First exit-time analysis for an approximate Barndorff-Nielsen and Shephard model with stationary self-decomposable variance process
FFirst exit-time analysis for an approximate Barndorff-Nielsenand Shephard model with stationary self-decomposablevariance process
Shantanu Awasthi ∗ , Indranil SenGupta † Department of MathematicsNorth Dakota State UniversityFargo, North Dakota, USA.June 16, 2020
Abstract
In this paper, an approximate version of the Barndorff-Nielsen and Shephard model,driven by a Brownian motion and a L´evy subordinator, is formulated. The first-exittime of the log-return process for this model is analyzed. It is shown that with certainprobability, the first-exit time process of the log-return is decomposable into the sumof the first exit time of the Brownian motion with drift, and the first exit time of aL´evy subordinator with drift. Subsequently, the probability density functions of the firstexit time of some specific L´evy subordinators, connected to stationary, self-decomposablevariance processes, are studied. Analytical expressions of the probability density functionof the first-exit time of three such L´evy subordinators are obtained in terms of variousspecial functions. The results are implemented to empirical S&P 500 dataset.
Key Words:
L´evy process, Subordinator, Self-decomposability, First-exit time, Laplacetransform.
The time required for a stochastic process, starting at a given initial state, to reach a thresholdfor the first time is referred to as the first-exit time or the first hitting time. It is typically veryuseful in determining expected lifetime of mechanical devices. The first-exit time processesare very useful for understanding various financial sectors, especially the insurance industryand investment firms. The first-exit time processes arise naturally in the studies of variousdisciplines. For example, this is used in [41] to model the death probability density functionfor a decaying stochastic process that represents either the end of functionality for a machine, ∗ Email: [email protected] † Email: [email protected] a r X i v : . [ q -f i n . M F ] J un r a zero health state for an organism. The paper [21] provides an expanded first-exit timedensity function that expresses the human death distribution. The first-exit time analysis ofa two-dimensional symmetric stable process is discussed in detail in the paper [13]. This isfurther developed in [23, 45] where the first-exit time process of an inverse Gaussian L´evyprocess is considered. The one-dimensional distribution function of the first-exit time processis obtained. The first-exit time analysis related to a geophysical data is provided in [15]. Thepaper [31], provides generalized notions and analysis methods for the exit-time of randomwalks on graphs.The first-exit time process of the standard Brownian motion is well-studied in the litera-ture (see [2]). The paper [25] studies the first-exit time of Brownian motion for a parabolicdomain. In [14], the Fokker-Planck equation is solved for the Brownian motion with drift, inthe presence of a fixed initial point and elastic boundaries. An explicit expression is obtainedfor the density of the first-exit time. The paper [19] studies the first-exit time problem for thesolutions of some stochastic differential equation for bounded or unbounded intervals. Studiesin [28, 43, 44] discuss the first-exit time process for strictly increasing L´evy processes. In thepioneering paper [22], the authors study the first-exit-time to flat boundaries for a doubleexponential jump diffusion process. The related stochastic process consists of a continuousBrownian motion-driven part, and a jump part with jump sizes satisfy a double exponen-tial distribution. In general, with the help of a fluctuation identity, the paper [1] provides,a generic link between a number of known identities for the first-exit time and overshootabove/below a fixed level of a L´evy process. In [30], a class of increasing L´evy processesperturbed by an independent Brownian motion is considered, and the problem of determin-ing the distribution of the first-exit time is addressed. The first-exit time analysis of theOrnstein-Uhlenbeck (OU) process to a boundary is a long-standing problem with no knownclosed-form solution for the general case. In [27] a general mean-reverting process is con-sidered to investigate the long-and short-time asymptotics using a combination of Hopf-Coleand Laplace transform techniques.Many problems in finance are related to the first-exit time processes. A deeper under-standing of such processes leads to a wiser estimation of fluctuations in the market. In[42], the first-exit time distributions of stock price returns in different time windows areanalyzed. The probability distribution obtained by such analysis is compared with thoseobtained from different models for stock market evolution. The paper [18] shows that forcontinuous time transformations, independent of the Brownian motion, analytical results forthe double-barrier problem can be obtained via the Laplace transform of the time change.The analysis provides a power series representation for the resulting first-exit time probabili-ties. In [26], explicit analytical characterizations are provided for the first-exit time densitiesfor the Cox-Ingersoll-Ross (CIR) and OU diffusions. Such characterizations are obtained interms of the relevant Sturm-Liouville eigenfunction expansions. In [48], a doubly skewedCIR process is studied. A modified spectral expansion is used to obtain the first-exit timedistribution of a doubly skewed CIR process. A detailed study of the first-exit times of dif-fusion processes and their applications to finance is provided in [24]. The studies in [33, 34]discuss the first-exit time analysis related to some financial processes from a data-scienceand sequential hypothesis testing perspective. In [8], the authors provide a solution to the2ptimal stopping problem of a Brownian motion subject to the constraint that the stoppingtime’s distribution is a given measure consisting of finitely many atoms. The distributionconstraints lead to an application in mathematical finance to model-free super-hedging withan outlook on volatility.Some analytically tractable formulas are available for the density of the first-exit timeprocess (see [45]). However, in general, an explicit expression for the density of the first-exittime process for a financial model is mostly unknown. In this paper, we analyze the first-exittime processes in connection to the Barndorff-Nielsen and Shephard (BN-S) model, a popu-larly used stochastic volatility model for financial analysis. In this paper we provide variousanalytical formulas related the distribution of the first-exit time processes in connection toan approximate version of the BN-S model. For this study we use various properties of theLaplace transform and their relations to special functions. In particular, the first-exit timeprocesses for some well-known self-decomposable L´evy subordinators are analyzed.The organization of the paper is as follows. In Section 2, a description of the BN-S modelis provided. An approximation of the BN-S model is formulated based on the stationary,self-decomposable distributions of the variance process. In Section 3, the first-exit timefor a