Stochastic Areas of Diffusions and Applications in Risk Theory

Noname manuscript No. (will be inserted by the editor) Stochastic areas of diffusions and applications in risk theory arXiv:1312.0283v1 [q-fin.RM] 1 Dec 2013 Zhenyu Cui Received: date / Accepted: date Abstract In this paper we study the stochastic area swept by a regular time-homogeneous diffusion till a stopping time. This unifies some recent literature in this area. Through stochastic time change we establish a link between the stochastic area and the stopping time of another associated time-homogeneous diffusion. Then we characterize the Laplace transform of the stochastic area in terms of the eigenfunctions of the associated diffusion. We also explicitly obtain the integer moments of the stochastic area in terms of scale and speed densities of the associated diffusion. Specifically we study in detail three stopping times: the first passage time to a constant level, the first drawdown time and the Azema-Yor stopping time. We also study the total occupation area of the diffusion below a constant level. We show applications of the results to a new structural model of default (Yildirim 2006), the Omega risk model of bankruptcy in risk analysis (Gerber, Shiu and Yang 2012), and a diffusion risk model with surplus-dependent tax (Albrecher and Hipp 2007, Li, Tang and Zhou 2013). JEL Classification C02 C63 G12 G13 Mathematics Subject Classification (2000) 60G44 · 91B70 · 91B25 Keywords Time-homogeneous diffusion · first passage time · occupation time · Azema-Yor stopping time · Omega risk model Z. Cui Corresponding Author. Department of Mathematics, Brooklyn College of the City University of New York, Tel.: +1718-951-5600, ext. 6892 Fax: +1718-951-4674 E-mail: [email protected] 2 Zhenyu Cui 1 Introduction The stochastic area swept by a stochastic process till a stopping time has an intuitive geometric meaning and has been applied to modeling objects of “accumulative nature" in physics, queueing theory, mathematical finance, actuarial risk theory and graph enumeration problems. Let (Vt ){t>0} denote the underlying stochastic process. In queueing theory, if Vt represents the length of a queue at time t, and denote the first passage time of V to the threshold l by τl , which is the time when the queue system first collapses(overflow time). Then the stochastic area swept by V until τl represents the cumulative waiting time experienced by all the users till the overflow of the system. In mathematical finance, there are two main approaches in the area of credit risk modeling. The structural approach(Merton 1974, Black and Cox 1976) assumes that the default event happens when the total value of the firm’s asset first goes below the face value of its debt. If we model the firm’s asset value as a diffusion V , then the default time is the first passage time of V to a fixed threshold. The second is the reduced-form model(Jarrow and Turnbull 1995, Duffie and Singleton 1999), where default is modeled by an exogenously-determined intensity or compensator process. The default time is the first jump time of a point process. Recently there is a third new approach by Yildirim 2006, which uses the information of the stochastic area swept by the firm’s asset value process. The default time is when the cumulative stochastic area below a threshold level exceeds an exogenous level. This model with stochastic area allows the firm to recover from financial distress and is flexible. In actuarial risk theory or ruin theory, recently there is some interest in using the total occupation time of a stochastic process below a constant level to model bankruptcy. The new model is named the “Omega risk model". The Omega risk model was first introduced by Albrecher, Gerber and Shiu 2011, and it distinguishes the ruin time(negative surplus) from the bankruptcy time(negative surplus area surpasses a specified level) of a company. In classical ruin theory, the ruin event occurs when the surplus of a company first becomes negative. The Omega risk model assumes the more realistic situation that a company can still do business even with a negative surplus. The bankruptcy time is linked to the total occupation R ∞ time of the firm value V below a certain threshold l. Here the total occupation time 0 1{Vt <l} dt is a special stochastic area with the integrand being an indicator function. In the applied probability literature, there are a few papers proposing to study the distribution of the area swept by a stochastic process till a stopping time. More specifically, Perman and Wellner 1996 study the distribution and moments of the stochastic area in the case of a standard Brownian motion. Kearney and Majumadar 2005, Kearney, Majumadar and Martin 2007 extend the study to a drifted Brownian motion. Knight 2000 considers the case of a reflected Brownian bridge. Janson 2007 provides a survey to this area and links it to the Wright’s constants in graph enumeration. Recently, Abundo 2013b considers a one-dimensional jump diffusion and show that the Laplace transform and moments of the associated stochastic area till the first