Convergence of hitting times for jump-diffusion processes
aa r X i v : . [ m a t h . P R ] O c t Modern Stochastics: Theory and Applications 2 (2015) 203–218DOI: 10.15559/15-VMSTA32
Convergence of hitting times for jump-diffusionprocesses
Georgiy Shevchenko
Taras Shevchenko National University of Kyiv, Mechanics and Mathematics Faculty,Volodymyrska 64, 01601 Kyiv, Ukraine [email protected] (G. Shevchenko)Received: 16 February 2015, Revised: 7 September 2015, Accepted: 7 September 2015,Published online: 23 September 2015
Abstract
We investigate the convergence of hitting times for jump-diffusion processes. Spe-cifically, we study a sequence of stochastic differential equations with jumps. Under reasonableassumptions, we establish the convergence of solutions to the equations and of the momentswhen the solutions hit certain sets.
Keywords
Stochastic differential equation, Poisson measure, jump-diffusion process,stopping time, convergence
In this article, we consider a sequence of stochastic differential equations with jumps X n ( t ) = X n (0) + Z t a n (cid:0) s, X n ( s ) (cid:1) ds + Z t b n (cid:0) s, X n ( s ) (cid:1) dW ( s )+ Z t Z R m c n (cid:0) s, X n ( s − ) , θ (cid:1)e ν ( dθ, ds ) , t ≥ , n ≥ . Here W is a standard Wiener process, e ν is a compensated Poisson random measure,and X n (0) is nonrandom (see Section 2 for precise assumptions). Assuming that a n → a , b n → b , c n → c , and X n (0) → X (0) as n → ∞ in an appropriatesense, we are interested in convergence of hitting times τ n → τ , n → ∞ , where τ n = inf (cid:8) t ≥ ϕ n (cid:0) t, X n ( t ) (cid:1) ≥ (cid:9) G. Shevchenko is the first time when the process X n hits the set G nt = { x : ϕ n ( t, x ) ≥ } .The study is motivated by the following observation. Jump-diffusion processesare commonly used to model prices of financial assets. When the parameters of ajump-diffusion process are estimated with the help of statistical methods, there is anestimation error. Thus, it is natural to investigate whether the optimal exercise strate-gies are close for two jump-diffusion processes with close parameters. Moreover, weshould study particular hitting times since, in the Markovian setting, the optimal stop-ping time is the hitting time of the optimal stopping set.There is a lot of literature devoted to jump-diffusion processes and their appli-cations in finance. The book [1] gives an extensive list of references on the subject.The convergence of stopping times for diffusion and jump-diffusion processes wasstudied in [2, 3, 6]. All these papers are devoted to the one-dimensional case, andthe techniques are different from ours. Here we generalize these results to the multi-dimensional case and also relax the assumptions on the convergence of coefficients.As an auxiliary result of independent interest, we prove the convergence of solutionsunder very mild assumptions on the convergence of coefficients. Let ( Ω, F , F , P ) be a standard stochastic basis with filtration F = {F t , t ≥ } satisfying the usual assumptions. Let { W ( t ) = ( W ( t ) , . . . , W k ( t )) , t ≥ } be astandard Wiener process in R k , and ν ( dθ, dt ) be a Poisson random measure on R m × [0 , ∞ ) . We assume that W and ν are compatible with the filtration F , that is, for any t > s ≥ and any A ∈ B ( R m ) and B ∈ B ([ s, t ]) , the increment W ( t ) − W ( s ) andthe value ν ( A × B ) are F t -measurable and independent of F s .Assume in addition that ν ( dθ, dt ) is homogeneous, that is, for all A ∈ B ( R m ) and B ∈ B ([0 , ∞ )) , E [ ν ( A × B )] = µ ( A ) λ ( B ) , where λ is the Lebesgue measure, µ is a σ -finite measure on R m having no atom at zero. Denote by e ν the correspondingcompensated measure, that is, e ν ( A × B ) = ν ( A × B ) − µ ( A ) λ ( B ) for all A ∈B ( R m ) , B ∈ B ([0 , ∞ )) .For each integer n ≥ , consider a stochastic differential equation in R d X ni ( t ) = X ni (0) + Z t a ni (cid:0) s, X n ( s ) (cid:1) ds + k X j =1 Z t b nij (cid:0) s, X n ( s ) (cid:1) dW j ( s )+ Z t Z R m c ni (cid:0) s, X n ( s − ) , θ (cid:1)e ν ( dθ, ds ) , t ≥ , i = 1 , . . . , d. (1)In this equation, the initial condition X n (0) ∈ R d is nonrandom, and the coefficients a ni , b nij : [0 , ∞ ) × R d → R , c ni : [0 , ∞ ) × R d × R m → R , i = 1 , . . . , d , j = 1 , . . . , k ,are