combination of Brownian motion and L´evy subordinator is analyzed. In Section 4,the first-exit time distribution is studied in connection to various self-decomposable L´evysubordinators. It is shown that the analysis is related to the first-exit time analysis of thelog-return for the approximation of the BN-S model presented in Section 2. In Section 5 somenumerical results are provided based on S&P 500 close price dataset for a period of ten years.Finally, a brief conclusion is provided in Section 6. Financial time series of different assets share many common features which are successfullycaptured by the stochastic model introduced in various works of Ole Barndorff-Nielsen andNeil Shephard. The model is known in modern literature as the Barndorff-Nielsen and Shep-hard (BN-S) model (see [4, 6, 7]). This model is revised and refined in various recent works inliterature such as [36, 37]. This model is successfully implemented in the commodity marketsas well (see [39, 46]). Recently, this model is improved using various machine-learning drivenalgorithms (see [38, 40]).For the BN-S model, a frictionless financial market is considered where a risk-less assetwith constant interest rate r , and a stock, are traded up to a fixed horizon date T . It isassumed that the price process of the stock S = ( S t ) t ≥ is defined on some filtered probabilityspace (Ω , F , ( F t ) ≤ t ≤ T , P ) and is given by: S t = S exp( X t ) , (2.1)where the log-return X t is given by dX t = ( µ + β σ t ) dt + σ t dW t + ρ dZ λt , (2.2)3ith the variance process dσ t = − λσ t dt + dZ λt , σ > , (2.3)where the parameters µ , β ∈ R with λ > ρ <
0. In (2.2) and (2.3), W t and Z t are Brownian motion and L´evy subordinator, respectively. The L´evy subordinator Z isreferred to as the background driving L´evy process (BDLP). Also W and Z are assumed tobe independent and ( F t ) is assumed to be the usual augmentation of the filtration generatedby the pair ( W, Z ). Without loss of generality, we assume W = Z = 0.We assume Z satisfies the assumptions as described in [29]. It follows that the cumulanttransform κ ( θ ) = log E [ e θZ ], where it exists, takes the form κ ( θ ) = (cid:82) R + ( e θx − w ( x ) dx ,where w ( x ) is the L´evy density for Z . It is shown in [29] (Theorem 3.2) that there existsan equivalent martingale measure (EMM) Q , under which equations (2.2) and (2.3) can bewritten as: dX t = b t dt + σ t dW t + ρ dZ λt , with b t = ( r − λκ ( ρ ) − σ t ) , (2.4) dσ t = − λσ t dt + dZ λt , σ > , (2.5)where W t and Z t are Brownian motion and L´evy process respectively with respect to Q . Forthe rest of this paper we assume that the risk-neutral dynamics (with respect to Q ) of thestock price is given by (2.1), (2.4) and (2.5). It is trivial to show that the solution of (2.5) isgiven by σ t = e − λt σ + (cid:90) t e − λ ( t − s ) dZ λs . (2.6)From (2.6), the positivity of the process σ t is obvious. In fact, σ t is bounded below by thedeterministic function e − λt σ . In addition, the instantaneous variance of log-return X t isgiven by ( σ t + ρ λ Var[ Z ]) dt . Consequently, the continuous realized variance in the interval[0 , T ], denoted as σ R , is given by σ R = T (cid:82) T σ t dt + ρ λ Var[ Z ]. Therefore, by (2.6) we obtain σ R = 1 T (cid:18) λ − (1 − e − λT ) σ + λ − (cid:90) T (cid:16) − e − λ ( T − s ) (cid:17) dZ λs (cid:19) + ρ λ Var[ Z ] . (2.7)We state some results for the analysis of the variance process σ t , when the process isstationary and self-decomposable. The results are motivated by [16, 17, 20]. The pricingformulas for various derivatives are dependent on the variance process. Definition 2.1.
The distribution of a random variable X is said to be self-decomposable iffor any constant c , < c < , there exists an independent random variable X ( c ) , such that X d = cX + X ( c ) , where d = stands for the equality in the distribution. For self-decomposable laws the associated densities are unimodal (see [12, 35]). It isproved in [5, 47] that, if X is self-decomposable then there exists a stationary stochasticprocess { σ ( t ) } t ≥ , and a L´evy process { Z t } t ≥ , independent of σ , such that σ t d = X for all t ≥ σ t = exp( − λt ) σ + (cid:90) t exp ( − λ ( t − s )) dZ λs , for all λ > . { σ t } t ≥ , is a stationary stochastic process and { Z t } t ≥ is a L´evy process inde-pendent of σ , such that { σ t } and { Z t } satisfy dσ t = − λσ t dt + dZ λt , σ > , for all λ >
0, then σ t is self-decomposable.It is clear from [35] (Theorem 17.5(ii)) that for any self-decomposable law D there existsa L´evy process Z such that the process of OU type driven by Z has invariant distributiongiven by D . The following theorem (see [16, 17, 37]) gives the relation between the L´evydensities of such process generated by σ t and Z in (2.5). Theorem 2.2.
A random variable X has law in L if and only if X has a representation ofthe form X = (cid:82) ∞ e − t dZ t , where Z t is a L´evy process. In this case the L´evy measure U and W of X and Z are related by U ( dx ) = (cid:82) ∞ W ( e t dx ) dt . In addition, if u ( x ) , the L´evy densityof U is differentiable, then the L´evy measure W has a density w , and u and w are related by w ( x ) = − u ( x ) − xu (cid:48) ( x ) . (2.8)There are many known self-decomposable distributions, such as inverse Gaussian (IG),Gamma, positive tempered stable (PTS), etc.Consequently, if the stationary distribution of σ t is given by IG( δ , γ ) law, with the L´evydensity u ( x ) = √ π δ x − / exp( − γ x/ x >
0, then by (2.8), the L´evy density of Z is givenby w ( x ) = δ √ π x − (1 + γ x ) e − γ x , x >
0. Alternatively, if the stationary distribution of σ t is given by gamma law Γ( ν, α ), where the L´evy density of Γ( ν, α ) is given by u ( x ) = νx − e − αx , x >
0, then by (2.8) we obtain w ( x ) = ναe − αx , x > κ, δ, γ ), where β >
0, 0 < γ <
1, and k ≥
0. ForPTS( κ, δ, γ ) process the L´evy density is simple and is given by (see [16, 17]) u ( x ) = βk − γ γ Γ( γ )Γ(1 − γ ) x − γ − exp (cid:18) − k x (cid:19) , x > . If the stationary distribution of σ t is given by PTS( κ, δ, γ ) law, then by (2.8) we obtain thatthe L´evy density of Z is given by w ( x ) = βk − γ γx − γ − e − k x Γ( γ )Γ(1 − γ ) (cid:18) γ + k x (cid:19) , x > . In the above discussions we find that the distribution of Z is analytically tractable whenthe stationary distribution of σ t in (2.5) is given by a stationary, self-decomposable distribu-tion. We denote (as σ t is stationary), σ = E Q ( σ ) , (2.9)and µ = r − λκ ( ρ ) − σ . (2.10)5e approximate for (2.4) by dX t = µ dt + σ dW t + ρ dZ λt . (2.11)We refer to (2.1), (2.11), and (2.5), as an approximation of the BN-S model (2.1), (2.4), and(2.5). For most of the empirical financial data µ ≤ X t = µt + σW t + ρZ t , with µ ∈ R , σ >
0, and ρ < t >
0. For financialapplications µ ≤
0. For the subsequent sections we develop a general procedure to computethe first-exit time of the stochastic process X t . In this section, we develop a couple of results related to the first-exit time analysis of log-return processes of the form (2.11). At first, we develop the result related to the first-exittime of a simpler process W t + Y t , where Y is a L´evy subordinator, with W = Y = 0. If X and X are independent random variables, we denote X | = X . Theorem 3.1.