passage time to zero are solutions to certain partial differential-difference equations(PDDE) with outer conditions. Abundo 2013a extends similar results to the stochastic area till the first passage time to a non-zero level. Stochastic area of diffusion 3 Starting from the seminal paper of Lehoczky 1977, there is a continued interest in the study of drawdown/drawup time of diffusion processes(Hadjiliadis and Vecer 2006, Pospisil, Vecer and Hadjiliadis 2009, Zhang and Hadjiliadis (2010, 2012)). For a pointer to the vast literature in this area, please refer to the Ph.D. thesis of Zhang 2010. The stochastic area till the first drawdown time represents the accumulated firm value till the drawdown event, and reflects the profitability and solvency of a firm during a financial crisis. The Azema-Yor stopping time is a generalized drawdown time and is a candidate solution to many optimal stopping problems(Graversen and Peskir 1997, Pedersen 2005, Shepp and Shiryaev 1993), and also to the Skorokhod embedding problem(Obloj 2004). There is some recent interest in the literature to study risk models with tax: risk models perturbed by a functional of the running maximum process. Albrecher and Hipp 2007 introduce a constant tax rate at profitable times of the compound Poisson risk model. Li, Tang and Zhou 2013 introduce a surplusdependent tax rate to a diffusion risk model and model the ruin time of the company by the two-sided exit time of the diffusion process. Refer to Kyprianou and Ott 2012 for a pointer to relevant literature. We make three contributions to the current literature. First, through stochastic time change we link the study of the stochastic area till a stopping time to the study of a related stopping time of another time-homogeneous diffusion, whose drift and diffusion coefficients can be explicitly determined. This provides a unified method to study the stochastic area and allows us to characterize the Laplace transform of the stochastic area in terms of eigenfunctions of diffusions. In some examples, we can explicitly compute the eigenfunctions and the Laplace transform. We provide an explicit closed-form formula for the integer moments of the first passage area and extend the approach in Abundo 2013b, 2013a, since we do not need to solve the associated partial differential-difference equations, but solve a Sturm-Liouville type ordinary differential equation(ODE) instead. There are a number of known examples where we can solve the ODE explicitly(see Borodin and Salminen 2002). Second, we compute the probability of bankruptcy and the expected time of ruin in the Omega risk model for diffusions with a general bankruptcy rate function ω (.). Previous literature considers the special case when ω (.) is a constant or a piecewise constant(Albrecher, Gerber and Shiu 2011, Gerber, Shiu and Yang 2012, and Li and Zhou 2013). We extend the literature to a more general bankruptcy rate function and this provides more flexibility in modeling. Third, we explicitly compute the expected time of ruin in a diffusion risk model with surplus-dependent tax rate. We also explicitly characterize the expected stochastic area till the Azema-Yor stopping time, which represents the accumulated firm value till the optimal execution time in some optimal stopping problems(e.g. Russian options). The paper is organized as follows: Section 2 presents the stochastic time change method and links the study of the stochastic area till a stopping time to the study of a related stopping time of another diffusion. Section 3 considers the stochastic area till the first passage time of the diffusion to a constant level. We compute the Laplace transform of the stochastic area in terms of eigenfunctions of the associated diffusions and express all its integer moments in terms of the scale and speed densities as an explicit recursion formula. We illustrate an example in the new structural model of default proposed in Yildirim 2006. Section 4 provides the Laplace transform of the stochastic occupation area, and apply it to calculating the probability of bankruptcy in the Omega risk model with a general bankruptcy rate function ω (.). We illustrate with a concrete example. Section 5 computes the Laplace transform of the stochastic 4 Zhenyu Cui area till the first drawdown/drawup time of a diffusion. We obtain explicit closed-form formula for the expected time of ruin in a diffusion risk model with surplus-dependent tax. Section 6 concludes the paper with future research directions. 