nonrandom and measurable.In what follows, we abbreviate Eq. (1) as X n ( t ) = X n (0) + Z t a n (cid:0) s, X n ( s ) (cid:1) ds + Z t b n (cid:0) s, X n ( s ) (cid:1) dW ( s )+ Z t Z R m c n (cid:0) s, X n ( s − ) , θ (cid:1)e ν ( dθ, ds ) , t ≥ . (2) onvergence of hitting times for jump-diffusion processes For the rest of the article, we adhere to the following notation. By | · | we denotethe absolute value of a number, the norm of a vector, or the operator norm of a matrix,and by ( x, y ) the scalar product of vectors x and y ; B k ( r ) = { x ∈ R k : | x | ≤ r } .The symbol C means a generic constant whose value is not important and may changefrom line to line; a constant dependent on parameters a, b, c, . . . will be denoted by C a,b,c,... .The following assumptions guarantee that Eq. (2) has a unique strong solution.(A1) For all n ≥ , T > , t ∈ [0 , T ] , x ∈ R d , (cid:12)(cid:12) a n ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) b n ( t, x ) (cid:12)(cid:12) + Z R m (cid:12)(cid:12) c n ( t, x, θ ) (cid:12)(cid:12) µ ( dθ ) ≤ C T (cid:0) | x | (cid:1) . (A2) For all n ≥ , T ≥ , t ∈ [0 , T ] , R > , and x, y ∈ B d ( R ) (cid:12)(cid:12) a n ( t, x ) − a n ( t, y ) (cid:12)(cid:12) + (cid:12)(cid:12) b n ( t, x ) − b n ( t, y ) (cid:12)(cid:12) + Z R m (cid:12)(cid:12) c n ( t, x, θ ) − c n ( t, y, θ ) (cid:12)(cid:12) µ ( dθ ) ≤ C T,R | x − y | . Moreover, under these assumptions, for any T ≥ , we have the following esti-mate: E h sup t ∈ [0 ,T ] (cid:12)(cid:12) X n ( t ) (cid:12)(cid:12) i ≤ C T (cid:0) (cid:12)(cid:12) X n (0) (cid:12)(cid:12) (cid:1) (3)(see, e.g., [5, Section 3.1]). From this estimate it is easy to see from Eq. (2) that forall t, s ∈ [0 , T ] , E (cid:2)(cid:12)(cid:12) X n ( t ) − X n ( s ) (cid:12)(cid:12) (cid:3) ≤ C T (cid:0) (cid:12)(cid:12) X n (0) (cid:12)(cid:12) (cid:1) | t − s | . (4)Now we state the assumptions on the convergence of coefficients of (2).(C1) For all t ≥ and x ∈ R d , a n ( t, x ) → a ( t, x ) , b n ( t, x ) → b ( t, x ) , Z R m (cid:12)(cid:12) c n ( t, x, θ ) − c ( t, x, θ ) (cid:12)(cid:12) µ ( dθ ) → , n → ∞ . (C2) X n (0) → X (0) , n → ∞ . First, we establish a result on convergence of solutions to stochastic differential equa-tions.
Theorem 3.1.
Let the coefficients of Eq. (2) satisfy assumptions (A1), (A2), (C1),and (C2). Then, for any
T > , we have the convergence in probability sup t ∈ [0 ,T ] (cid:12)(cid:12) X n ( t ) − X ( t ) (cid:12)(cid:12) P −→ , n → ∞ . G. Shevchenko
If additionally the constant in assumption (A2) is independent of R , then for any T > , E h sup t ∈ [0 ,T ] (cid:12)(cid:12) X n ( t ) − X ( t ) (cid:12)(cid:12) i → , n → ∞ . Proof.
Denote ∆ n ( t ) = sup s ∈ [0 ,t ] | X n ( t ) − X ( t ) | , a n,ms = a n ( s, X m ( s )) , b n,ms = b n ( s, X m ( s )) , c n,ms ( θ ) = c n ( s, X m ( s − ) , θ ) , I na ( t ) = Z t a n,ns ds, I nb ( t ) = Z t b n,ns dW ( s ) ,I nc ( t ) = Z t Z R m c n,ns ( θ )˜ ν ( dθ, ds ) . It is easy to see that I nb and I nc are martingales.Write ∆ n ( t ) ≤ C (cid:16)(cid:12)(cid:12) X n (0) − X (0) (cid:12)(cid:12) + sup s ∈ [0 ,t ] (cid:12)(cid:12) I na ( s ) − I a ( s ) (cid:12)(cid:12) + sup s ∈ [0 ,t ] (cid:12)(cid:12) I nb ( s ) − I b ( s ) (cid:12)(cid:12) + sup s ∈ [0 ,t ] (cid:12)(cid:12) I nc ( s ) − I c ( s ) (cid:12)(cid:12) (cid:17) . For N ≥ , define σ nN = inf (cid:8) t ≥ (cid:12)(cid:12) X ( t ) (cid:12)(cid:12) ∨ (cid:12)(cid:12) X n ( t ) (cid:12)(cid:12) ≥ N (cid:9) and denote t = t ≤ σ nN . Then E (cid:2) ∆ n ( t ) t (cid:3) ≤ E (cid:2) ∆ n (cid:0) t ∧ σ nN (cid:1) (cid:3) ≤ C (cid:18)(cid:12)(cid:12) X n (0) − X (0) (cid:12)(cid:12) + X x ∈{ a,b,c } E h sup s ∈ [0 ,t ∧ σ nN ] (cid:12)(cid:12) I nx ( s ) − I x ( s ) (cid:12)(cid:12) i(cid:19) . We estimate E h sup s ∈ [0 ,t ∧ σ nN ] (cid:12)(cid:12) I na ( s ) − I a ( s ) (cid:12)(cid:12) i ≤ E " sup s ∈ [0 ,t ] Z s (cid:12)(cid:12) a n,nu − a , u (cid:12)(cid:12) u du ! ≤ E " Z t (cid:12)(cid:12) a n,nu − a , u (cid:12)(cid:12) u du ! ≤ t Z t E (cid:2)(cid:12)(cid:12) a n,nu − a , u (cid:12)(cid:12) u (cid:3) du ≤ C t Z t (cid:0) E (cid:2)(cid:12)(cid:12) a n,nu − a n, u (cid:12)(cid:12) u (cid:3) + E (cid:2)(cid:12)(cid:12) a n, u − a , u (cid:12)(cid:12) u (cid:3)(cid:1) du. (5)In turn, Z t E (cid:2)(cid:12)(cid:12) a n,nu − a n, u (cid:12)(cid:12) u (cid:3) du = Z t E (cid:2)(cid:12)(cid:12) a n (cid:0) u, X n ( u ) (cid:1) − a n (cid:0) u, X ( u ) (cid:1)(cid:12)(cid:12) u (cid:3) du ≤ C N,t Z t E (cid:2)(cid:12)(cid:12) X n ( u ) − X n (0) (cid:12)(cid:12) u (cid:3) du ≤ C N,t Z t E (cid:2) ∆ n ( u ) u (cid:3) du. onvergence of hitting times for jump-diffusion processes By the Doob inequality and Itô isometry we obtain E h sup s ∈ [0 ,t ∧ σ nN ] (cid:12)(cid:12) I nb ( s ) − I b ( s ) (cid:12)(cid:12) i ≤ C E (cid:2)(cid:12)(cid:12) I nb (cid:0) t ∧ σ nN (cid:1) − I b (cid:0) t ∧ σ nN (cid:1)(cid:12)(cid:12) (cid:3) = C Z t E (cid:2)(cid:12)(cid:12) b n,ns − b , s (cid:12)(cid:12) s (cid:3) ds. Estimating as in (5), we arrive at Z t E (cid:2)(cid:12)(cid:12) b n,ns − b , s (cid:12)(cid:12) s (cid:3) ds ≤ C N,t Z t E (cid:2) ∆ n ( s ) s (cid:3) ds + C Z t E (cid:2)(cid:12)(cid:12) b n, s − b , s (cid:12)(cid:12) s (cid:3) ds. Finally, the Doob inequality yields E h sup s ∈ [0 ,t ∧ σ nN ] (cid:12)(cid:12) I nc ( s ) − I c ( s ) (cid:12)(cid:12) i ≤ C E (cid:2)(cid:12)(cid:12) I nc (cid:0) t ∧ σ nN (cid:1) − I c (cid:0) t ∧ σ nN (cid:1)(cid:12)(cid:12) (cid:3) = C Z t Z R m E (cid:2)(cid:12)(cid:12) c n,ns ( θ ) − c , s ( θ ) (cid:12)(cid:12) s (cid:3) µ ( dθ ) ds ≤ C Z t Z R m (cid:0) E (cid:2)(cid:12)(cid:12) c n,ns ( θ ) − c n, s ( θ ) (cid:12)(cid:12) s (cid:3) + E (cid:2)(cid:12)(cid:12) c n, s ( θ ) − c , s ( θ ) (cid:12)(cid:12) s (cid:3)(cid:1) µ ( dθ ) ds. By (A2) we have C Z t Z R m E (cid:2)(cid:12)(cid:12) c n,ns ( θ ) − c n, s ( θ ) (cid:12)(cid:12) s (cid:3) µ ( dθ ) ds ≤ C N,t Z t E (cid:2)(cid:12)(cid:12) X n ( s ) − X ( s ) (cid:12)(cid:12) s (cid:3) ds ≤ C N,t Z t E (cid:2) ∆ n ( s ) s (cid:3) ds. Collecting all estimates, we arrive at the estimate E (cid:2) ∆ n ( t ) t (cid:3) ≤ C (cid:12)(cid:12) X n (0) − X (0) (cid:12)(cid:12) + C N,t Z t E (cid:2) ∆ n ( s )1 s (cid:3) ds + C t Z t E (cid:2)(cid:12)(cid:12) ˜ a n, s − ˜ a , s (cid:12)(cid:12) s (cid:3) ds + C Z t E (cid:2)(cid:12)(cid:12) b n, s − b , s (cid:12)(cid:12) s (cid:3) ds + C Z t Z R m E (cid:2)(cid:12)(cid:12) c n, s ( θ ) − c , s ( θ ) (cid:12)(cid:12) s (cid:3) µ ( dθ ) ds, where we can assume without loss of generality that the constants are nondecreasingin t . The application of the Gronwall lemma leads to G. Shevchenko E (cid:2) ∆ n ( T ) T (cid:3) ≤ C N,T (cid:12)(cid:12) X n (0) − X (0) (cid:12)(cid:12) + Z T E (cid:2)(cid:12)(cid:12) ˜ a n, s − ˜ a , s (cid:12)(cid:12) s (cid:3) ds + Z T E (cid:2)(cid:12)(cid:12) b n, s − b , s (cid:12)(cid:12) s (cid:3) ds + Z T Z R m E (cid:2)(cid:12)(cid:12) c n, s ( θ ) − c , s ( θ ) (cid:12)(cid:12) s (cid:3) µ ( dθ ) ds ! . We claim that the right-hand side of the latter inequality vanishes as n → ∞ . Indeed,the integrands are bounded by C T (1 + | X ( s ) | ) due to (A1) and vanish pointwise dueto (C1). Hence, the convergence of integrals follows from the dominated convergencetheorem. The first term vanishes due to (C2); thus, E (cid:2) ∆ n ( T ) T (cid:3) → , n → ∞ . Now to prove the first statement, for any ε > , write P (cid:0) ∆ n ( T ) > ε (cid:1) ≤ ε E (cid:2) ∆ n ( T ) T (cid:3) + P (cid:0) σ nN < T (cid:1) ≤ ε E (cid:2) ∆ n ( T ) T (cid:3) + P (cid:16) sup t ∈ [0 ,T ] (cid:12)(cid:12) X n (0) (cid:12)(cid:12) ≥ N (cid:17) + P (cid:16) sup t ∈ [0 ,T ] (cid:12)(cid:12) X (0) (cid:12)(cid:12) ≥ N (cid:17) . This implies lim n →∞ P (cid:0) ∆ n ( T ) > ε (cid:1) ≤ n ≥ P (cid:16) sup t ∈ [0 ,T ] (cid:12)(cid:12) X n (0) (cid:12)(cid:12) ≥ N (cid:17) . By the Chebyshev inequality we have lim n →∞ P (cid:0) ∆ n ( T ) > ε (cid:1) ≤ N sup n ≥ E h sup t ∈ [0 ,T ] (cid:12)(cid:12) X n (0) (cid:12)(cid:12) i . Therefore, using (3) and letting N → ∞ , we get lim n →∞ P (cid:0) ∆ n ( T ) > ε (cid:1) = 0 , as desired.In order to prove the second statement, we repeat the previous arguments with σ nN ≡ T , getting the estimate E (cid:2) ∆ n ( T ) (cid:3) ≤ C T (cid:12)(cid:12) X n (0) − X (0) (cid:12)(cid:12) + Z T E (cid:2)(cid:12)(cid:12) ˜ a n, s − ˜ a , s (cid:12)(cid:12) (cid:3) ds + Z T E (cid:2)(cid:12)(cid:12) b n, s − b , s (cid:12)(cid:12) (cid:3) ds + Z T Z R m E (cid:2)(cid:12)(cid:12) c n, s ( θ ) − c , s ( θ ) (cid:12)(cid:12) (cid:3) µ ( dθ ) ds ! . Hence, we get the required convergence as before, using the dominated convergencetheorem. onvergence of hitting times for jump-diffusion processes