For a Brownian motion W t and a L´evy subordinator Y t , and a, b > , inf { τ > W τ + Y τ ≥ a + b } = inf { t > W t ≥ a } + inf { α > Y α ≥ b } , (3.1) with probability P = (cid:90) ∞ (cid:90) ∞ (cid:90) ∞−∞ P ( (cid:15) ; t, α ) P ( (cid:15) ; t, α ) d(cid:15) dt dα, (3.2) where P ( (cid:15) ; t, α ) = (cid:90) ∞−∞ e − τ α √ πα (cid:90) ∞ max( a,a − (cid:15) − τ ) e − s t √ πt ds dτ, (3.3) and P ( (cid:15) ; t, α ) = (cid:90) ∞ f Y t ( β ) (cid:32)(cid:90) ∞ max(max( b,b + (cid:15) − β ) , f Y α ( s ) ds (cid:33) dβ, (3.4) where the probability density function of Y t is given by f Y t ( · ) .Proof. We consider the set A = { t > W t ≥ a } , where W t is a standard Brownian motion.In addition, we consider another set B = { α > Y α ≥ b } , where Y t is a L´evy subordinator.Clearly, the first-exit time of a combination of W t and Y t , in the sense that its value is morethan a + b , is given byinf( A + B ) = inf { τ > W τ + Y τ ≥ a + b } = inf { t + α > W t + α + Y t + α ≥ a + b, t > , α > } . For a fixed (cid:15) ∈ R , we define P ( (cid:15) ; t, α ) = P ( W t + α ≥ a − (cid:15), W t ≥ a ) , (3.5)6 ( (cid:15) ; t, α ) = P ( Y t + α ≥ b + (cid:15), Y α ≥ b ) . (3.6)We proceed to compute P ( (cid:15) ; t, α ) and P ( (cid:15) ; t, α ). We observe, P ( (cid:15) ; t, α ) = P ( W t + α ≥ a − (cid:15), W t ≥ a )= P ( W t + α − W t ≥ a − (cid:15) − W t , W t ≥ a )= P ( W t ≥ a − (cid:15) − ( W t + α − W t ) , W t ≥ a )= P ( W t ≥ max( a, a − (cid:15) − χ )) , χ ∼ N (0 , α ) , χ | = W t = (cid:90) ∞−∞ e − τ α √ πα (cid:90) ∞ max( a,a − (cid:15) − τ ) e − s t √ πt ds dτ. On the other hand, P ( (cid:15) ; t, α ) = P ( Y t + α ≥ b + (cid:15), Y α ≥ b )= P ( Y t + α − Y α ≥ b + (cid:15) − Y α , Y α ≥ b )= P ( Y α ≥ b + (cid:15) − ( Y t + α − Y α ) , Y α ≥ b )= P ( Y α ≥ max( b, b + (cid:15) − η )) , η ∼ the distribution of Y t , η | = Y α . As the probability density function of Y t is given by f Y t ( · ), therefore we obtain P ( (cid:15) ; t, α ) = (cid:90) ∞ f Y t ( β ) (cid:32)(cid:90) ∞ max(max( b,b + (cid:15) − β ) , f Y α ( s ) ds (cid:33) dβ. Clearly, { t > W t ≥ a } + { α > Y α ≥ b } = { t + α > W t + α + Y t + α ≥ a + b, t > , α > } ,with probability P , where P is given by (3.2), and P ( (cid:15) ; t, α ) and P ( (cid:15) ; t, α ) are obtained by(3.3) and (3.4), respectively. Consequently, inf( A + B ) = inf( A ) + inf( B ). Hence the theoremis proved.Next, we generalize the result in Theorem 3.1 for the log-return stochastic process (2.11) inthe approximation of the BN-S model . In the BN-S model ρ < leverage effect of the market. Typically in a derivative market, a significant fluctu-ation always corresponds to a “big-downward-movement” of the asset prices. Consequently,for the next theorem we focus on the first-exit time corresponding to a “downward-movement”of the log-return process (2.11). For the following theorem we assume W = Z = 0. Theorem 3.2.
For a Brownian motion W t and a L´evy subordinator Z t , if µ ∈ R , σ > , ρ < , and a, b > , then inf { τ > µτ + σW τ + ρZ τ ≤ − a − b } = inf { t > µt + σW t ≤ − a } + inf { t > µt + ρZ t ≤ − b } , (3.7) with probability P = (cid:90) ∞ (cid:90) ∞ (cid:18)(cid:90) ∞−∞ P ( (cid:15) ; t, α ) P ( (cid:15) ; t, α ) d(cid:15) (cid:19) dt dα, (3.8)7 here P ( (cid:15) ; t, α ) = (cid:90) ∞−∞ e − τ t √ πt (cid:90) min (cid:16) ( − a − µt σ , ( − a − (cid:15) ) σ − τ − µ ( t t σ (cid:17) −∞ e − s t √ πt ds dτ, (3.9) and P ( (cid:15) ; t , t ) = (cid:90) ∞ f Z t ( β ) (cid:32)(cid:90) ∞ max (cid:16) max (cid:16) ( b − µt ρ , ( b + (cid:15) ) ρ − β − µ ( t t ρ (cid:17) , (cid:17) f Z t ( s ) ds (cid:33) dβ, (3.10) where the probability density function of Z t is given by f Z t ( · ) .Proof. For fixed (cid:15) ∈ R , we define and compute the following joint probabilities. At first, wecompute, for a > P ( (cid:15) ; t , t ) = P ( W t + t + µ ( t + t )2 σ ≤ − aσ − (cid:15)σ , W t + µt σ ≤ − aσ )= P ( W t + t − W t + µt σ ≤ − aσ − (cid:15)σ − W t − µt σ , W t + µt σ ≤ − aσ )= P ( W t + µt σ ≤ − aσ − (cid:15)σ − ( W t + t − W t ) − µt σ , W t + µt σ ≤ − aσ )= P (cid:18) W t ≤ ( − a − (cid:15) ) σ − ( W t + t − W t ) − µt σ − µt σ , W t ≤ − aσ − µt σ (cid:19) = P (cid:18) W t ≤ min (cid:18) ( − a − µt ) σ , ( − a − (cid:15) ) σ − χ − µ ( t + t )2 σ (cid:19)(cid:19) , χ ∼ N (0 , t ) , χ | = W t ! = (cid:90) ∞−∞ e − τ t √ πt (cid:90) min (cid:16) ( − a − µt σ , ( − a − (cid:15) ) σ − τ − µ ( t t σ (cid:17) −∞ e − s t √ πt ds dτ. With ρ <
0, we compute for b > P ( (cid:15) ; t , t ) = P ( Z t + t + µ ( t + t )2 ρ ≥ − bρ + (cid:15)ρ , Z t + µt ρ ≥ − bρ )= P ( Z t + t + µ ( t + t )2 ρ − Z t ≥ − bρ + (cid:15)ρ − Z t , Z t + µt ρ ≥ − bρ )= P (cid:18) Z t ≥ ( − b + (cid:15) ) ρ − (( Z t + t − Z t ) + µ ( t + t )2 ρ ) , Z t ≥ − bρ − µt ρ (cid:19) = P (cid:18) Z t ≥ max (cid:18) ( − b − µt ) ρ , ( − b + (cid:15) ) ρ − η − µ ( t + t )2 ρ (cid:19)(cid:19) , where η ∼ the distribution of Z t . Since η | = Z t , therefore we obtain (3.10). For a, b >
0, wedefine a set A = { τ > µτ + σW τ + ρZ τ ≤ − a − b } = { t + t > µ ( t + t ) + σW t + t + ρZ t + t ≤ − a − b, t > , t > } . A = { t + t > µ ( t + t ) + σW t + t + ρZ t + t ≤ − a − b } = { t > µt + σW t ≤ − a } + { t > µt + ρZ t ≤ − b } , with probability P given by (3.8). Consequently,inf A = inf { t > µt + σW t ≤ − a } + inf { t > µt + ρZ t ≤ − b } . This proves (3.7).The purpose of Theorem 3.1 and Theorem 3.2 is to decompose the first-exit time processof a linear combination of Brownian motion and L´evy subordinator into the individual first-exit time processes of the Brownian motion and a L´evy subordinator. However, as observedin both of the theorems, such decomposition holds only with certain probability.