2 Stochastic time change and stochastic area till a stopping time Given a complete filtered probability space (Ω, F, Ft , P ) with state space J = (l, r ), −∞ 6 l < r 6 ∞, and assume that the J-valued diffusion V = (Vt ){t∈[0,∞)} satisfies the SDE dVt = µ(Vt ) dt + σ (Vt ) dWt , V0 = v0 ∈ J. (1) where W is a Ft -Brownian motion and µ, σ : J → R are Borel functions satisfying the Engelbert-Schmidt conditions ∀x ∈ J, σ (x) 6= 0, 1 σ 2 (·) , µ(·) ∈ L1loc (J ), σ 2 (·) (2) where L1loc (J ) denotes the class of locally integrable functions, i.e. the functions J → R are integrable on compact subsets of J. This condition (2) guarantees that the SDE (1) has a unique in law weak solution that possibly exits its state space J(see Theorem 5.15, p341, Karatzas and Shreve 1991). Denote the possible explosion time of V from its state space by ζ, i.e. ζ = inf {u > 0, Vu 6∈ J}, which means that P -a.s. on {ζ = ∞} the trajectories of V do not exit J, and P -a.s. on {ζ < ∞}, we have lim Vt = r or lim Vt = l. V is defined such that it stays t→ζ t→ζ at its exit point, which means that l and r are absorbing boundaries. The following terminology is used: V exits the state space J at r means P (ζ < ∞, lim Vt = r ) > 0. t→ζ In what follows, λ(.) denotes the Lebesgue measure on B (R). Let b be a Borel function such that λ(x ∈ (l, r ) : b2 (x) = 0) = 0, and assume the following local integrability condition ∀x ∈ J, σ (x) 6= 0, b2 (·) ∈ L1loc (J ). σ 2 (·) (3) Rt Lemma 1 Define the function ϕt := 0 b2 (Vu )du, for t ∈ [0, ζ ]. Then ϕt is a nondecreasing and continuously differentiable function for t ∈ [0, ζ ] with positive derivative P -a.s. Proof Recall that b(.) is a positive Borel function, thus ϕt is an increasing function for 0 6 t 6 ζ. For t ∈ [0, ζ ), it is clear that ϕt is a continuous function, and the continuity of ϕt at t = ζ on the set {ζ < ∞} follows from the Dambis-Dubins-Schwartz theorem(see the proof of Theorem 4.6, p175, Karatzas and Shreve 1991). Also note that ϕt is represented as a time integral and is thus differentiable with derivative b2 (Vt ), which is positive P -a.s. from the definition of the function b(.). ⊔ ⊓ The following result is about stochastic time-change, and for completeness we provide its proof. Stochastic area of diffusion 5 Theorem 1 (Theorem 3.2.1 of Cui 2013) Assume the conditions (2) and (3) (i) Define τ (t) := τt := ( inf {u > 0 : ϕu∧ζ > t}, ∞,  0 6 t < ϕζ ,  on ϕζ 6 t < ∞ . on (4) Define  a new filtration Gt = Fτt , t ∈ [0, ∞), and a new Gt -adapted process Xt := Vτt , on 0 6 t < ϕζ . Then we have the stochastic representation Vt = XR t b2 (Vs )ds = Xϕt , 0 P − a.s., on {0 6 t < ζ} . (5) and the process X is a time-homogeneous diffusion, which solves the following SDE under P dXt = µ(Xt ) σ (Xt ) 1 dt + 1 dBt , b2 (Xt ) {t∈[0,ϕζ )} b(Xt ) {t∈[0,ϕζ )} X0 = v0 . (6) where Bt is the Gt -adapted Dambis-Dubins-Schwartz Brownian motion defined in the proof. (ii) Define ζ X := inf {u > 0 : Xu 6∈ J}, then ζ X = ϕζ = we can rewrite the SDE (6) as dXt = Rζ 0 σ (Xt ) µ(Xt ) 1 dt + 1 dBt , X X b2 (Xt ) {t∈[0,ζ )} b(Xt ) {t∈[0,ζ )} m2 (Vs )ds, P -a.s., and X0 = v0 . (7) Rτ (iii) Let τ denote a Ft stopping time1 of Vt , t ∈ [0, ζ ), then ϕτ := 0 b2 (Vs )ds is a Gt stopping time and τ X = ϕτ , P-a.s., where τ X is the corresponding stopping time for Xt , t ∈ [0, ζ X ). Proof Since λ(x ∈ (ℓ, r ) : b2 (x) = 0) = 0, ϕs is an increasing and continuous function on [0, ζ ], from Problem 3.4.5  (ii), p174 of Karatzas and Shreve 1991, ϕτt ∧ζ = t ∧ ϕζ , P -a.s. for 0 6 t < ∞. On 0 6 t < ϕζ , when u = ζ, ϕζ∧ζ = ϕζ > t holds P -a.s. according to the assumption. Then τt 6 ζ, P -a.s. due to the definition in (4). Thus  ϕτt = t, P -a.s. on 0 6 t < ϕζ . On {0 6 s < ζ}, choose t = ϕs , then 0 6 t < ϕζ , P -a.s. Substituting this t into the definition of the process X, Xϕs = Xt := Vτt = Vτϕs = Vs , P -a.s. For the last equality, note that τϕs = inf {u > 0 : ϕu∧ζ > ϕs } = inf {u > 0 : u ∧ ζ > s} = s, P -a.s., on {0 6 s < ζ}. Then we have proved the representation Vs = Xϕs , on {0 6 s < ζ}. For X satisfying the relation (5), we aim to show that X satisfies the SDE (6), where B is the Dambis-Dubins-Schwartz Brownian motion adapted to Gt constructed as R t∧ζ follows: Note that Mt∧ζ = 0 b(Vu )dWu , t ∈ [0, ∞) is a continuous local martingale, R t∧ζ with quadratic variation ϕt∧ζ = 0 b2 (Vu )du, t ∈ [0, ∞). Then limt→∞ ϕt∧ζ = ϕζ , P -a.s. due to the left continuity of ϕs at s = ζ. From the Dambis-Dubins-Schwartz theorem(Ch.V, Theorem 1.6 and Theorem 1.7 of Revuz and Yor 1999), there exists a 1 It is understood that τ = ζ, P -a.s on {τ > ζ}. 6 Zhenyu Cui possible enlargement (Ω̄, G¯t , P̄ ) of (Ω, Gt , P ) and a standard Brownian motion β̄ on Ω̄ independent of M with β̄0 = 0, such that the process (R τ t b(Vu )dWu , Bt := R0ζ b(Vu )dWu + βet−ϕζ , 0  t < ϕζ ,  on t > ϕζ . on (8) is a standard linear Gt -Brownian motion. Our construction of τt , t ∈ [0, ∞) agrees with that in Problem 3.4.5, p174 of Karatzas and Shreve 1991. From Problem 3.4.5 (ii) and the construction (8), Bϕs = Ms , P -a.s. on {0 6 s < ζ}. On {s = ζ}, Bϕζ := Rζ 0 b(Vu )dWu +βe0 = Rζ 0 b(Vu )dWu =: Mζ , P -a.s. Thus Bϕt = Mt , P -a.s. on {0 6 t 6 ζ}. For the convenience of exposition, denote µ1 (.) = µ(.)/b2 (.), and σ1 (.) = σ (.)