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APPROXIMATION OF FIRST PASSAGE TIMES OF SWITCHING DIFFUSION JAROSLAV KRYSTUL AND ARUNABHA BAGCHI

Abstract. This paper studies the approximation of first passage time of a multi-dimensional switching diffusion process to a given target domain. We propose a discrete-time strong approximation scheme for switching diffusion processes with state-dependent switching rates. If τ and τ h are the first passage times of the continuous and discretized processes respectively, then under some conditions we show that τ h converges in distribution to τ as the discretization step tends to zero.

1. Introduction In this paper we study the approximation of first passage time of a multidimensional switching diffusion process with state-dependent switching rates to a given target set. The problem of finding the probability of hitting the target set before some fixed time is of great importance in many applications, e.g. air traffic management [2], reliability analysis [1], finance applications etc. An analytical solution is available only in some special simple cases. Thus, approximate method is needed. Weak approximations of killed (or stopped) diffusions were studied in [5], [6], [7], [8] and [10]. They develop and prove the convergence of numerical schemes that approximate the expected value E[g(x(τ ), τ )] of a given function g depending on the solution x of an Itˆo stochastic differential equation and on the first exit time τ from a given domain. We will consider one hybrid model, also called a switching diffusion, where switching rates of the discrete component may depend on continuous component [3]. Let {xt , θt } be the switching diffusion taking its values in Rn × M defined by (1.1) (1.2)

dxt = a(θt , xt )dt + b(θt , xt )dWt , Pθt+δ |θt ,xt (θ|η, x) = ληθ (x)δ + o(δ), η 6= θ,

where M is a finite set of modes and (Wt )t≥0 is a Brownian motion in Rn . We set 4

τ = inf{t > 0 : xt ∈ D} for the first passage time to the cylinder set D × M, where D ⊂ Rn is a closed connected set. We want to estimate (1.3)

P (τ ≤ T )

Date: May 14, 2004. Key words and phrases. First passage time, strong approximation, switching diffusion. This research has been performed with support of the European Commission through the HYBRIDGE project. 1

where T is a fixed time. We propose to approximate the switching diffusion (1.1)(1.2) by a discrete-time strong approximation (Xthi , θthi )ti ∈I (where I is a time discretization), and approximate probability (1.3) by P (τ h ≤ T ),

(1.4) 4

where τ h = inf{t > 0 : Xth ∈ D}. Time discrete approximations of an Ito diffusion are well explained in [9]. A discretization scheme for jump-diffusion process with state-dependent intensities was considered in [4]. We propose an Euler-type discretization scheme for hybrid model (1.1)-(1.2) and prove its convergence. Following the approach of Gobet (1999) [5], we then show that P (τ ≤ T ) − P (τ h ≤ T ) −→ 0 as the discretization step tends to zero. Note that probability (1.4) can be estimated by Monte Carlo methods whatever the dimension n is. The organization of the paper is the following. Section 2 describes the hybrid model in more detail. The strong approximation scheme and its convergence proof are presented in section 3. The approximation of first passage time is discussed in section 4. 2. Stochastic hybrid model Let M = {e1 , e2 , . . . , eN } be a finite set of unit vectors, i.e. ei ∈ M is a i-th unit vector in RN . Let a : b : λij :

Rn × M → Rn Rn × M → Rn×n Rn → R, i, j = 1, 2, . . . , N

For each θ ∈ M , a(·, θ) and b(·, θ) are assumed to be bounded, continuous and Lipschitz. For all i, j ∈ {1, . . . , N } λij (·) is assumed to be bounded, continuous and PN Lipschitz, λij (·) ≥ 0 for i 6= j and j=1 λij (·) = 0 for any i ∈ {1, . . . , N }. For i, j ∈ {1, . . . , N }, i 6= j, x ∈ Rn we construct the intervals ∆ij (x) of the real line in the following manner [3]: ∆12 (x) ∆13 (x)

= [0, λ12 (x)) = [λ12 (x), λ12 (x) + λ13 (x)) .. . h ´ PN −1 PN ∆1N (x) = λ (x), λ (x) 1j 1j j=2 hPj=2 ´ PN N ∆21 (x) = j=2 λ1j (x), j=2 λ1j (x) + λ21 (x) and so on. Thus, in general, ∆ij (x) =

i−1 X N hX i0 =1

j 0 =1 j 0 6=i0

λi0 j 0 (x) +

j−1 X

λij 0 (x),

j 0 =1 j 0 6=i

i−1 X N X i0 =1

2

j 0 =1 j 0 6=i0

λi0 j 0 (x) +

j X j 0 =1 j 0 6=i

´ λij 0 (x) .