Remark 3.3.
It is well known (see [2, 23]) that for the process G t = inf { s > W s + γs ≥ δ t } , with γ, δ > , known as the inverse Gaussian (IG) process, G t follows IG ( δ t, γ ) distribution. As the process W s + γs is continuous, we also have G t = inf { s > W s + γs = δ t } . The distribution IG ( δ , γ ) is concentrated on R + and has probability density: p ( x ) = 1 √ π δ e δ γ x − / exp (cid:18) − δ x − + γ x (cid:19) , γ, δ > . Consequently, for Theorem 3.2 with µ < and a > , the first term on the right hand side of (3.7) has the distribution inf { t > µt + σW t ≤ − a } d = inf { t > − µt + σW t ≥ a } ∼ IG (cid:0) aσ , − µσ (cid:1) .The case is not the same if the Brownian motion does not have any drift term. In thatcase, it is known (see [3]) that inf { s > σW s ≥ a } , with a > , satisfies a L´evy distributionwith the probability density function aσ √ πx exp (cid:18) − a σ x (cid:19) , x > . Consequently, for Theorem 3.1, the first term on the right hand side of (3.1) , i.e., inf { t > W t ≥ a } = inf { t > W t = a } , with a > , has the probability density function a √ πx exp (cid:16) − a x (cid:17) , x > . Similar result holds for the first term on the right hand side of (3.7) in Theorem 3.2 with µ = 0 .Note that, for the case when µ = 0 and a > , inf { s > σW s ≤ − a } = inf { s > σW s = − a } d = inf { s > σW s = a } = inf { s > σW s ≥ a } . We note that for Theorem 3.1, if a, b ≤
0, then (3.1) is trivially satisfied. Similarly, forTheorem 3.2, if a, b ≤
0, then (3.7) is trivially satisfied. As W = Z = 0, therefore all therelated first-exit times are zero in those cases.9 First-exit time distribution for some self-decomposable pro-cesses
Consider the log-return dynamics X t given by (2.11), in the approximation of the BN-Smodel (2.1), (2.11), and (2.5). In Theorem 3.2, it is shown that with certain probability,the first-exit time process inf { t > X t ≤ − a − b } , is decomposable into the sum ofthe first exit time of two processes- (1) the Brownian motion with drift, and (2) a L´evysubordinator with drift. We denote three stochastic processes: A a + b = inf { t > X t ≤− a − b } = inf { t > µt + σW t + ρZ t ≤ − a − b } , B a = inf { t > µt + σW t ≤ − a } , and C b = inf { t > µt + ρZ t ≤ − b } , with ρ <
0, and a, b >
0. In these expressions σ and µ are given by (2.9) and (2.10), respectively. Thus, σ >
0. Also, in general, for financialapplications µ ≤
0. With these notations, from Theorem 3.2 we obtain that A a + b = B a + C b .The probability density function of the process B , with µ ≤
0, is discussed in Remark 3.3.In this section we discuss the probability density function of the process C for some specialcases. Accordingly, with probability P given by (3.8), the probability density function of theprocess A is equal to the convolution of the probability density functions of the processes B and C .The goal of this section is to analyze the first-exit time distribution for the L´evy subordina-tor in the decompositions provided in Theorem 3.1 and Theorem 3.2. For simplicity we assume µ = 0. We consider the distribution of the corresponding process C b = inf { s > Z s ≥ − bρ } ,for three self-decomposable distributions. As b > ρ <
0, in general, C can be writtenas the stochastic process T t = inf { s > Z s ≥ t } , t > T t in relation to Gamma, IG, and PTS subordinators, respectively. At first, we describe some special functions necessary for the development of the rest of thispaper. • MacRobert E -function is denoted as E ( m ; a : n ; b j : x ) = E ( a , · · · , a m : b , · · · , b n : x ) . For m ≥ n + 1, with | x | <
1, MacRobert E -function is defined as m (cid:88) i =1 (cid:81) mj =1 ˜ ∗ Γ( a j − a i )Γ( a i ) x a i (cid:81) nk =1 Γ( b k − a i ) n +1 ˜ F m − (cid:20) a i , a i − b + 1 , · · · , a i − b n + 1; a i − a + 1 , · · · , ˜ ∗ , · · · , a i − a m + 1;( − m + n x (cid:21) . For m ≤ n + 1, with | x | >
1, MacRobert E -function is defined as (cid:81) mi =1 Γ( a i ) (cid:81) nj =1 Γ( b j ) m ˜ F n (cid:20) a , · · · , a m ; b , · · · , b n ; − x (cid:21) . n = 0, the notation E ( · :: · ) is used. The ˜ ∗ denotes that the term containing a j − a i corresponding to j = i is omitted. Here m ˜ F n [ · ] is generalized hypergeometric functions,defined as m ˜ F n (cid:20) a , · · · , a m ; b , · · · , b n ; x (cid:21) = ∞ (cid:88) n =0 ( a ) · · · ( a m ) n x n ( b ) · · · ( b n ) n n ! , where ( · ) n is the Pochhammer symbol. • Gauss hypergeometric function F ( a, b, c ; x ) is defined as F ( a, b, c ; x ) = ∞ (cid:88) n =0 ( a ) n ( b ) n x n ( c ) n n ! , where ( · ) n is the Pochhammer symbol, c (cid:54) = 0 , − , − , . . . ;, and | x | ≤
1. For x ∈ C ,with | x | ≥
1, the series can be analytically continued along any path in the complexplane that avoids the branch points 1 and infinity. An integral representation of thehypergeometric function is given by F ( a, b, c ; x ) = Γ( c ) (cid:82) t b − (1 − t ) c − b − (1 − xt ) − a dt Γ( b )Γ( c − b ) . • Modified Bessel functions are solutions of the modified Bessel equation. The modifiedBessel function of the first kind is defined by I ν ( z ) = i − ν J ν ( iz ), with ν ∈ R , and J ν ( · )is the Bessel function of the first kind. • Upper incomplete gamma function is given byΓ( a, x ) = (cid:90) ∞ x t a − e − t dt. For x >
0, Γ( a, x ) converges for all real a . In particular, Γ(0 , x ) is the exponentialintegral (cid:82) ∞ x t − e − t dt .Next, we describe some results related to the Laplace transform. For t ≥
0, and s ∈ C , wedenote the Laplace transform of f ( t ) by L ( f ( t )) = F ( s ), where f ( t ) is piecewise continuousfunction on every finite interval in [0 , ∞ ) satisfying | f ( t ) | < M e at , for some M > t ∈ [0 , ∞ ). The Laplace transform and the inverse Laplace transform are related by: F ( s ) = (cid:90) ∞ f ( t ) e − st dt, and f ( t ) = 12 πi (cid:90) x + i ∞ x − i ∞ e st F ( s ) ds, for some x ∈ R , where x is greater than the real part of all singularities of F ( s ), and F ( s ) isbounded on the line Re( s ) = x in the complex-plane. We list some useful properties relatedto the Laplace transform. The following result is elementary and can be found in [32].11 emma 4.1. The following results hold: (1) L − ( aF ( as − b )) = e bta f ( ta ) , with a > , b ∈ R ;(2) L − (cid:16) − dF ( s ) ds (cid:17) = tf ( t ) ; (3) L − (cid:16) F ( s ) s (cid:17) = (cid:82) t f ( u ) du ; (4) L − ( sF ( s ) − f (0)) = df ( t ) dt . The following results provide various relations between the Laplace transform and specialfunctions. These results can be found in [32].