/b(.). Integrate the SDE in (1) from 0 to t ∧ ζ Vt∧ζ − V0 = = Z Z t∧ζ µ(Vu )du + 0 t∧ζ Z t∧ζ σ (Vu )dWu 0 µ1 (Vu )b2 (Vu )du + 0 Z t∧ζ σ1 (Vu )b(Vu )dWu . (9) 0 Apply the change of variables formula similar to Problem 3.4.5 (vi), p174 of Karatzas and Shreve 1991, and note the relation (5) Z t∧ζ µ1 (Vu )b2 (Vu )du = 0 Z t∧ζ µ1 (Xϕu )dϕu = 0 Z ϕt∧ζ µ1 (Xu )du, (10) 0 and similarly Z t∧ζ σ1 (Vu )b(Vu )dWu = 0 Z t∧ζ σ1 (Xϕu )dBϕu = 0 Z ϕt∧ζ σ1 (Xu )dBu (11) 0 Ru where the first equality in (11) is due to the relationship Bϕu = Mu = 0 b(Vs )dWs , P a.s. on {0 6 u 6 t ∧ ζ}, which we have established above. Also notice the representation Vt∧ζ = Xϕt∧ζ , P -a.s. and V0 = X0 , then Xϕt∧ζ − X0 =  Z ϕt∧ζ µ1 (Xu )du + 0 Z ϕt∧ζ σ1 (Xu )dBu (12) 0 Then on 0 6 s 6 ϕt∧ζ Xs − X0 = Z s µ1 (Xu )du + 0 Z s σ1 (Xu )dBu . (13) 0 Note that for 0 6 t < ∞, we have s ∈ [0, ϕζ ], P -a.s. From (13), and recall the definition of µ1 (.) and σ1 (.), we have the following SDE for X dXs = σ (Xs ) µ(Xs ) 1 ds + 1 dBs , b2 (Xs ) {s∈[0,ϕζ )} b(Xs ) {s∈[0,ϕζ )} X0 = V0 = v0 . This completes the proof of statement (i). Statement (ii) and (iii) are direct consequences of the stochastic representation Vt∧ζ = Xϕt∧ζ , P -a.s. in statement (i), because ϕt is an increasing function with respect to t. This completes the proof. ⊔ ⊓ Stochastic area of diffusion 7 3 Stochastic first passage area and moments In this section, we consider the two-sided exit time of the diffusion in (1) from an open interval (a, c) ⊂ J¯ such that a < v0 < c. Define τx = inf {t > 0 : Vt = x}, ¯ x ∈ J, (14) where inf ∅ = ∞ by convention. Now we recall some classical theory on diffusion exit times. Define the scale density of the diffusion V in (1)  Z s(x) := exp − x .  2µ(u) du , σ 2 (u) x ∈ J¯, (15) and the scale function is S (x) := Z x s(y )dy = . Z x .  Z exp − y .  2µ(u) du dy, σ 2 (u) x ∈ J¯. (16) The Laplace transforms of τa and τc of the two-sided exit problem for a diffusion process V were first solved by Darling and Siegert 1953. Consider the following SturmLiouville ordinary differential equation 1 2 σ (x)g ′′(x) + µ(x)g ′(x) = λg (x), 2 λ > 0, (17) and from classical diffusion theory it can be shown that (17) always has two independent, positive and convex solutions2 . Here we denote the decreasing solution as g−,λ (.), and the increasing solution as g+,λ (.). Based on this pair of solutions, define the auxiliary function fλ (y, z ) = g−,λ (y )g+,λ(z ) − g−,λ (z )g+,λ (y ), (18) We have the following lemma. Lemma 2 (Theorem 3.2 of Darling and Siegert 1953) With a < v0 < c, and λ > 0, we have the following Laplace transforms Ev0 [e−λτa ; τa < τc ] = fλ (v0 , c) , fλ (a, c) (19) Ev0 [e−λτc ; τc < τa ] = fλ (a, v0 ) . fλ (a, c) (20) and The following result gives the Laplace transform of the stochastic area till the first passage time. 2 See Borodin and Salminen 2002 for a collection of explicit examples of diffusions. 8 Zhenyu Cui Proposition 1 With a < v0 < c, and λ > 0, we have the following Laplace transforms Ev0 [e−λ Ev0 [e−λ Rτ a 0 Rτ c 0 b2 (Vs )ds ; τa < τc ] = fλ∗ (v0 , c) , fλ∗ (a, c) b2 (Vs )ds ; τc < τa ] = fλ∗ (a, v0 ) , fλ∗ (a, c) (21) and ∗ g−,λ (.) ∗ ∗ ∗ ∗ fλ∗ (y, z ) = g−,λ ( y ) g+ ,λ(z ) − g−,λ (z )g+,λ (y ), (22) ∗ where and g+,λ (.) are respectively the decreasing and increasing solutions of the following Sturm-Liouville type ordinary differential equation µ(x) ′ 1 σ 2 (x) ′′ g (x) + 2 g (x) = λg (x), 2 b2 ( x ) b (x) λ > 0, (23) Proof From Theorem 1 (i), we have Vt = XR t b2 (Vs )ds , P -a.s. on {0 6 t < ζ}. Define 0 τyX = inf {t > 0 : Xt = y}, (24) R τ then from Theorem 1 (iii), we have τyX = 0 y b2 (Vs )ds, P -a.s. Also notice the equivao n lence of events: {τa < τc } and τaX < τcX . Then we have i h Rτ 2 X a Ev0 e−λ 0 b (Vs )ds ; τa < τc = Ev0 [e−λτa ; τaX < τcX ]; Ev0 [e−λ Rτ c 0 b2 (Vs )ds X ; τc < τa ] = Ev0 [e−λτc ; τcX < τaX ], (25) and we have transformed the study of the stochastic area till the first passage time to a related problem of the first passage time of the diffusion X. Note that X is also a time-homogeneous diffusion with SDE given in (7). Then (25) combined with Lemma 2 completes the proof. ⊔ ⊓ Remark 1 Theorem 2.3 of Abundo 2013b provides the partial differential-difference equation(PDDE) that the Laplace transform of the stochastic area should satisfy. In the case of diffusions with no jumps, the formulation in (3.4) of Abundo 2013b includes outer conditions. Our approach is directly based on the classical diffusion theory developed in Darling and Siegert 1953, and the Laplace transform is expressed using the eigenfunctions of diffusions. He considers the case of the first passage time to the level 0, and we generalize it to a possibly non-zero constant here. Note that Abundo 2013a considers the case of first passage time to a possibly non-zero constant level, but the method still involves solving an associated PDDE with outer conditions(see Theorem 2.3, p5 of Abundo 2013a). In subsequent discussion, consider the two-sided exit time τ = τa ∧ τc , and the Rτ stochastic area Aτ := 0 b2 (Vs )ds. We aim to link the integer moments of the stochastic area to the moments of the two-sided exit time τ of the diffusion X with SDE in (7). Define the speed density of the diffusion V in (1) as m( x ) = 2 σ 2 ( x ) s( x ) , ¯ x ∈ J, (26) where s(.) is the scale density defined in (15). First recall the following lemma using our notation. Stochastic area of diffusion 9 Lemma 3 (Corollary 2.1, Wang and Yin 2008) Define µn (x) = Ex [τ n ], then there is the following recursive relation µn (x) = n +n S ( x ) − S ( a) S (c) − S (a) S (c) − S (x) S (c) − S (a) Z c Z xx a (S (c) − S (y ))µn−1(y )m(y )dy (S (y ) − S (a))µn−1 (y )m(y )dy, n = 1, 2, ... (27) and Ex [ τ ] = S ( x ) − S ( a) S (c) − S (a) Z c x (S (c) − S (y ))m(y )dy + S (c) − S (x) S (c) − S (a) Z x a (S (y ) − S (a))m(y )dy. (28) Remark 2 Letting a → −∞ in (27), one obtains the Siegert’s recursive formula3 for the moments of the first passage time of Vt through c. We have the following result for the integer moments of the stochastic area. Proposition 2 Define µ∗n (x) = Ex [ following recursive relation µ∗n (x) = n S ( x ) − S ( a) S (c) − S (a) S (c) − S (x) +n S (c) − S (a) Z c Z xx a Rτ 0 b2 (Vs )ds
