Parisian ruin probability for two-dimensional Brownian risk model

Let $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $\rho \in (-1,1).$ Parisian ruin is defined as a classical ruin that happens over an extended period of time, the so-called time-in-red. We derive exact asymptotics for the non-simultaneous Parisian ruin of the company conditioned on the event of non-simultaneous ruin happening. We are interested in finding asymptotics of such problem as $u \to \infty$ and with the length of time-in-red being of order $\frac{1}{u^2},$ where $u$ represents initial capital for the companies. Approximation of this problem is of interest for the analysis of Parisian ruin probability in bivariate Brownian risk model, which is a standard way of defining prolonged ruin models in the financial markets.


Introduction
Consider the following Brownian risk model for two portfolios where the claims W i (t), t ≥ 0 are modeled by two standard dependent Brownian motions, initial capitals u i > 0 and premium rates c i . The following representation of the dependence between the claims has been proposed in [1] and [2] (W 1 (s), W 2 (t)) = (B 1 (s), ρB 1 (t) + 1 − ρ 2 B 2 (t)), s, t ≥ 0, where B 1 , B 2 are two independent standard Brownian motions and ρ ∈ [−1, 1]. The ruin probability of a single portfolio in the time horizon [0, T ], T > 0 is given by (see e.g., [3]) π T (c i , u) := P inf Date: June 28, 2021. 1 for i = 1, 2, u ≥ 0, with Φ the distribution function of an N (0, 1) random variable. Since from self-similarity of Brownian motion we have the following equalities in distribution for c ′ then without loss of generality one can assume T = 1. There are at least two different approaches on how to define the extension of the above to the two-dimensional model. Denote W * i (s) = W i (s) − c i s, i = 1, 2. Define the simultaneous ruin probability as π A,ρ (c 1 , c 2 , u, v) = P{∃ s∈A : W * 1 (s) > u, W * 2 (s) > v} which has been recently studied in [4] for A = [0, 1]. Similarly, define non-simultaneous ruin probability as π A×B,ρ (c 1 , c 2 , u, v) = P{∃ s∈A,t∈B : W * 1 (s) > u, W * 2 (t) > v} which has been studied for the case A = B = [0, 1] in [5]. In this contribution we focus on an extensions of the non-simultaneous results of ruin for two-dimensional risk portfolios. In [6] Loeffen, Czarna and Palmowski studied the so-called Parisian ruin of a single portfolio, which is defined as for some H(u) ≥ 0 and A = [0, T ]. This model defines the concept of the ruin as crossing the barrier over the extended period of time, the so-called time in red. It seems more natural than the classical ruin approach, since it allows for easier practical investigations whether the ruin has occurred. This model has also been studied for various sets A and various processes in many other contributions, e.g. [7], [8], [9]. To analyse the model in two-dimensional framework we use the following definition of the ruin probability P * A×B,H(u) (c 1 , c 2 , u, v) := P ∃ s ′ ∈A,t ′ ∈B ∀ s∈[s ′ ,s ′ +H 1 (u)] ∀ t∈[t ′ ,t ′ +H 2 (u)] W * 1 (s) > u, W * 2 (t) > v , for some H 1 (u), H 2 (u) ≥ 0 and intervals A, B. We refer to [10], where one can find an application of Parisian ruin to actuarial risk theory, where R i is treated as a surplus process of an insurance company with initial capital u i . For more general intervals A, B we have the following comparison between Parisian and classical ruin π A×B,ρ (c 1 , c 2 , u, au) ≥ P * A×B,H(u) (c 1 , c 2 , u, au). Since Parisian ruin probability cannot be determined explicitely for general Gaussian risks, our aim is to investigate the asymptotic behaviour of the Parisian ruin conditioned on the classical ruin occurring, for which the results are known. Hence we calculate for u → ∞ and also find for which H(u) we have that for some C > 0 We prove that the above is true for for some S 1 , S 2 > 0. To simplify notation we denote and similarly (c 1 , c 2 , u, au).
is closely connected to the length of the intervals with comparable variance for the Brownian motion (see [11]). For the choice of H(u) = o( 1 u 2 ) following the same line of proof we have that On the other hand, if we choose H(u) such that u 2 H(u) → ∞, H(u) < 1, then the methods employed in this contribution are not sufficient and the asymptotics are of different order, even in the one-dimensional setting.

Main results
Based on the relation between a and ρ, either both of the coordinates impact the asymptotics, or one of the coordinates is negligible (up to a constant). We begin with cases where one of the coordinates dominates the other one and hence the results can be derived from one-dimensional models. Denote by Ψ the survival function of a standard Normal random variable and by φ t * the probability density function of Our next results are separated into different cases, based on a relative relation between ρ and A a = 1 4a (1 − √ 8a 2 + 1). Function A a has been found by analytical calculations. Heuristically, when ρ < 0 is relatively big compared to a (in terms of absolute value), then it is less likely that the ruin will occur simultaneously and the asymptotics should be significantly different than the ones that have been discovered for simultaneous ruin in [12].
Denote t * = a ρ(2aρ−1) and introduce the following notation for the one-dimensional constants In each particular case, finintess and positivity of P and H has been proven in Lemma 3.5.