Lemma 4.2.
The following results hold.(1) L (cid:18) t − (cid:82) ∞ ue − u t f ( u ) du (cid:19) = 2( √ π ) F ( √ s ) .(2) L − (cid:16) e as s (cid:17) = I (2 √ at ) , where I ( x ) is the modified Bessel function of the first kind, andRe ( s ) > .(3) L − (cid:16) e − a √ s (cid:17) = ae − a t √ πt , Re ( a ) > , Re ( s ) > .(4) L (Γ( v, at )) = Γ( v ) s [1 − (1 + sa ) − v ] , where Γ( v, at ) is upper incomplete gamma function,and Re ( ν ) > , Re ( s ) > − Re ( a ) .(5) L (cid:0) erf( √ at ) (cid:1) = √ as √ s + a , where erf( x ) = √ π (cid:82) x e − t dt , Re ( s ) > max(0 , − Re ( a )) .(6) L − (cid:16) √ s + a (cid:17) = e − at √ πt , Re ( s ) > − Re ( a ) .(7) L − (cid:16) √ s + as (cid:17) = e − at √ πt + √ a erf[ √ at ] , Re ( s ) > max(0 , − Re ( a )) .(8) L − (cid:16) s c − e − ( bs ) m (cid:17) = m
12 + mc (2 π ) m +12 b c (cid:80) i, − i i E (cid:16) c, c + m , . . . , c + m − m :: be iπ m m t (cid:17) , where E ( · : · : · ) is the MacRobert E -function, Re ( s ) > , Re ( c ) > , Re ( b ) > , m = 2 , , . . . . Inthe above expression (cid:80) i, − i denotes that in expression following the summation sign, i is to be replaced by − i and two expressions are to be added. In the two-dimension, for x ∈ R , let F ( x, s ) = (cid:82) ∞ f ( x, t ) e − st dt , be the Laplace transformof function f ( x, t ) with respect to the t variable. Note that, for a subordinator X t , withprobability density function f X t ( · ), and L´evy measure π X , the L´evy-Khinchin representationgives (see[9]) (cid:90) ∞ e − zt f X s ( t ) dt = e − sψ X ( z ) (4.1)where ψ X ( · ) is the Laplace exponent of X and is given by ψ X ( z ) = (cid:82) ∞ (1 − e − zu ) π X ( du ),where π X is the L´evy measure of X . The following result can be found in [9]. Theorem 4.3.
The L´evy density w ( x ) and L´evy measure π X ( t, ∞ ) of the subordinator X (with π X ( t, ∞ ) = (cid:82) ∞ t w ( x ) dx ) satisfy L ( π X ( t, ∞ )) = ψ X ( s ) s , where ψ X ( s ) is the Laplaceexponent of the subordinator X . Theorem 4.4.
Let X = { X t } t> be a subordinator with the probability density function p ( x, t ) . Suppose p ( x, t ) admits continuous partial derivatives. Let T t = inf { τ > X τ ≥ t } ,for t > , represents the first-exit time process of X . Denote the probability density functionof T t by h t ( · ) = h ( · , t ) . Then, L ( h ( x, t )) = ψ X ( s ) e − xψ X ( s ) s , (4.2) where ψ X ( · ) is the Laplace exponent of the subordinator X . Theorem 4.5.
Denote the q -th moment of the first-exit time of the subordinator X by M q ( x, t ) . Then, L ( M q ( x, t )) = q Γ(1 + q ) s ( ψ X ( s )) q . (4.3) Let X t be a Gamma subordinator with L´evy density given by w X ( x ) = νe − αx x , x >
0, with ν, α >
0. In this case, the Laplace exponent of X is given by ψ X ( s ) = ν ln (cid:0) sα (cid:1) (see[16, 29]). Theorem 4.6.
For xν = n +1 , n = 0 , , , . . . , the probability density function of the first-exittime of X is given by h ( x, t ) = (cid:90) t e − uα α xc ( ν ( − u ) νx F ( − νx, − νx, − νx ; 1) + νu νx )( xν − du, (4.4) where F ( − νx, − νx, − νx ; 1) is the hypergeometric function.Proof. By Theorem 4.4, the Laplace transform of probability density of the first-exit time ofGamma subordinator is given as L ( h ( x, t )) = ln(1 + sα ) ν s (1 + sα ) xν = K ( x, s ) s , where K ( x, s ) = F ( x, s ) G ( x, s ), with F ( x, s ) = ν ln(1 + sα ),and G ( x, s ) = sα ) xν . Then L ( h ( x, t )) = K ( x,s ) s . Let the inverse Laplace transforms for F ( x, s ) and G ( x, s ) be f ( x, t ) and g ( x, t ), respectively.Note that L − (ln(1 + s )) = − e − t t , (see [32]). Using Lemma 4.1(1), we obtain, f ( x, t ) = − νe − tα t . xν = n + 1, where n is a non-negative integer, L − (cid:16) − s +1) xν (cid:17) = t xν − e − t ( xν − . Hence, byusing Lemma 4.1(2), we obtain g ( x, t ) = α xν t xν − e − tα ( xν − . Consequently, by standard convolutionprocedure, we obtain k ( x, t ) = (cid:90) t f ( x, τ ) g ( x, t − τ ) dτ = (cid:90) t − νe − τα α xν ( t − τ ) xν − e − α ( t − τ ) ( xν − dτ = e − tα α xν [ ν ( − t ) νx F ( − νx, − νx, − νx ; 1) + νt νx ]( xν − . Hence, with the application of Lemma 4.1(3), we obtain (4.4).The next result provides the first and the second order moment of the first-exit time ofGamma subordinator.