For each n ≥ , define the stopping time τ n = inf (cid:8) t ≥ ϕ n (cid:0) t, X n ( t ) (cid:1) ≥ (cid:9) (6)with the convention inf ∅ = + ∞ ; ϕ n is a function satisfying certain assumptions tobe specified later. In this section, we study the convergence τ n → τ as n → ∞ .The motivation to study stopping times of the form (6) comes from the financialmodeling. Specifically, let a financial market model be driven by the process X n solving Eq. (2), and q > be a constant discount factor. Consider the problem ofoptimal exercise of an American-type contingent claim with payoff function f andmaturity T , that is, the maximization problem E (cid:2) e − qτ f (cid:0) X n ( τ ) (cid:1)(cid:3) → max , where τ is a stopping time taking values in [0 , T ] . Define the value function v n ( t, x ) = sup τ ∈ [ t,T ] E (cid:2) e − q ( τ − t ) f (cid:0) X n ( τ ) (cid:1) | X n ( t ) = x (cid:3) as the maximal expected discounted payoff provided that the price process X n startsfrom x at the moment t ; the supremum is taken over all stopping times with values in [ t, T ] .Then it is well known that the minimal optimal stopping time is given as τ ∗ ,n = inf (cid:8) t ≥ v n (cid:0) t, X n ( t ) (cid:1) = f (cid:0) X n ( t ) (cid:1)(cid:9) , that is, it is the first time when the process X n hits the so-called optimal stopping set G n = (cid:8) ( t, x ) ∈ [0 , T ] × R d : v n ( t, x ) = f ( x ) (cid:9) . Note that τ ∗ ,n ≤ T since v ( T, x ) = g ( x ) . Since, obviously, v n ( t, x ) ≥ f ( x ) , wemay represent τ ∗ ,n in the form (6) with ϕ n = f ( x ) − v n ( t, x ) . Let
T > be a fixed number playing the role of finite maturity of an Americancontingent claim. Let also the stopping times τ n , n ≥ , be given by (6) with ϕ n : [0 , T ] × R d → R satisfying the following assumptions.(G1) ϕ ∈ C ([0 , T ) × R d ) , and the derivative D x ϕ is locally Lipschitz continuousin x , that is, for all t ∈ [0 , T ) , R > , s ∈ [0 , t ] , and x, y ∈ B d ( R ) , (cid:12)(cid:12) D x ϕ ( s, x ) − D x ϕ ( s, y ) (cid:12)(cid:12) ≤ C t,R | x − y | . (G2) For all n ≥ and x ∈ R d , ϕ n ( T, x ) = 0 .(G3) For all t ∈ [0 , T ) and x ∈ R d , (cid:12)(cid:12) b ( t, x ) ⊤ D x ϕ ( t, x ) (cid:12)(cid:12) > . (7) G. Shevchenko
Here by b ( t, x ) ⊤ D x ϕ ( t, x ) we denote the vector in R k with j th coordinate equalto d X i =1 b ij ( t, x ) ∂ x i ϕ ( t, x ) , j = 1 , . . . , k. Remark 5.1.
Assumption (7) means that the diffusion is acting strongly enoughtoward the border of the set G t := { x ∈ R d : ϕ ( t, x ) ≤ } . In which situa-tions does this assumption hold, will be studied elsewhere. Here we just want toremark that it is more delicate than it might seem. For example, consider the opti-mal stopping problem described in the beginning of this section with n = 0 in (2).Then, under suitable assumptions (see, e.g., [4, 7]), we have the smooth fit principle: ∂ x v ( t, x ) = ∂ x f ( x ) on the boundary of the optimal stopping set. This means thatwe cannot set ϕ ( t, x ) = f ( x ) − v ( t, x ) in order for (7) to hold, contrary to whatwas proposed in the beginning of the section.We will also assume the locally uniform convergence ϕ n → ϕ .(G4) For all t ∈ [0 , T ) and R > , sup ( s,x ) ∈ [0 ,t ] × B d ( R ) (cid:12)(cid:12) ϕ n ( s, x ) − ϕ ( s, x ) (cid:12)(cid:12) → , n → ∞ . Remark 5.2.
The convergence of value functions in optimal stopping problems usu-ally holds under fairly mild assumptions on the convergence of coefficients and pay-offs. However, as we explained in Remark 5.1, we cannot use the value functionfor ϕ n . This means that we should find a function ϕ n defining G different from v n ( t, x ) − f ( x ) , but it still should satisfy the convergence assumption (G4).The question in which cases such functions exist and the convergence assumption(G4) takes places will be a subject of our future research.In the case where ν has infinite activity, that is, µ ( R m ) = ∞ , we will also needsome additional assumptions on the components of Eq. (2).(A3) For each r > , µ ( R m \ B m ( r )) < ∞ .(A4) For all t ≥ , x ∈ R d , and θ ∈ R m , (cid:12)(cid:12) c ( t, x, θ ) (cid:12)(cid:12) ≤ h ( t, x ) g ( θ ) , where the functions g, h are locally bounded, g (0) = 0 , and g ( θ ) → , θ → . Remark 5.3.