Proofs
We recall that Therefore in the proofs we can also focus on investigating the asymptotic behaviour of P S 1 ,S 2 (c 1 , c 2 , u, au), since the asymptotics for π [0,1] 2 ,ρ (c 1 , c 2 , u, au) has been calculated in [5].
3.1. Proof of Theorem 2.1. We divide the proof into two partsa < ρ and a = ρ, since the methods used are quite different. Further define S 1,2 = max(S 1 , S 2 ), which will be commonly used notation in both parts of the proof.
Case (i): a < ρ. First note that On the other hand Since a < ρ, we have that Further from independence of increments of Brownian motion we have for B a Brownian motion indepen- Finally we have that and from [8][Cor 3.5] we have This completes the proof of case (i).
Case (ii): a = ρ. Notice that for ∆ > 0 Next observe that Notice that with [8] we have for some and with [3] we have for some Since from [5] [Thm 2.1] we have Finally, following calculations from case (i) we have that for B a Brownian motion independent of B 1 , B 2 Further we have On the other hand with self-similarity and independence of increments of Brownian motion we have that Hence the claim follows from (3.1) and from [5][Thm 2.1], which gives

3.2.
Proof of Theorem 2.2. We again recall that and as in the proof of the Theorem 2.1 we focus on investigating the asymptotic behaviour of P S 1 ,S 2 (c 1 , c 2 , u, au). Before we begin the proof we need few technical lemmas. First let be the covariance matrix of (W 1 (s), W 2 (t)). In [5] it was noted that the drift has a significant impact on the optimization problem that was used to determine asymptotics for the classical ruin. We denote below Note that for a > ρ and large enough u we have b(s, t) ∼ ( t−aρ min(s,t) st−ρ 2 (min(s,t)) 2 , as−ρ min(s,t) st−ρ 2 (min(s,t)) 2 ) > 0. From [13] we have that for any s, t positive the following logarythmic asymptotics occurs Hence we will use the function q * a (s, t) to reflect the asymptotics of P{W * 1 (s) > u, W * 2 (t) > au}. Below we present the main lemma that solves the optimization problem stated above and was first derived in [5].
In the rest of the paper we denote where t u is defined as in Lemma 3.1. We moreover recall the comparison between the behaviour of variance Remark 3.2. For a ∈ (max(0, ρ), 1] and t u < 1 as in Lemma 3.1 we have Define also η u,k,l (s, t) := (η 1,u,k (s), η 2,u,l (t)) : The following lemma is used to calculate the ruin probability on an interval of size of order O( 1 u 2 ).
) and ∆, S 1 , S 2 > 0 be given constants. The following lemmas are used to show that the constants I 1 in Lemma 3.3 are positive and finite.
Lemma 3.5. i) For any b, c > 0, S ≥ 0 such that 2b > c we have Proof of Lemma 3.5 Ad i) We have that This constant is the same as in [8] This constant is the same as in [7](2.5) for α = 1 and Lemma 3.6. Take any a > max(0, ρ), S 1 , S 2 ≥ 0. Then Proof of Lemma 3.6 Positivity comes of the constant from the fact that the function that we integrate is positive on a set of positive mass. Finitness of the constant follows straightforwardly from (c 1 , c 2 ; u, au).
We have . Using Bonferroni inequality we have that where Using Taylor expansion we have that for k < l From Lemma 3.5 we have that lim ∆→∞ C 2,P C From the proof of [5][Theorem 2.2, case (ii)] we have that for C = C 2,P C i.e. the double sum is negligible compared to the single sum. , where from c 1 − ρc 2 > 0 we obtain that . The proof follows the path of the proof of case (ii). Again from Lemma 3.3 we have Observe that the Taylor expansions mentioned in case (ii) are independent on c 1 , c 2 and hence remain the same here. Therefore with [5][Lem 3.6] we have From Lemma 3.5 we have that lim ∆→∞ C Since D u,∆ is the same as [5](3.16), we have from [5] that With that the proof of case (ii) is complete.
Case (iii): ρ = − 1 2 , a = 1. According to Lemma 3.1 t * = 1. The proof is analogous to case (ii). We use (3.6) and (3.5) with Notice that using calculations from case (ii) and (iii) for a = 1, ρ = − 1 2 we have that which leads us to Similarly, by interchanging the roles of c 1 and c 2 and using previous results we have that as u → ∞ Therefore with Lemma 3.5 we can write that 4,P C D u,∆ is the same as [5](3.15) or (3.16), exactly in the same way as in the proof of [5][Thm. 2.2, case (iv)] and hence With that the proof of case (iii) is complete.
) and for large enough u we have t u < 1. The proof is analogous to case (ii) with Using Taylor expansion we have From Lemma 3.5 we have that lim ∆→∞ C 5,P C D u,∆ is exactly the same as [5](3.18) and hence which means that the double sum is negligible. With that the proof of case (iv) is complete.
Case (v): a = 1, ρ < A a . According to Lemma 3.1, there are two optimal points: where for large enough u we have We can use (3.5) and (3.6) with Using the same calculations as in case (iv) for a = 1 we have that 6,P C 6,P C By interchanging the roles of c 1 and c 2 we can analogously get From the proof of [5] [Thm 2.2, case (vi)] we have With that we obtain that D u,∆ is the same as [5](3.19) and hence With that the proof of case (v) is complete.
Hence it remains to investigate Since the interplay between k u and l u influences the behaviour of the integrals above, we split the proof into three parts : k u = l u , k u < l u , k u > l u .
The finitness of (3.4) and the application of the dominated convergence theorem can be proven identically as in the previous case. This completes the proof.