n ] = Ex [(Aτ )n ], then there is the (S (c) − S (y ))µ∗n−1(y )m∗(y )dy (S (y ) − S (a))µ∗n−1(y )m∗ (y )dy, n = 1, 2, ... (29) and Ex [ A τ ] = S ( x ) − S ( a) S (c) − S (a) Z c x (S (c) − S (y ))m∗(y )dy + S (c) − S (x) S (c) − S (a) Z x a (S (y ) − S (a))m∗(y )dy, where m∗ ( x ) = 2 b2 ( x ) , σ 2 ( x ) s( x ) ¯ x ∈ J, (30) Proof From Theorem 1 (i), we have Vt = XR t b2 (Vs )ds , P -a.s. on {0 6 t < ζ}. Define 0 τ X = τaX ∧ τbX , then from Theorem 1 (iii), we have τ X = 2 µ(.)/b (.) Rτ 0 b2 (Vs )ds, P -a.s. Note that X and V share the same scale density since σ2 (.)/b2 (.) = σµ2((..)) , but the speed density for diffusion X is different and given as m∗ (x) in (30). This combined with Lemma 53 completes the proof. ⊔ ⊓ Remark 3 Abundo 2013b derives the recursive ODEs for the moments of Aτ in equation (3.5), p94 of his paper. Here we have derived explicit recursive relations of all the integer moments in terms of scale and speed densities, and do not need to solve an ODE. 3 equation (3.14) of Siegert 1951 10 Zhenyu Cui Example: case of the geometric Brownian motion in a new structural model of default The structural approach to credit risk modeling assumes that the default event happens when the total value of the firm’s asset first goes below the face value of its debt. The reduced-form approach assumes that default is modeled by an exogenouslydetermined intensity or compensator process, and the default time is the first jump time of a point process. Recently there is a third new approach by Yildirim 2006, which uses the information of the stochastic area under the firm’s asset value process. The default time is when the following two events both happen: the firm value process hits an default level and the stochastic area till this hitting time exceeds an exogenous level. In order to determine the probability of default for this new structural model, we need to determine the distribution of the stochastic area till the first passage time. If we model the firm’s asset value as a geometric Brownian motion, then we aim to calculate the Laplace transform of the stochastic passage area. Assume that Vt , t > 0 is a geometric Brownian motion with state space J = (0, ∞) dVt = µVt dt + σVt dWt , V0 = v0 ∈ J, (31) where µ 6= 0. Choose b2 (x) = x2 , and the SDE governing the diffusion X is dXt = µ dt + σdWt , Xt X0 = v0 . (32) (ν ) We recognize (32) as the SDE of a (scaled) standard Bessel process, and Xt = σRt , t > 0 where R(ν ) is a standard Bessel process with index ν = 2σµ2 − 1. For convenience, we assume that 2σµ2 > 1, thus ν > 0. From classical diffusion theory, the associated ODE (23) has two fundamental solutions(Proposition 6.2.3.1, p345 of Jeanblanc, Yor and Chesney 2009): √ √ ν ∗ ∗ 1− ν2 g+ ; g−,λ (x) = c2 K ν−2 (x 2λ)x1− 2 , x ∈ J¯, (33) ,λ (x) = c1 I ν−2 (x 2λ)x 2 2 with two constants c1 and c2 , where I (.) and K (.) are respectively the modified Bessel functions of the first and second kinds. Compute the auxiliary functions s(x) = c3 x−ν−1 , S (x) = −c3 x−ν , ν m∗ ( x ) = 1 2xν +1 c3 σ2 x ∈ J¯, (34) where c3 is a constant. From Proposition 1, we can compute √ √ √ √  v 1− ν2 K ν−2 (v0 2λ)I ν−2 (c 2λ) − K ν−2 (c 2λ)I ν−2 (v0 2λ) R 0 −λ 0τa b2 (Vs )ds 2 2 2 2 √ √ √ √ ; τa < τc ] = Ev0 [e , a K ν−2 (a 2λ)I ν−2 (c 2λ) − K ν−2 (c 2λ)I ν−2 (a 2λ) 2 and Ev0 [e −λ Rτ c 0 b2 (Vs )ds 2 √ ; τc < τa ] = 2 √ 2 √ √ √ √ √ . K ν−2 (a 2λ)I ν−2 (c 2λ) − K ν−2 (c 2λ)I ν−2 (a 2λ) 2 c 2 2 2 2 2 The first moment of the stochastic area is as follows. Ev0 Z τ 0  b2 (Vs )ds = √  v 1− ν2 K ν−2 (a 2λ)I ν−2 (v0 2λ) − K ν−2 (v0 2λ)I ν−2 (a 2λ) 0 νx2 (c−ν − a−ν ) − νa2 c2 (c−ν−2 − a−ν−2 ) − νx−ν (c2 − a2 ) , (c−ν − a−ν )σ 2 (ν + 2) and the other higher order moments can be similarly obtained from Proposition 2. 