Assumption (A3) means that only small jumps of µ can accumulateon a finite interval; assumption (A4) means that small jumps of µ are translated byEq. (2) to small jumps of X n . An important and natural example of a situation wherethese assumptions are satisfied is an equation X ( t ) = X (0) + Z t a (cid:0) s, X ( s ) (cid:1) ds + Z t b (cid:0) s, X ( s ) (cid:1) dW ( s )+ Z t h (cid:0) s, X ( s − ) (cid:1) dZ ( s ) , t ≥ , driven by a Lévy process Z ( t ) = R t R R m θ e ν ( dθ, ds ) . onvergence of hitting times for jump-diffusion processes Now we are in a position to state the main result of this section.
Theorem 5.1.
Assume (A1)–(A4), (C1), (C2), (G1)–(G4). Then we have the followingconvergence in probability: τ n P −→ τ , n → ∞ . Proof.
Let ε, δ be small positive numbers. We are to show that for all n large enough, P (cid:0)(cid:12)(cid:12) τ n − τ (cid:12)(cid:12) > ε (cid:1) < δ. (8)Using estimate (3) and the Chebyshev inequality, we obtain that for some R > , P (cid:16) sup t ∈ [0 ,T ] (cid:12)(cid:12) X ( t ) (cid:12)(cid:12) ≥ R (cid:17) < δ . Denote K = [0 , T − ε/ × B d ( R + 2) , M = 1 + R + C T,R +2 + C T + C T − ε/ ,R +2 + sup ( t,x ) ∈K (cid:0)(cid:12)(cid:12) a ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) b ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) ∂ t ϕ ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) D x ϕ ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) b ( t, x ) ⊤ D x ϕ ( t, x ) (cid:12)(cid:12) − (cid:1) , where, with some abuse of notation, C T,R +2 is the constant from (A2) correspondingto T and R + 2 , C T is the sum of constants from (A1) and (4), and C T − ε/ ,R +2 isthe constant from (G1) corresponding to T − ε/ and R + 2 .Let κ ∈ (0 , M ] be a number, which we will specify later. Now we claim that thereexists a function ϕ ∈ C , ([0 , T ) × R d ) such that sup ( t,x ) ∈K (cid:12)(cid:12) ϕ ( t, x ) − ϕ ( t, x ) (cid:12)(cid:12) < κ / and, moreover, sup t ∈ [0 ,T − ε/ x ∈ B d ( R +1) (cid:0)(cid:12)(cid:12) ∂ t ϕ ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) D x ϕ ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) D xx ϕ ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) b ( t, x ) ⊤ D x ϕ ( t, x ) (cid:12)(cid:12) − (cid:1) ≤ C T − ε/ ,R +2 + sup ( t,x ) ∈K (cid:0)(cid:12)(cid:12) ∂ t ϕ ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) D x ϕ ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) b ( t, x ) ⊤ D x ϕ ( t, x ) (cid:12)(cid:12) − (cid:1) ≤ M. Indeed, we can take the convolution ϕ ( t, x ) = ( ϕ ( t, · ) ⋆ ψ )( x ) with a delta-likesmooth function ψ , supported on a ball of radius less than .Further, by (G4) there exists n ≥ such that for all n ≥ n , sup ( t,x ) ∈K (cid:12)(cid:12) ϕ n ( t, x ) − ϕ ( t, x ) (cid:12)(cid:12) < κ / . (9)On the other hand, by Theorem 3.1 there exists n ≥ such that for all n ≥ n , P (cid:18) sup t ∈ [0 ,T ] (cid:12)(cid:12) X n ( t ) − X ( t ) (cid:12)(cid:12) ≥ κ M (cid:19) < δ . (10)In what follows, we consider n ≥ n ∨ n . G. Shevchenko
Define the stopping time σ n = inf (cid:26) t ≥ (cid:12)(cid:12) X n ( t ) − X ( t ) (cid:12)(cid:12) ≥ κ M or (cid:12)(cid:12) X ( t ) (cid:12)(cid:12) ≥ R (cid:27) ∧ T. Write P (cid:0)(cid:12)(cid:12) τ n − τ (cid:12)(cid:12) > ε (cid:1) ≤ P (cid:0)(cid:12)(cid:12) τ n − τ (cid:12)(cid:12) > ε, σ n > T − ε/ (cid:1) + P (cid:16) sup t ∈ [0 ,T ] (cid:12)(cid:12) X ( t ) (cid:12)(cid:12) ≥ R (cid:17) + P (cid:18) sup t ∈ [0 ,T ] (cid:12)(cid:12) X n ( t ) − X ( t ) (cid:12)(cid:12) ≥ κ M (cid:19) < P (cid:0)(cid:12)(cid:12) τ n − τ (cid:12)(cid:12) > ε, σ n > T − ε/ (cid:1) + δ . (11)For any t ≤ σ n , (cid:12)(cid:12) X n ( t ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) X ( t ) (cid:12)(cid:12) + κ M < R + 1 , and hence, (cid:12)(cid:12) ϕ n (cid:0) t, X n ( t ) (cid:1) − ϕ (cid:0) t, X ( t ) (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) ϕ n (cid:0) t, X n ( t ) (cid:1) − ϕ (cid:0) t, X n ( t ) (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) ϕ (cid:0) t, X n ( t ) (cid:1) − ϕ (cid:0) t, X ( t ) (cid:1)(cid:12)(cid:12) ≤ κ + M (cid:12)(cid:12) X n ( t ) − X ( t ) (cid:12)(cid:12) ≤ κ . Now take some η ∈ (0 , ε/ whose exact value will be specified later and write theobvious inequality P (cid:0) τ ∗ , T + ε < τ ∗ ,nT , σ n > T − ε/ (cid:1) ≤ P (cid:0) τ < T − ε, τ + η < τ n , σ n > T − ε/ (cid:1) . (12)Assume that τ < T − ε , τ + η < τ n , σ n > T − ε/ . Then, for all t ∈ [ τ , τ + η ] =: I η , (cid:12)(cid:12) ϕ n (cid:0) s, X ( s ) (cid:1) − ϕ (cid:0) s, X ( s ) (cid:1)(cid:12)(cid:12) ≤ κ , ϕ n (cid:0) t, X n ( t ) (cid:1) < . Therefore, in view of the inequality ϕ ( τ , X ( τ )) ≥ , we obtain inf t ∈I η ϕ (cid:0) t, X ( t ) (cid:1) ≥ ϕ (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) − κ . (13)Further, we will work with the expression ϕ ( t, X ( t )) − ϕ ( τ , X ( τ )) for t ∈I η . For convenience, we will abbreviate f s = f ( s, X ( s )) ; for example, ϕ s = ϕ ( s, X ( s )) .Let r > be a positive number, which we will specify later, and assume that ν does not have jumps on I η greater than r , that is, ν (( R m \ B m ( r )) × I η ) = 0 . Write,using the Itô formula, ϕ (cid:0) t, X ( t ) (cid:1) − ϕ (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) = Z tτ L s ϕ s ds + Z tτ (cid:0) D x ϕ s , b s dW ( s ) (cid:1) + Z tτ Z B m ( r ) ∆ s ( θ ) e ν ( dθ, ds )=: I ( t ) + I ( t ) + I ( t ) , where onvergence of hitting times for jump-diffusion processes L t ϕ t = ∂ t ϕ t + (cid:0) D x ϕ t , a t (cid:1) + 12 tr (cid:0) b t (cid:0) b t (cid:1) ⊤ D xx ϕ t (cid:1) + Z B m ( r ) (cid:0) ∆ s ( θ ) − (cid:0) D x ϕ s , c (cid:0) s, X ( s − ) , θ (cid:1)(cid:1)(cid:1) µ ( dθ ) ,∆ s ( θ ) = ϕ (cid:0) s, X ( s − ) + c (cid:0) s, X ( s − ) , θ (cid:1)(cid:1) − ϕ (cid:0) s, X ( s − ) (cid:1) . Start with estimating I ( t ) . Since t ≤ σ n ∧ ( T − ε/ for any t ∈ I η , by thedefinition of M and σ n we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ t ϕ t + (cid:0) D x ϕ t , a t (cid:1) + 12 tr (cid:0) b t (cid:0) b t (cid:1) ⊤ D xx ϕ t (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M + M + M ≤ M . Further, by (A4), for t ∈ I η and θ ∈ B r , | c ( t, X ( t − ) , θ ) | ≤ h ( t, X ( t − )) g ( θ ) ≤ K m r , where K = sup t ∈ [0 ,T ] , | x |≤ R h ( t, x ) and m r = sup θ ∈ B m ( r ) g ( θ ) . Since m r → , r → , we can assume that r is such that m r ≤ /K . Then, for t ∈ I η , bythe Taylor formula (cid:12)(cid:12)(cid:12)(cid:12)Z B m ( r ) (cid:0) ∆ t ( θ ) − (cid:0) D x ϕ t , c (cid:0) t, X ( t − ) , θ (cid:1)(cid:1)(cid:1) µ ( dθ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤
12 sup ( u,x ) ∈ [0 ,T ] × B d ( R +1) (cid:12)(cid:12) D xx ϕ ( u, x ) (cid:12)(cid:12) Z B m ( r ) (cid:12)(cid:12) c (cid:0) t, X ( t − ) , θ (cid:1)(cid:12)(cid:12) µ ( dθ ) ≤ M (cid:0) (cid:12)(cid:12) X ( t ) (cid:12)(cid:12) (cid:1) ≤ M (cid:0) R (cid:1) ≤ M . Summing up the estimates, we get (cid:12)(cid:12) I ( t ) (cid:12)(cid:12) ≤ (cid:0) M + M (cid:1) η ≤ M η. (14)Now proceed to I ( t ) . By the Doob inequality, for any a > , P (cid:16) sup t ∈I η (cid:12)(cid:12) I ( t ) (cid:12)(cid:12) ≥ a, σ n > T − ε/ (cid:17) ≤ P (cid:16) sup t ∈ [ τ , ( τ + η ) ∧ σ n ] (cid:12)(cid:12) I ( t ) (cid:12)(cid:12) ≥ a (cid:17) ≤ Ca − E " Z T Z B m ( r ) ∆ s ( θ )1 [ τ , ( τ + η ) ∧ σ n ] ( s ) e ν ( dθ, ds ) ! = Ca − Z T Z B m ( r ) E (cid:2) ∆ s ( θ ) [ τ , ( τ + η ) ∧ σ n ] ( s ) (cid:3) µ ( dθ ) ds ≤ Ca − M Z T Z B m ( r ) E (cid:2)(cid:12)(cid:12) c (cid:0) s, X ( s − ) , θ (cid:1)(cid:12)(cid:12) [ τ , ( τ + η ) ∧ σ n ] ( s ) (cid:3) µ ( dθ ) ds ≤ Ca − M Z T Z B m ( r ) E (cid:2) K m r [ τ , ( τ + η ) ∧ σ n ] ( s ) (cid:3) µ ( dθ ) ds ≤ K a − m r η with some constant K . Further, we fix a = δ η / and some r > such that m r ≤ δ / (16 K ) and m r ≤ /K . Then P (cid:16) sup t ∈I η (cid:12)(cid:12) I ( t ) (cid:12)(cid:12) ≥ δ η / , σ n > T − ε/ (cid:17) ≤ δ . G. Shevchenko