Theorem 4.7.
The first order moment (mean) and the second order moment of the first-exittime of Gamma subordinator X t is given by m ( x, t ) = t (cid:90) ∞ (cid:90) αe − λα ( λα ) u − du dλν Γ( u ) , (4.5) and t ( x, t ) = t (cid:90) ∞ (cid:90) λe − λα ( λα ) u − du dλν Γ( u ) , (4.6) respectively.Proof. Using Theorem 4.5, we obtain the Laplace transform of the q -th moment of the first-exit time of the Gamma subordinator as q Γ(1+ q ) s ( ψ X ( s )) q . Consequently, the Laplace transform ofthe first order moment of the first-exit time of the Gamma subordinator is given by M ( x, s ) = Γ(2) sν ln(1 + sα ) . We observe that L − (cid:16) Γ(2)ln( s ) (cid:17) = (cid:82) ∞ t u − Γ( u ) du . Consequently, using Lemma 4.1(1), we obtain L − (cid:18) Γ(2) ν ln(1 + sα ) (cid:19) = (cid:90) ∞ α Γ(2) e − tα ( tα ) u − ν Γ( u ) du. Therefore, L − ( M ( x, s )) can be computed using Lemma 4.1(3) as (4.5). Once again, byTheorem 4.5, the Laplace transform of second order moment of the first-exit time of theGamma subordinator is given by T ( x, s ) = 2Γ(3) s ( ν ln(1 + sα )) . − d ln(1+ sα ) − ds = α (ln(1+ sα )) , we obtain by Lemma 4.1(2), L − (cid:18) α ( ν ln(1 + sα )) (cid:19) = t (cid:90) ∞ α Γ(2) e − tα ( tα ) u − ν Γ( u ) du. Thus we obian (4.6).We conclude this subsection by considering the case when the subordinator Z , given by(2.5), is related to the Gamma subordinator in the BN-S model. As observed in Section 2, ifthe stationary distribution of σ t is given by gamma law Γ( ν, α ), then the L´evy density of Z is given by w ( x ) = ναe − αx , x > Theorem 4.8.
The probability density function of the first-exit time of a subordinator Z with L´evy density w ( x ) = ναe − αx , is given by h ( x, t ) = νe − xν I (cid:16) √ xναt (cid:17) e − αt , where I ( · ) is the modified Bessel function of the first kind.Proof. For this case, the L´evy measure of Z is given by π Z ( t, ∞ ) = (cid:82) ∞ t ναe − αx dx = ναe − αt .Using Theorem 4.3, we obtain ψ Z ( s ) s = νs + α . Consequently, ψ Z ( s ) = νss + α . The Laplacetransform of the probability density function of the first-exit time of Z is given by H ( x, s ) = νe − xνss + α s + α . Consequently, the probability density function of the first-exit time of Z is given by h ( x, t ) = L − ( H ( x, s )), where H ( x, s ) = νe − xνss + α s + α = νe − xν (1 − αs + α ) s + α = νe − xν e xναs + α s + α . Using Lemma 4.2(2), we obtain h ( x, t ) = νe − xν I (cid:0) √ xναt (cid:1) e − αt . The first-exit time of IG processes are described in [45]. In this subsection we consider thesubordinator Z , given by (2.5), that is related to the IG subordinator in the BN-S model.As observed in Section 2, if the stationary distribution of σ t is given by IG( δ , γ ) law, thenthe L´evy density of Z is given by w ( x ) = δ √ π x − (1 + γ x ) e − γ x , x >
0, and δ , γ > convolution of two functions p ( x, t ) and q ( x, t ) by p ( x, t ) ∗ q ( x, t ) = (cid:90) t p ( x, τ ) q ( x, t − τ ) dτ. Consequently, for three functions p ( x, t ), q ( x, t ), and r ( x, t ),( p ( x, t ) ∗ q ( x, t )) ∗ r ( x, t ) = (cid:90) t (cid:90) u p ( x, τ ) q ( x, u − τ ) r ( x, t − u ) dτ du. heorem 4.9. The probability density function of the first-exit time of a subordinator Z withL´evy density w ( x ) = δ √ π x − (1 + γ x ) e − γ x , is given by h ( x, t ) = ( p ( x, t ) ∗ q ( x, t )) ∗ r ( x, t ) ,where p ( x, t ) = − δ γ erf( γ √ t √ )2 + δ γ − , γ t ) δ γ √ π , (4.7) q ( x, t ) = e − xγ δ √ e − tγ t − √ π (cid:90) ∞ ue − u t (cid:32) I (cid:48) (cid:32) (cid:115) xγ δ √ u (cid:33) (cid:32)(cid:115) xγ δ √ u (cid:33) + δ ( u ) (cid:33) du, (4.8) where δ ( · ) is the Dirac delta function, I ( · ) is the modified Bessel function of the first kind,and r ( x, t ) = xδ γ e − γ t e xδ γ e − δ x γ t √ π (2 t ) . (4.9) Proof.
