Hence the LDP for s (0) n with rate function I (0) implies LDP for s n with rate function I. Theorem 3.1 is proved.
We proceed to prove Lemma 3.1. P r o o f of Lemma 3.1 consists in checking conditions I−III of Theorem 2.1. Condition I follows from the LDP for s n in C[0, 1] (e.g. [6] or [9] or [10]).
To bound P 1 (n) use the exponential Chebyshev's (Chernoff's) inequality (see e.g. [6]): where R := x k (T + k n ). Since for all n and T large enough Therefore Similarly we obtain the bound for some δ 2 > 0, C 2 < ∞. Hence condition III holds and proof of Lemma 3.1 is complete.

Moderate Deviation
Principle for Random Walks on half-line. Let random piece-wise linear function s n = s n (·) ∈ C be defined as before by the sums S k of independent random variables distributed as ξ. Let ξ have zero mean Eξ = 0 and assume Cramer's condition Let a sequence x = x(n), used in the construction of s n , satisfy The rate function for s n , is defined as Similarly to the proof of Lemma 3.1, the proof of Theorem 3.2 consists in checking that I − III hold, replacing n by x 2 n . In all other details the proof is the same. Condition I is verified with help of [11] and [12]. Condition II is obvious. Only condition III requires a clarification, done by using the following form of Kolmogorov's inequality ( [13], p. 295, lemma 11.2.1): .
Since conditions I III are easily checked, then LDP follows from Theorem 2.1 with rate function Theorem 3.3. The process w n satisfies LDP in (C, ρ) with κ = 0 with good rate function I.
Next result is given in [4] (Theorem 2.7) Теорема 3.4. The process w n satisfies LDP in (C, ρ (P ) ) with good rate function I. To compare Theorems 3.3 and 3.4 take a set B as Since it is a compliment to an open set (f 0 ) 1 , it is closed in (C, ρ), and therefore Infimum is taken over absolutely continuous functions in B. Therefore by using Cauchy-Bunyakovski inequality This gives that I(g) ≥ 2 for all g ∈ B.
We show next that there is an f ∈ B such that I(f ) = 2. This f is given by Consider now [B] (P ) , the closure of B in metric ρ (P ) . By taking g n (t) = t 2 n it is easy to see that g n ∈ B for all n and lim n→∞ ρ (P ) (g n , f 0 ) = 0. Therefore, f 0 ∈ [B] (P ) . Therefore I([B] (P ) ) = 0, and the upper bound in Theorem 3.4 for the set B is trivial, which does not allow to find logarithmic asymptotic of the required probability. §4. Large Deviations for Diffusion Processes on half-line 4.1. Zero drift. Consider a stochastic process X n (t), t ≥ 0, defined on the stochastic basis (Ω, F, F t , P) that is an Itô integral with respect to Wiener process w(t).
where σ n (ω, t) is F t -adapted and such that the Itô integral is defined. Lemma 4.1. Let for some λ > 0 and all t ≥ 0, n ≥ 0 Then for any N < ∞ and ε > 0 there exists T = T N,ε < ∞ such that We bound P r from above as follows. For any c > 0 we have exp c  We proceed to bound P r,1 . For ease of notation we drop arguments in σ n (ω, t). Using (4.1) we have By Doob's martingale inequality for Taking c = √ nεT r 2λT (r+1) we obtain P r,2 is bounded exactly the same. It follows from (4.3), (4.4), (4.5) that Using inequality (4.6) we have It follows now that for which proves (4.2). Let now X n solve a stochastic differential equation (SDE) on half-line [0, ∞). Теорема 4.1. Let σ(x) be a measurable function of real argument x, such that for some λ ≥ 1 and all Let the Lebesgue measure of discontinuities of σ be zero. Then X n satisfies LDP in space (C, ρ κ ) with κ = 0 and good rate function: P r o o f. Existence of weak solution in (4.7) follows e.g. from Proposition 1 of [14].
Condition I holds by Theorem 1 in [14] and by extending f for t ≥ T by its value at T . Consider the rate function We verify condition II. Using (4.8) and applying Cauchy-Bunyakovskii inequality, we have   Then for any κ > 0, N < ∞ and ε > 0 there exists T = T N,ε,κ < ∞ such that Further argument repeats verbatim Lemma 4.1. Thus (4.11) is proved.
P r o o f. Existence of strong solution in (4.9) is assured by Theorem 1 in [15]. Condition I follows from [16] and that it is possible to extend f for t ≥ T by solution of differential equation: Condition II is verified similarly to that in Theorem 4.1. From (4.12), (4.13) and Lemma 4.2 follows condition III. The Theorem is now proved. §5. Large Deviations for CEV model on half-line Consider X n (t), t ≥ 0, that solves the following SDE (also known as the Constant Elasticity of Variance model, CEV). X n (t) = 1 + t 0 µX n (s)ds + 1 n 1−γ t 0 σ(X n (s)) γ dw(s), where µ and σ are arbitrary constants, γ ∈ [1/2, 1), n > 0. Existence and uniqueness of strong solution is given e.g. in [17] and [18].  Since X n (t) ≡ 0 for t ≥ τ , the above inequality trivially holds for t ∈ [0, ∞).
Solving, we have  Condition III follows from Lemma 5.1. Theorem 5.1 is now proved.