Hence, in view of (12)–(14), we obtain P (cid:0) τ + ε < τ n , σ n > T − ε/ (cid:1) ≤ P (cid:16) inf t ∈I η I ( t ) ≥ − κ − M η − δ η / , σ n > T − ε/ (cid:17) + P (cid:16) sup t ∈I η (cid:12)(cid:12) I ( t ) (cid:12)(cid:12) ≥ δ η / , σ n > T − ε/ (cid:17) + P (cid:0) ν (cid:0)(cid:0) R m \ B m ( r ) (cid:1) × I η (cid:1) > (cid:1) ≤ P (cid:16) inf t ∈I η I ( t ) ≥ − κ − M η − δ η / , σ n > T − ε/ (cid:17) + η µ (cid:0) R m \ B m ( r ) (cid:1) + δ . Assume further that η ≤ η := δ µ ( R m ) / (not yet fixing its exact value). Setting κ = ( ηM ) ∧ M , we get P (cid:0) τ + ε < τ n , σ n > T − ε/ (cid:1) ≤ P (cid:16) inf t ∈I η I ( t ) ≥ − ηM − δ η / , σ n > T − ε/ (cid:17) + δ . (15)Write I ( t ) = J ( t ) + J ( t ) + J ( t ) , where J ( t ) = Z tτ (cid:0) D x ϕ s − D x ϕ τ , b s dW ( s ) (cid:1) ,J ( t ) = Z tτ (cid:0) D x ϕ τ , (cid:0) b s − b τ (cid:1) dW ( s ) (cid:1) ,J ( t ) = (cid:0) D x ϕ τ , b τ (cid:0) W ( t ) − W (cid:0) τ (cid:1)(cid:1)(cid:1) = (cid:0) u τ , W ( t ) − W (cid:0) τ (cid:1)(cid:1) ; u s = b (cid:0) s, X ( s ) (cid:1) ⊤ D x ϕ (cid:0) s, X ( s ) (cid:1) . Taking into account that ( s, X ( s )) ∈ K for s ≤ σ n , we estimate with the helpof Doob’s inequality E h sup t ∈I η J ( t ) σ n >T − ǫ/ i ≤ E h sup t ∈ [ τ , ( τ + η ) ∧ σ n ] J ( t ) i ≤ C E " Z ( τ + η ) ∧ σ n τ (cid:0) D x ϕ s − D x ϕ τ , b s dW ( s ) (cid:1)! ≤ C E "Z ( τ + η ) ∧ σ n τ | D x ϕ s − D x ϕ τ | (cid:12)(cid:12) b s (cid:12)(cid:12) ds ≤ CM E "Z τ + ητ (cid:12)(cid:12) X ( s ) − X (cid:0) τ (cid:1)(cid:12)(cid:12) ds ≤ CM (cid:0) (cid:12)(cid:12) X (0) (cid:12)(cid:12) (cid:1) η ≤ CM (cid:0) R (cid:1) η ≤ CM η . Similarly, using (A2), we get E h sup t ∈I η J ( t ) σ n >T − ǫ/ i ≤ CM η . onvergence of hitting times for jump-diffusion processes The Chebyshev inequality yields P (cid:16) sup t ∈I η (cid:0)(cid:12)(cid:12) J ( t ) (cid:12)(cid:12) + (cid:12)(cid:12) J ( t ) (cid:12)(cid:12)(cid:1) ≥ η / , σ n > T − ε/ (cid:17) ≤ K M η / with certain constant K . Assume further that η ≤ η := (cid:18) δ K M (cid:19) / , in which case the right-hand side of the last inequality does not exceed δ/ , and that η ≤ η := 1125 M , so that η / ≥ ηM . Hence, in view of (15), we obtain P (cid:0) τ + ε < τ n , σ n > T − ε/ (cid:1) ≤ P (cid:16) inf t ∈I η J ( t ) ≥ − ηM − η / − δ η / , σ n > T − ε (cid:17) + 3 δ ≤ P (cid:16) inf t ∈I η J ( t ) ≥ − η / − δ η / , (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) ∈ K (cid:17) + 3 δ . (16)Further, due to the strong Markov property of W , P (cid:16) inf t ∈I η J ( t ) ≥ − η / − δ η / , (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) ∈ K (cid:17) = E h K (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) P (cid:16) inf t ∈I η J ( t ) ≥ − η / − δ η / | F τ (cid:17)i = E h K (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) × P (cid:16) inf z ∈ [0 ,η ] (cid:0) u ( s, x ) , W ( s + z ) − W ( s ) (cid:1) ≥ − η / − δ η / (cid:17) | ( s,x )=( τ ,X ( τ )) i , where u ( s, x ) = b ( s, x ) ⊤ D x ϕ ( s, x ) . Observe now that { ( u ( s, x ) , W ( z + s ) − W ( s )) ,z ≥ } is a standard Wiener process multiplied by | u ( s, x ) | . Therefore, P (cid:16) inf z ∈ [0 ,η ] (cid:0) u ( s, x ) , W ( s + z ) − W ( s ) (cid:1) ≥ − η / − δ η / (cid:17) = 1 − P (cid:0)(cid:0) u ( s, x ) , W ( s + η ) − W ( s ) (cid:1) < − η / − δ η / (cid:1) = 1 − Φ (cid:18) − η / + ∆ η / | u ( s, x ) | η / (cid:19) = 1 − Φ (cid:18) − η / + δ | u ( s, x ) | (cid:19) , where Φ is the standard normal distribution function. Thus, G. Shevchenko P (cid:16) inf t ∈I η J ( t ) ≥ − η / − δ η / , (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) ∈ K (cid:17) ≤ E (cid:20) K (cid:0) τ , X (cid:0) τ (cid:1)(cid:1)(cid:18) − Φ (cid:18) − η / + δ | u ( τ , X ( τ )) | (cid:19)(cid:19)(cid:21) ≤ − Φ (cid:0) − M (cid:0) η / + δ (cid:1)(cid:1) ≤ M √ √ π (cid:0) η / + δ (cid:1) . Note that the definition of M does not depend on δ . Thus, we can assume withoutloss of generality that δ ≤ √ π/ (32 M √ . Finally, if η ≤ η := (cid:18) δ √ π √ (cid:19) , then P (cid:16) inf t ∈I η J ( t ) ≥ − η / − ∆ η − / , (cid:0) τ , X (cid:0) τ (cid:1)(cid:1) ∈ K (cid:17) ≤ δ . (17)Now we can fix η = min { ε/ , η , η , η , η } , making all previous estimates to hold.Combining (16) with (17), we arrive at P (cid:0) τ + ε < τ n , σ n > T − ε/ (cid:1) ≤ δ . Similarly, P (cid:0) τ n + ε < τ , σ n > T − ε/ (cid:1) ≤ δ , and hence P (cid:0)(cid:12)(cid:12) τ n − τ (cid:12)(cid:12) > ε, σ n > T − ε/ (cid:1) ≤ δ . Plugging this estimate into (11), we arrive at the desired inequality (8).