We obtain the L´evy measure for Z as π Z ( t, ∞ ) = (cid:90) ∞ t w ( x ) dx = (cid:90) ∞ t δ x − e − γ x + δ ( γ ) x − e − γ x √ π dx = − δ γ erf( γ √ t √ )2 + δ γ − , γ t ) δ γ √ π . Using Theorem 4.3, Lemma 4.2(4), and Lemma 4.2(5) we obtain, L ( π Z ( t, ∞ )) = ψ Z ( s ) s = δ γ s − δ γ √ s ( (cid:113) s + γ ) − δ γ [1 − (cid:113) (1 + sγ )]2 s . (4.10)Consequently, by Theorem 4.4, we obtain that the Laplace transform of the probabilitydensity function of the first-exit time of Z is given by H ( x, s ) = δ γ s − δ γ √ s ( (cid:113) s + γ ) − δ γ [1 − (cid:113) (1 + sγ )]2 s e − x δ γ − δ γ √ (cid:114) s + γ − δ γ [1 − (cid:114) (1+ 2 sγ = P ( x, s ) Q ( x, s ) R ( x, s ) , where P ( x, s ) = δ γ s − δ γ √ s (cid:113) s + γ − δ γ [1 − (cid:113) (1 + sγ )]2 s ,Q ( x, s ) = exp − xγδ xγ δ √ (cid:113) s + γ , R ( x, s ) = exp xδ γ − xδ γ (cid:113) (1 + sγ )2 . We denote the inverse Laplace transforms of P ( x, s ), Q ( x, s ), and R ( x, s ) by p ( x, t ), q ( x, t ),and r ( x, t ), respectively.We have p ( x, t ) = L − (cid:32) δ γ s − δ γ √ s ( (cid:113) s + γ ) − δ γ [1 − (cid:113) (1+ sγ )]2 s (cid:33) . From this, comparing with(4.10), we note that p ( x, t ) = L − ( ψ Z ( s ) s ). Hence p ( x, t ) is given by (4.7).Next, we compute q ( x, t ) using Lemma 4.2(2), Lemma 4.2(1), and Lemma 4.1(4). Lemma4.2(2) gives L − (cid:16) e as s (cid:17) = I (2 √ at ).With L ( s ) = e as s , we find l ( t ) = L − ( L ( s )) = I (2 √ at ). We notice I (0) = 1. Con-sequently, using Lemma 4.1(4), we have L − ( sL ( S ) − l (0)) = l (cid:48) ( t ). Hence, we obtain, L − ( e as ) − L − (1) = I (cid:48) (2 √ at )( (cid:112) at ), and thus L − ( e as ) = I (cid:48) (2 √ at )( (cid:112) at ) + δ ( t ), where δ ( · ) isthe Dirac delta-function.Using Lemma 4.2(1), we obtain L − ( e as − / ) = t − √ π (cid:82) ∞ ue − u t ( I (cid:48) (2 √ au )( (cid:112) au ) + δ ( u )) du .Therefore, q ( x, t ) = e − xγ δ √ e − tγ t − √ π (cid:90) ∞ ue − u t (cid:32) I (cid:48) (cid:32) (cid:115) xγ δ √ u (cid:33) (cid:32)(cid:115) xγ δ √ u (cid:33) + δ ( u ) (cid:33) du. Finally, r ( x, t ) = L − ( R ( x, s )) = L − e xδ γ e − xδ γ (cid:114) (1+ 2 sγ . Using Lemma 4.2(3), we obtain L − ( e xδ γ e − xδ γ √ s ) = e xδ γ ( xδ γ ) e − δ x γ t √ πt . Consequently,using Lemma 4.1(1), we obtain (4.9).Finally, if h ( x, t ) is the probability density function of the first-exit time of Z , then h ( x, t ) = L − ( H ( x, s )) = L − ( P ( x, s ) Q ( x, s ) R ( x, s )) = ( p ( x, t ) ∗ q ( x, t )) ∗ r ( x, t ). Let X t be a positive tempered stable (PTS) subordinator with L´evy density given by u ( x ) = βk − γ γ Γ( γ )Γ(1 − γ ) x − γ − exp (cid:18) − k x (cid:19) , x > , with β >
0, 0 < γ <
1, and k ≥
0. 17 heorem 4.10.
The probability density function of the first-exit time of X is given by h ( x, t ) = p ( x, t ) ∗ q ( x, t ) , where p ( x, t ) = a Γ (cid:18) − γ, k t (cid:19) , (4.11) where a = βγ γ Γ( γ )Γ(1 − γ ) , and q ( x, t ) = e − xa Γ( − γ ) e (( − xa )Γ( − γ )) γ ) t ( γ ) γ +22 γ (2 π ) γ +12 γ (cid:88) i, − i i E (cid:32) , γ, . . . , − γ :: (( k ) γ ( − xa )Γ( − γ )) γ e iπ γ − γ t (cid:33) . (4.12) In (4.12) , E ( · : · : · ) is the MacRobert E -function, and (cid:80) i, − i denotes that in expressionfollowing the summation sign, i is to be replaced by − i and two expressions are to be added.Proof. We have π X ( t ) = (cid:90) ∞ t u ( x ) dx = (cid:90) ∞ t βk − γ γx − γ − e − k x Γ( γ )Γ(1 − γ ) dx = βγ Γ( − γ, k t )Γ( γ )Γ(1 − γ )2 γ . We compute L ( π X ( t )) using Theorem 4.3 to obtain the Laplace exponent of density functionof X as ψ X ( s ) = βγ Γ( − γ )[1 − (1 + sk ) γ ]Γ( γ )Γ(1 − γ )2 γ . Now using Theorem 4.4, we obtain the Laplace transform of the probability density functionof the first-exit time of X as H ( x, s ) = L ( h ( x, t )) = (cid:32) a Γ( − γ )[1 − (1 + sk ) γ ] s (cid:33) e − ax Γ( − γ )[1 − (1+ sk ) γ ] , where a = βγ Γ( γ )Γ(1 − γ )2 γ . To compute h ( x, t ), the probability density function of the first-exittime of X , we find p ( x, t ) = L − (cid:32) a Γ( − γ )[1 − (1 + sk ) γ ] s (cid:33) , and q ( x, t ) = L − (cid:16) e − ax Γ( − γ )[1 − (1+ sk ) γ ] (cid:17) , and use the convolution result. By using Lemma 4.2(4), we obtain, the expression of p ( x, t )as (4.11). 18ext, compute q ( x, t ). Denote Q ( x, s ) = exp (cid:0) − ax Γ( − γ )[1 − (1 + sk ) γ ] (cid:1) . We observe Q ( x, s ) = e − xa Γ( − γ ) exp (cid:18) − (cid:18) ( − xa Γ( − γ )) γ + 2 sk ( − xa Γ( − γ )) γ (cid:19) γ (cid:19) = e − xa Γ( − γ ) exp (cid:32) − (cid:32) ( − xa Γ( − γ )) γ + s (cid:18) − xa (cid:18) k (cid:19) γ Γ( − γ ) (cid:19) γ (cid:33) γ (cid:33) . Hence, by using Lemma 4.2(8) and Lemma 4.1(1), we obtain the expression of q ( x, t ) as(4.12).We conclude this subsection by considering a subordinator Z related to the PTS subor-dinator in the BN-S model. If the stationary distribution of σ t is given by PTS( κ, δ, γ ) law,then that the L´evy density of Z is given by w ( x ) = βk − γ γx − γ − e − k x Γ( γ )Γ(1 − γ ) (cid:18) γ + k x (cid:19) , x > , β > , < γ < , k ≥ . (4.13)As in the previous sections, Z is the subordinator given by (2.5). Theorem 4.11.