2 2 Stochastic area of diffusion 11 4 Stochastic occupation area and the Omega risk model Classical ruin theory assumes that the ruin will occur at the first time when the surplus of a company is negative. For a pointer to the literature in this area, please refer to Gerber and Shiu 1998. Recently, a new concept of ruin has been proposed and studied in a series of papers starting with Albrecher, Gerber and Shiu 2011. They coined the name “the Omega risk model", and within this model there is a distinction between ruin(negative surplus) and bankruptcy(going out of business). The company continues operation even with a period of negative surplus, and they introduce a bankruptcy rate function ω (x) with x denoting the value of negative surplus. ω (.) can be treated as an intensity of bankruptcy, and for x 6 0, ω (x)dt is the probability of bankruptcy within dt time units. Assume that the value of the company is modeled by a time-homogeneous diffusion Vt , t ∈ [0, ζ ) with SDE (1), and assume that the state space is J = (l, r ) with −∞ 6 l < r 6 ∞. Assume that the initial value of the company satisfies v0 > 0. If we introduce an auxiliary “bankruptcy monitoring" process N on the same probability space(with a possibly enlarged filtration), and assume that conditional on V , N follows a Poisson process with state-dependent intensity ω (Vt )1{Vt <0} , t > 0. Then we define the time of bankruptcy τω as the first arrival time of the Poisson process N , i.e. τω := inf  t>0: Z t 0 ω (Vs )1{Vs <0} ds > e1  (35) , where e1 is an independent exponential random variable with unit rate. Define eλ is another independent exponential random variable with rate λ. We can express the Laplace transform of the bankruptcy time as i h R eλ Ev0 [e−λτω ] = Pv0 (τω < eλ ) = 1 − Ev0 e− 0 ω(Vs )1{Vs <0} ds . (36) for λ > 0. Similar as in Gerber, Shiu and Yang 2012, define the (total) exposure as E := Z 0 ∞ ω (Vs )1{Vs <0} ds. (37) Let λ → 0+ in (44), we can calculate the probability of bankruptcy as h ψ (v0 ) = P (τω < ∞ | V0 = v0 ) = 1 − Ev0 e− R∞ 0 ω (Vs )1{Vs <0} ds i = 1 − Ev0 [e−E ], (38) In the literature, people have considered the following special cases(see Sec. 4, Gerber, Shiu and Yang 2012): (i) ω (x) = c with a constant c ; (ii)4 ω (x) = ηk , if ck−1 < x < ck , k = 1, 2, ..., n for the constants c0 = −∞ < c1 < ... < cn−1 < cn = 0; (iii)5 ω (x) = −ηx, x < 0 for some η > 0. Intuitively, (37) represents the “stochastic occupation area", and it measures the area swept by the sample path of V that lies under the level zero. This motivates us to study the general case and we assume an arbitrary bankruptcy rate function such 4 The assumption of ω(x) being a piecewise constant is also employed in Li and Zhou 2013. In this case, Gerber, Shiu and Yang 2012 manage to express the probability of bankruptcy in terms of Airy functions, but this is possible only in their setting of modeling firm value V as an arithmetic Brownian motion with drift. 5 12 Zhenyu Cui that ω (x) > 0, x 6 0, ω (x) = 0, x > 0 and ω (.) is a decreasing function. From (38), to calculate the probability of bankruptcy, we aim to calculate the Laplace transform of the total exposure E . Based on the solutions to the ODE in (17), define a pair of Laplace exponents for λ>0 ψλ± (x) = ± ′ g±,λ (x) g±,λ (x) ¯ x ∈ J, , (39) From properties of the solutions to the ODE (17), we have(see Li and Zhou 2013) s( x ) ψ0− (x) = R ∞ , s(y )dy x s( x ) , s(y )dy −∞ ψ0+ (x) = R x and (40) Recall the following result from Li and Zhou 2013. Lemma 4 (Corollary 3.2 of Li and Zhou 2013) For v0 > 0 i g (v ) h R eδ −,δ 0 Ev0 e−λ 0 1{Vs <0} ds = g−,δ (0) + − δ δ +λ ψδ +λ (0) + ψδ (0) + − ψδ+λ (0) + ψδ (0) +1− g−,δ (v0 ) , g−,δ (0) (41) and for v0 6 0, i g h R eδ +,δ +λ (v0 ) Ev0 e−λ 0 1{Vs <0} ds = g+,δ+λ (0) + − δ δ +λ ψδ +λ (0) + ψδ (0) ψδ++λ (0) + ψδ− (0) +   g+,δ+λ (v0 ) 1− . δ+λ g+,δ+λ (0) δ (42) Recall from Li and Zhou 2013 the following property of the function g−,δ (x) as δ → 0+. g−,0 (x) = 1, g−,0 (x) = Z if ∞ Z ∞ x s(y )dy = ∞; s(y )dy, if x Z ∞ x s(y )dy < ∞. (43) Now by taking δ → 0+ in the above (41) and (42), we have Lemma 5 For v0 > 0, if6 S (∞) < ∞, then h Ev0 e −λ R∞ and if S (∞) = ∞, then 0 1{Vs <0} ds i R∞ 0 = 1 − Rv∞ 0 i h R∞ Ev0 e−λ 0 1{Vs <0} ds = s(y )dy ψλ+ (0) s(y )dy ψλ+ (0) + ψ0− (0) ψ0− (0) ψλ (0) + ψ0− (0) + (44) , (45) , For v0 < 0, i g (v ) h R∞ ψ0− (0) +,λ 0 Ev0 e−λ 0 1{Vs <0} ds = . g+,λ (0) ψλ+ (0) + ψ0− (0) 6 Since s(y) > 0 on the compact interval [0, v0 ], v0 > 0 (or [v0 , 0], v0 < 0), R is equivalent to S(∞) = 0∞ s(y)dy < ∞. Similar for the other case. (46) R∞ v0 s(y)dy < ∞ Stochastic area of diffusion 13 For the diffusion X with SDE in (7), from the solutions to the associated SturmLiouville ODE in (23), we define a pair of Laplace exponents for λ > 0 ′,∗ ψλ±,∗ (x) = ± g±,λ (x) ∗ (x) g±,λ , ¯ x ∈ J, (47) From properties of the solutions to the ODE (23), we have s( x ) , ψ0−,∗ (x) = R ∞ x s(y )dy s( x ) , −∞ s(y )dy ψ0+,∗ (x) = R x and (48) and note that ψ0±,∗ (x) = ψ0± (x) because the diffusions V and X share the same scale density s(.). However, ψλ±,∗ (x) 6= ψλ± (x), λ > 0 in general. The following result gives the Laplace transform of the total occupation area. Proposition 3 For v0 > 0, if S (∞) < ∞, then h Ev0 e −λ R∞ 0 b2 (Vs )1{Vs <0} ds and if S (∞) = ∞, then i R∞ s(y )dy 0 = 1 − Rv∞ +,∗ s(y )dy ψλ (0) + ψ0−,∗ (0) 0 i h R∞ 2 Ev0 e−λ 0 b (Vs )1{Vs <0} ds = ψλ+,∗ (0) ψ0−,∗ (0) ψλ+,∗ (0) + ψ0−,∗ (0) , (49) (50) , For v0 6 0, i g ∗ (v0 ) h R∞ 2 ψ0−,∗ (0) +,λ . Ev0 e−λ 0 b (Vs )1{Vs <0} ds = ∗ + ,∗ g+,λ (0) ψ (0) + ψ0−,∗ (0) (51) λ Proof From Theorem 1 (i), we have Vt = XR t b2 (Vs )ds = Xϕt , P -a.s. on {0 6 t < ζ}. 