Remark 5.4.
It is easy to modify the proof for the case where (7) holds for all ( t, x ) ∈G := { ( t, x ) ∈ [0 , T ) × R d : ϕ ( t, x ) = 0 } . Indeed, the continuity would imply that(7) holds in some neighborhood of G , which is sufficient for the argument. Remark 5.5.
As we have already mentioned, assumptions (A3) and (A4) are notneeded in the case µ ( R m ) < ∞ . Indeed, we can set r = 0 in the previous argumentand skip the estimation of I ( t ) . Nevertheless, these assumptions does not seem veryrestrictive, as we pointed out in Remark 5.3. Here we extend the results of the previous subsection to the case of infinite timehorizon. Let, as before, the stopping times τ n , n ≥ , be given by (6). We impose thefollowing assumptions.(H1) ϕ ∈ C ([0 , ∞ ) × R d ) , and D x ϕ is locally Lipschitz continuous in x , that is,for all T > , R > , t ∈ [0 , T ] , and x, y ∈ B d ( R ) , (cid:12)(cid:12) D x ϕ ( t, x ) − D x ϕ ( t, y ) (cid:12)(cid:12) ≤ C T,R | x − y | . onvergence of hitting times for jump-diffusion processes (H2) τ < ∞ a.s.(H3) For all t ≥ and x ∈ R d , (cid:12)(cid:12) D x ϕ ( t, x ) b ( t, x ) ⊤ (cid:12)(cid:12) > . (H4) For all t ≥ and R > , sup ( s,x ) ∈ [0 ,t ] × B d ( R ) (cid:12)(cid:12) ϕ n ( t, x ) − ϕ ( t, x ) (cid:12)(cid:12) → , n → ∞ . Theorem 5.2.
Assume (A1), (A2), (C1), (C2), (H1)–(H4). Then we have the followingconvergence in probability: τ n P −→ τ , n → ∞ . Proof.
Fix arbitrary ε ∈ (0 , and δ > . Since τ < ∞ a.s., P ( τ > T − ≤ δ forsome T > . For n ≥ , t ∈ [0 , T ] , and x ∈ R d , define ˜ ϕ n ( t, x ) = ϕ n ( t, x ) [0 ,T ) ( t ) , τ nT = τ n ∧ T . Then the functions ˜ ϕ n , n ≥ , satisfy (G1)–(G3) and τ nT = inf { t ≥ ϕ n ( t, X n ( t )) ≥ } . Therefore, in view of Theorem 5.1, P (cid:0)(cid:12)(cid:12) τ nT − τ ∗ , T (cid:12)(cid:12) > ε (cid:1) → , n → ∞ . We estimate P (cid:0)(cid:12)(cid:12) τ n − τ (cid:12)(cid:12) > ε (cid:1) ≤ P (cid:0)(cid:12)(cid:12) τ nT − τ T (cid:12)(cid:12) > ε (cid:1) + P (cid:0) τ > T − (cid:1) ≤ P (cid:0)(cid:12)(cid:12) τ nT − τ T (cid:12)(cid:12) > ε (cid:1) + δ. Hence, lim n →∞ P (cid:0)(cid:12)(cid:12) τ n − τ (cid:12)(cid:12) > ε (cid:1) ≤ δ. Letting δ → , we arrive at the desired convergence. Example 5.1.
Let d = k = m = 1 and for all t ≥ , x, θ ∈ R , a n ( t, x ) = a n , b n ( t, x ) = b n , c n ( t, x, θ ) = c n θ , where a n , b n , c n ∈ R . Then we have a sequence ofLévy processes X n ( t ) = X n (0) + a n t + b n W ( t ) + c n Z t Z R θ e ν ( ds, dθ ) . Consider the following times: τ n = inf (cid:8) t ≥ X n ( t ) ≥ h n ( t ) (cid:9) ∧ T, n ≥ , of crossing some curve h ∈ C ([0 , T )) .Assume that a n → a , b n → b = 0 , c n → c , and X n (0) → X (0) as n → ∞ and, for any t ∈ [0 , T ) , sup s ∈ [0 ,t ] | h n ( t ) − h ( t ) | → as n → ∞ . Then τ n P −→ τ , n → ∞ . Indeed, setting ϕ n ( t, x ) = ( h n ( t ) − x ) [0 ,T ) ( t ) , we can check that allassumptions of Theorem 5.1 are in force. G. Shevchenko
Example 5.2.
Let d = k = m = 1 . Suppose that the coefficients a n , b n , c n satisfy(A1), (A2) and that the convergence (C1)–(C3) takes place. Assume that b ( t, x ) > for all t ≥ and x ∈ R . Define τ n = inf (cid:8) t ≥ X n ( t ) / ∈ (cid:0) l n , r n (cid:1)(cid:9) , n ≥ . It is not hard to check that, due to the nondegeneracy of b , τ < ∞ a.s. Assume that l n → l , r n → r , n → ∞ . Then, setting ϕ n ( t, x ) = ( x − l n )( r n − x ) and usingTheorem 5.2, we get the convergence τ n P −→ τ , n → ∞ . Acknowledgments
The author would like to thank the anonymous referee whose remarks led to a sub-stantial improvement of the manuscript.
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