The probability density function of the first-exit time of a subordinator Z ,with L´evy density (4.13) , is given by h ( x, t ) = ( p ( x, t ) ∗ q ( x, t )) ∗ r ( x, t ) , where p ( x, t ) = a (cid:18) γ Γ( − γ, k t − γ, k t (cid:19) , (4.14) where a = βγ γ Γ( γ )Γ(1 − γ ) , and q ( x, t ) = e − xaγ Γ( − γ ) e (( − xaγ )Γ( − γ )) γ ) t ( γ ) γ +22 γ (2 π ) γ +12 γ (cid:88) i, − i i E (cid:32) , γ, . . . , − γ :: (( k ) γ ( − xaγ )Γ( − γ )) γ e iπ γ − γ t (cid:33) , (4.15) r ( x, t ) = e − xa Γ(1 − γ ) e (( − xa )Γ(1 − γ )) γ − ) t ( γ − ) γ +12 γ − (2 π ) γ γ − (cid:88) i, − i i E , γ, . . . , − γ :: (( γ ) γ − ( − xa )Γ(1 − γ )) γ − e iπ ( γ − − γ − t . (4.16) In (4.15) and (4.16) , E ( · : · : · ) is the MacRobert E -function, and (cid:80) i, − i denotes that inexpression following the summation sign, i is to be replaced by − i and two expressions are tobe added. roof. We obtain π Z ( t, ∞ ) as π Z ( t, ∞ ) = (cid:90) ∞ t w ( x ) dx = (cid:90) ∞ t βk − γ γx − γ − e − k x Γ( γ )Γ(1 − γ ) ( γ + k x dx = βγ γ Γ( γ )Γ(1 − γ ) (cid:18) Γ( − γ, k t γ + Γ(1 − γ, k t (cid:19) . We use Theorem 4.3 to obtain ψ Z ( s ) s = βγ γ Γ( γ )Γ(1 − γ ) (cid:18) γ Γ( − γ ) s (1 − (1 + 2 sk ) γ ) + Γ(1 − γ ) s (1 − (1 + 2 sk ) γ − ) (cid:19) . Consequently, ψ Z ( s ) = a (cid:18) γ Γ( − γ )(1 − (1 + 2 sk ) γ ) + Γ(1 − γ )(1 − (1 + 2 sk ) γ − ) (cid:19) , where a = βγ γ Γ( γ )Γ(1 − γ ) . Using Theorem 4.4, we obtain the Laplace transform of the proba-bility density function of the first-exit time of Z as H ( x, s ) = L ( h ( x, t )) = P ( x, s ) Q ( x, s ) R ( x, s ) , where P ( x, s ) = a (cid:18) γ Γ( − γ ) s − γ Γ( − γ ) s (1 + 2 sk ) γ + Γ(1 − γ ) s − Γ(1 − γ ) s (1 + 2 sk ) γ − (cid:19) ,Q ( x, s ) = exp (cid:18) − xa (cid:18) γ Γ( − γ )(1 − (1 + 2 sk ) γ ) (cid:19)(cid:19) , and R ( x, s ) = exp (cid:18) − xa (cid:18) Γ(1 − γ )(1 − (1 + 2 sk ) γ − (cid:19)(cid:19) . We denote the inverse Laplace transform for P ( x, s ), Q ( x, s ), and R ( x, s ), by p ( x, t ), q ( x, t ),and r ( x, t ), respectively. Using Lemma 4.2(4), we obtain L − ( γ Γ( − γ ) s (1 − (1 + sk ) γ ) = γ Γ( − γ, k t ). Also, using Lemma 4.2(4), we obtain L − ( Γ(1 − γ ) s − Γ(1 − γ ) s (1 + sk ) γ − ) =Γ(1 − γ, k t ). Hence, we obtain p ( x, t ) as given by (4.14).Next, we observe that Q ( x, s ) can be written as: Q ( x, s ) = e − xaγ Γ( − γ ) exp (cid:18) − (cid:18) ( − xaγ Γ( − γ )) γ + 2 sk ( − xaγ Γ( − γ )) γ (cid:19) γ (cid:19) = e − xaγ Γ( − γ ) exp (cid:32) − (cid:32) ( − xaγ Γ( − γ )) γ + s (cid:18) − xaγ (cid:18) k (cid:19) γ Γ( − γ ) (cid:19) γ (cid:33) γ (cid:33) . R ( x, s ) can be written as R ( x, s ) = e − xa Γ(1 − γ ) exp (cid:32) − (cid:18) ( − xa Γ(1 − γ )) γ − + 2 sk ( − xa Γ(1 − γ )) γ − (cid:19) γ − (cid:33) = e − xa Γ(1 − γ ) exp − ( − xa Γ(1 − γ )) γ − + s (cid:32) − xa (cid:18) k (cid:19) γ − Γ(1 − γ ) (cid:33) γ − γ − . Hence by using Lemma 4.2(8) and Lemma 4.1(1), we obtain (4.16). Finally, by convolutiontheorem, we obtain the probability density function of the first-exit time of Z as h ( x, t ) = L − ( H ( x, s )) = ( p ( x, t ) ∗ q ( x, t )) ∗ r ( x, t ). For this section, we use the S&P 500 daily close price dataset for the period May 11, 2010 toMay 8, 2020. Table 1 summarizes some features of this empirical dataset.Table 1: Properties of the empirical dataset.S&P 500 daily close priceMean 2027.003Median 2036.709Maximum 3386.149Minimum 1022.580Figure 1 shows a line plot of the empirical dataset. The log-return process for the cor-responding dataset is shown in Figure 2. Figure 3 and Figure 4 show the histograms of theS&P 500 daily close price, and corresponding log-returns respectively.For the empirical dataset we consider the log-return process X t , with X = 0. For thefirst-exit time process of the log-return, inf { s > X s ≥ t } , we consider the associated first-exit time processes of the Brownian motion inf { s > W s = t } , and the L´evy subordinatorinf { s > Z s ≥ t } . In the plots in Figure 5, we provide the histograms corresponding tothe first-exit time of X t for the empirical dataset for various values of t . In the plots ofFigure 6, we use Remark 3.3 to plot the probability density functions of inf { s > W s = t } ,for t = 1 , , ,
4. Finally, we use Gamma-type subordinators described in Section 4.2 withL´evy density w ( x ) = ναe − αx . After finding appropriate parameter values, in the plots ofFigure 7, we use Theorem 4.8 to plot the probability density functions of inf { s > Z s ≥ t } ,for t = 1 , , ,
4. From these figures, it is clear that for the time duration when there isno big fluctuation of the empirical dataset, inf { s > W s = t } plays the dominant rolein determining the distribution of inf { s > X s ≥ t } . However, for the time durationof big fluctuation of the empirical dataset, inf { s > Z s ≥ t } plays the dominant role indetermining the distribution of inf { s > X s ≥ t } .21igure 1: S&P 500 daily close price from May, 2010 -May, 2020.Figure 2: S&P 500 log-returns from May, 2010 -May, 2020.Figure 3: Histogram for the S&P 500 daily close price.22igure 4: Histogram for the log-return.Figure 5: Histograms corresponding to inf { s > X s ≥ t } , for (left to right) t = 1 , , , { s > W s = t } , for (left to right) t = 1 , , , { s > Z s ≥ t } , for (left to right) t = 1 , , , It is shown in this paper that an analytically tractable expression can be obtained for theprobability density function of the first-exit time process of an approximate BN-S process.For the financial data, the density function of the first-exit time of the corresponding log-return process provides an important insight. In particular, such density function facilitates24he understanding of a “crash-like” future fluctuation of the market. In addition, this analysishas two-fold advantages. Firstly, based on the insight from the probability density functionof the first-exit time process, the empirical data analysis for the future market is improved.Secondly, and more importantly, this provides a concrete way to improve existing stochasticmodels. For example, most of the existing financial models suffer from the lack of long-range dependence problem. An understanding of the density function of the first-exit timeof stochastic models driven by a general L´evy process can contribute positively to mitigatethis issue.In the numerical results, we show various plots in support of the theoretical analysisprovided in this paper. However, the analysis is dependent on the accurate estimation ofmodel parameters for the empirical dataset. At present, we are implementing various machinelearning based calibration techniques to improve the estimates of the parameter values forthe empirical dataset. In effect, this may significantly improve the numerical results. Theseconcepts, along with their connection to the first-exit time analysis, will be developed in asequel of this paper.
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