0 Since we assume that l and r are absorbing boundaries, thus it is understood that R∞ Rt 2 Rζ 2 b (Vs )ds = 0 b (Vs )ds, P -a.s. on {∞ > t > ζ}. Thus we have 0 b2 (Vs )1{Vs <0} ds = 0 Rζ b2 (Vs )1{Vs <0} ds, P -a.s. Apply the change of variables formula similar as Problem 0 3.4.5 (vi), p174 of Karatzas and Shreve 1991, we have Z ζ 0 b2 (Vs )1{Vs <0} ds = Z ζ 0 1{Xϕs <0} dϕs = Z ϕζ 0 1{Xu <0} du = Z ζX 0 1{Xu <0} du, P-a.s (52) and the last equality is due to Theorem 1 (ii). This combined with Lemma 5 applied to the diffusion X completes the proof. ⊔ ⊓ Remark 4 Take the bankruptcy rate function as ω (x) = b2 (x), x < 0, then from (38), we can obtain the probability of bankruptcy in terms of auxiliary functions defined as combinations of solutions to the Sturm-Liouville ODE in (23). 14 Zhenyu Cui An explicit example with a general bankruptcy rate function: Now we show an example where we can explicitly compute the probability of bankruptcy. Assume that the company value is modeled as the following SDE with state space J = (−∞, ∞) dVt = µVt2 dt + Vt dWt , V0 = v0 ∈ J, (53) and µ 6= 0. Proposition 4 For the diffusion V in (53) with the bankruptcy rate function7 as ω (x) = x2 , x < 0 and ω (x) = 0, x > 0, we have that the probability of bankruptcy is given by √ µ2 +2−µ −2µv0   e , if v0 > 0, µ > 0 √ 2 ψ (v0 ) =  µ +2+µ e−2µv0 , 1 − √ 22µ  µ +2+µ   1, (54) if v0 6 0, µ > 0 µ<0 Remark 5 For the company value modeled by the diffusion V in (53), from the above result we can see that it will eventually go bankrupt with probability 1 if µ < 0. For µ > 0, we can explicitly determine the probability of bankruptcy. Proof From Theorem 1, the SDE governing the diffusion X is dXt = µdt + dWt , X0 = v0 . (55) or equivalently Xt = v0 + Wt + µt. From classical diffusion theory, the associated ODE (23) has two fundamental solutions ± ∗ g±,λ ( x ) = eβλ , where βλ± = −µ ± p ¯ x ∈ J, (56) µ2 + 2λ. We can also compute S (x) = 1 − e−2µx ; 2µ ±,∗ ψλ (x) = ±βλ± . (57) For v0 > 0, if µ > 0, we have S (∞) < ∞. Take λ = 1, then from Proposition 3 i h R∞ 2 Ev0 e− 0 b (Vs )1{Vs <0} ds = 1 − e−2µv0 ψ1+,∗ (0) ψ1+,∗ (0) + −,∗ ψ0 (0) = 1− (−µ + p µ+ µ2 + 2)e−2µv0 p µ2 + 2 (58) If µ 6 0, we have S (∞) = ∞, then from Proposition 3 i h R∞ 2 Ev0 e− 0 b (Vs )1{Vs <0} ds = ψ0−,∗ (0) ψ1+,∗ (0) + ψ0−,∗ (0) = 0. (59) 7 It is clear that ω(.) is non-negative and decreasing when x < 0, which comply with practical applications. In our notation, we shall have b2 (x) = ω(x) = x2 . , Stochastic area of diffusion 15 For v0 6 0, if µ > 0, we have S (∞) < ∞ and i g ∗ (v0 ) h R∞ 2 ψ0−,∗ (0) 2µe−2µv0 +,λ p . = Ev0 e− 0 b (Vs )1{Vs <0} ds = ∗ g+,λ (0) ψ1+,∗ (0) + ψ0−,∗ (0) µ + µ2 + 2 (60) If µ 6 0, we have S (∞) = ∞ and i h R∞ 2 Ev0 e− 0 b (Vs )1{Vs <0} ds = 0. (61) ⊔ ⊓ The above calculations combined with (38) completes the proof. 5 Stochastic drawdown area and risk model with tax Assume that the stock price is modeled by a regular time-homogeneous diffusion given in (1). Denote Mt = sup06u6t Vu as the running maximum of the stock price. Define the first drawdown time of a units as τDD = inf {t > 0 : Mt − Vt > a}. (62) This stopping time is of importance in modeling stock prices during downturn of the financial market(e.g. the 2008 financial crisis), and is an inherent constraint in some portfolio optimization problems. The drawdown constraint allows us to encode risk attitudes into the portfolio optimization problem, and is of both practical and theoretical interests. It was first introduced by Grossman and Zhou 1993 in a continuous-time framework , and studied by Cvitanic and Karatzas 1995 and Cherny and Obloj 2013. The seminal paper Lehoczky 1977 provides a closed-form expression for the joint Laplace transform of the first drawdown time and the running maximum stopped at the first drawdown time. Lemma 6 (equation (4), p602 of Lehoczky 1977) The joint Laplace transform of τDD and MτDD is E [e−αMτDD −βτDD ] = Z ∞ e−αu− Ru 0 d(z )dz c(u)du. (63) 0 for α, β > 0, where d( z ) = c(x) = g (z − a)h′ (z ) − h(z − a)g ′ (z ) ; g (z − a)h(z ) − g (z )h(z − a) g (x)h′(x) − g ′ (x)h(x) . g (x − a)h(x) − g (x)h(x − a) (64) and here g (.) and h(.) are two independent solutions to the following Sturm-Liouville ODE associated with diffusion V . 1 2 σ (x)f ′′ (x) + µ(x)f ′ (x) = βf (x), 2 x ∈ [−a, ∞). (65) Rτ Denote the stochastic area till the first drawdown time of a units as 0 DD b2 (Vs )ds. Rτ Similar as above, we can derive the joint Laplace transform of MτDD and 0 DD b2 (Vs )ds. 16 Zhenyu Cui Proposition 5 The joint Laplace transform of MτDD and E [e −αMτDD −β R τDD 0 b2 (Vs )ds ]= Z ∞ e−αx− Rx 0 R τDD 0 b2 (Vs )ds is d∗ (z )dz ∗ c (x)dx. (66) 0 for α, β > 0, where d∗ ( z ) = c∗ (x) = g ∗ (z − a)h′∗ (z ) − h∗ (z − a)g ′∗(z ) ; g ∗ (z − a)h∗ (z ) − g ∗ (z )h∗ (z − a) g ∗ (x)h′∗ (x) − g ′∗ (x)h∗ (x) . − a) g ∗ (x − a)h∗ (x) − g ∗ (x)h∗ (x (67) and here g ∗ (.) and h∗ (.) are any two independent solutions to the following ODE µ(x) ′ 1 σ 2 (x) ′′ f (x) + 2 f (x) = βf (x), 2 b2 ( x ) b (x) x ∈ [−a, ∞). (68) Proof From Theorem 1 (i), we have Vt = XR t b2 (Vs )ds , P -a.s. on {0 6 t < ζ}. Define 0 X τDD = Rinf {t > 0 : max06u6t Xt − Xt > a}, then from Theorem 1 (iii), we have τ X τDD = 0 DD b2 (Vs )ds, P -a.s., and max06u6τDD Vu = max06u6τ X Xu , P -a.s. This DD combined with Lemma 6 completes the proof. ⊔ ⊓ Azema and Yor 1979 introduced a family of simple local martingales and proposed a solution to the Skorokhod embedding problem. These processes are later named AzemaYor processes and the associated passage time is called the Azema-Yor stopping time. Their applications range from solving the Skorokhod embedding problem(Obloj 2004), pricing capped Russian options(Ott 2013) and portfolio optimization with drawdown constraints(El Karoui and Meziou 2006) Consider the diffusion V as defined in (1) with initial value v0 > 0, and define the running maximum process associated with V as Mt = ( max Vu ) ∨ s. 06u6t (69) started at s > v0 > 0. Define the Azema-Yor stopping time as τAY = inf {t > 0 : Vt 6 g (Mt )}, (70) for any continuous function g defined on [0, ∞) satisfying 0 < g (x) < x for x > 0. In the following we show another application of the Azema-Yor stopping time in a diffusion risk model where we assume that there is a loss-carry-forward taxation. It was introduced into the risk theory by Albrecher and Hipp 2007 in the Levy insurance model framework. The basic idea is to allow the tax to be paid at a certain fixed rate immediately when the surplus of the company is at a running maximum. For a pointer to the literature in this area, please refer to Albrecher, Renaud and Zhou 2008 and the references therein. We cast our model in a regular time-homogeneous diffusion setting(Li, Tang and Zhou 2013). Assume that the value of the firm is modeled by the diffusion V in (1). Now introduce a surplus-dependent tax rate: whenever the process Vt coincides with Stochastic area of diffusion 17 its running maximum Mt , the firm pays tax at rate γ (Mt ) and γ (.) : [v0 , ∞) → [0, 1) is a measurable function. The value process after taxation satisfies dUt = dVt − γ (Mt )dMt , t > 0, (71) with U0 = V0 = v0 . For a default threshold a(conventionally assigned 0 in the ruin theory), define the time of default with tax as T U (a) = inf {t > 0 : Ut = a}, (72) and inf ∅ = ∞ by convention. Note that a < v0 . Now we want to compute E[T U (a)], which represents the expected time of ruin. Introduce the following function γ̄ (x) = x − Z x γ (z )dz = v0 + v0 Z x v0 (1 − γ (z ))dz, (73) x > v0 . Notice that v0 < γ̄ (x) 6 x. We have the following representation Ut = Vt −Mt +γ̄ (Mt ), then we have T U (a) = inf {t > 0 : Ut = a} = inf {t > 0, Vt = g (Mt )}, where g (x) = x − γ̄ (x) + a = Z (74) x γ (z )dz + a. (75) v0 We have that a 6 g (x) < x − v0 + a < x because γ (.) : [v0 , ∞) → [0, 1). Thus we can see that equation (74) represents an Azema-Yor stopping time. Our objective is to calculate the expectation of the expected time of ruin T U (a) R T U ( a) and the stochastic area till ruin 0 b2 (Vs )ds. We first recall a lemma. Lemma 7 (Theorem 4.1 of Pedersen and Peskir 1998) Here s > v0 is the initial value of the running maximum process M . If g (s) < v0 6 s, then E[τAY ] = 2 + Z S (s) − S (v0 ) S (s) − S (g (s)) ∞ s Z v0 g(s) s( t) S (t) − S (g (t)) S (t) − S (g (s)) S (v0 ) − S (g (s)) dt + 2 σ 2 ( t) s( t) S (s) − S (g (s)) Z t g(t) S (r ) − S (g (t)) dr σ 2 ( r ) s( r ) !  Z exp − t s Z s v0 S ( s) − S ( t) dt σ 2 ( t) s( t)  s( r ) dr dt S (r ) − S (g (r )) (76) ) . and E[τAY ] = 0 for 0 < v0 6 g (s). For the diffusion risk model with tax, we have the following result on the expected value of the ruin time and the stochastic area till ruin. Proposition 6 (Expected Ruin Time and Ruin Area with Tax) With g (.) defined in (75) (i) The expected time of ruin with tax is U E[T (a)] = 2 Z ∞ v0 s( t) S (t) − S (g (t)) Z t g(t) S (r ) − S (g (t)) dr σ 2 ( r ) s( r ) !  Z exp − t v0  s( r ) dr dt. S (r ) − S (g (r )) (77) 18 Zhenyu Cui (ii) The expected stochastic area till the ruin time with tax is E "Z =2 T U ( a) 2 b (Vs )ds 0 Z ∞ v0 # s( t) S (t) − S (g (t)) Z t g(t) b2 (r )(S (r ) − S (g (t))) dr σ 2 ( r ) s( r ) !  Z exp − t v0  s( r ) dr dt. S (r ) − S (g (r )) (78) Proof For (i), note that U0 = v0 means that we consider the case when the running maximum starts at the same initial value of the process V , i.e. s = v0 in (69). Taking s = v0 in (76), we arrive at the expression (77). For (ii), from Theorem 1 (i), we have Vt = XR t b2 (Vs )ds , P -a.s. on {0 6 t < ζ}. 0 U Define TX (a) = inf {t > 0, Xt 6 g (MtX )}, then from Theorem 1 (iii), we have R T U ( a) U TX ( a) = 0 b2 (Vs )ds, P -a.s., and max06u6T U (a) Vu = max06u6T U (a) Xu , P X a.s. This combined with part (i) completes the proof. ⊔ ⊓ 6 Conclusion and future research In this paper we have studied the stochastic area of a time-homogeneous diffusion till a stopping time. Through stochastic time change we explicitly express the Laplace transform of the stochastic area in terms of eigenfunctions of an associated diffusion. We also obtain the integer moments of the stochastic area explicitly in terms of scale and speed densities, and generalize the work of Abundo 2013a 2013b in the case of time-homogeneous diffusions, since his approach requires solving a partial differentialdifference equation with outer conditions. 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