For fixed x these are disjoint intervals, and the length of ∆ij (x) is λij (x). Now we define a function c : Rn × M × R → RN : ½ (2.1)

c(x, ei , z) =

ej − ei 0

if z ∈ ∆ij (x) otherwise.

Then the (Rn × M)-valued switching diffusion process (1.1)-(1.2) can be represented as a solution of the following SDE [3]: (2.2) (2.3)

dxt = a(xt , θt )dt + b(xt , θt )dWt Z c(xt , θt− , z)p(dt, dz) dθt = R

for t ≥ 0, with (x0 , θ0 ) a prescribed (Rn × M)-valued random variable; p(dt, dz) is a Poisson random measure with intensity dt · dz; (Wt ) is an n-dimensional Wiener process independent of (x0 , θ0 ) and p(dt, dz). Under assumption on Wt , p(dt, dz), (x0 , θ0 ), and on functions a, b and λ, equation (2.2)-(2.3) admits an a.s. pathwise unique solution. Define the following interval: 4

U (x) =

N ³ [ N [ i=1

´ ∆ij (x) ,

j=1 j6=i

it includes all intervals ∆ij (x), i, j = 1, . . . , N, i 6= j. Since the length of each interval ∆ij (x) is λij (x), and this is continuous and bounded function for i, j = 1, . . . , N, i 6= j, it follows that the length of interval U (x) (denote l(U (x))) is bounded and is a continuous function of x. Therefore, it has a maximum at some point x? : l(U (y)) ≤ l(U (x? )) for all y ∈ Rn . Then we denote the interval of maximum length as follows 4

Umax = U (x? ) 4

and the length of Umax is denoted as λmax = l(Umax ). We can rewrite equation (2.3) as follows Z dθt =

c(Xt , θt− , z)p(dt, dz). Umax

We can think of a Poisson random measure p(dt, dz) as assigning unit mass to (τn , zn ) if there is a jump at time τn of size zn . Let N (t) be a standard Poisson process with intensity λmax . We denote by τn , n = 1, 2, . . . the jump times of N (t). Let Umax be the ”mark” space, and (Zn )n≥1 be a sequence of i.i.d. random variables with uniform distribution on Umax , independent of N (t). In this special case we can represent the random Poisson measure p(dt, dz) with intensity dt · dz as a random counting measure associated to the marked point process (τn , Zn )n≥0 , i.e. for each Lebesgue measurable A ⊂ Umax X (2.4) p((0, t], A) = 1{τn ≤t} · 1{Zn ∈A} . n≥1 3

We check that E[p((0, t], A)] = λmax · t · P(Zn ∈ A) = λmax · t ·

l(A) = t · l(A). λmax

The representation (2.4) is very convenient for practical problems. We see that p(dt, dz) can be generated just by sampling independent random variables τn and Zn , n = 1, 2, . . . . 3. Strong approximation of switching diffusion 3.1. Discretization Scheme. Now we turn our attention to numerical solution of SDE (2.2)-(2.3). We will develop an Euler type discretization scheme which allows to obtain a strong approximation process to the solution of switching diffusion precess. To start with, we should define © the appropriateª discretization of time interval [0, T ]. Let us denote by Id = tdn : n = 0, 1, . . . , L the usual equidistant time discretization of a bounded interval [0, T ] with discretization step h = T /L. Suppose τ1 , τ2 , . . . are the jump times of the component θt driven by equations (2.2-2.3). Then we take a new time discretization I = {tn : n = 0, 1, . . . } which is the superposition of the random jump times τn of a component θt on interval [0, T ] and a deterministic grid Id . For a given time discretization I an Euler type approximation is a continuous time stochastic process {Xth , θth } satisfying the following equation with “delayed” coefficients1: Z t Z t (3.1) Xth = X0 + ah (s, X h , θh )ds + bh (s, X h , θh )dWs 0 0 Z tZ (3.2) θth = θ0 + ch (s−, X h , θh , z)p(ds, dz) 0

Umax

here 4

ah (s, X h , θh ) = h

h

h

4

b (s, X , θ ) = h

h

h

4

c (s−, X , θ , z) =

a(Xtk , θtk ),

s ∈ [tk , tk+1 ),

b(Xtk , θtk ),

s ∈ [tk , tk+1 ),

c(Xtk , θtk − , z), s ∈ [tk , tk+1 ).

The corresponding recursive discretization scheme (3.3) (3.4)

Xthi = Xthi−1 + a(Xthi−1 , θthi−1 )(ti − ti−1 ) + b(Xthi−1 , θthi−1 )(Wti − Wti−1 ), Z h h θti = θti−1 + c(Xthi , θthi−1 , z)p({ti }, dz), Umax

determines values of the approximating process (3.1)-(3.2) at discretization times only. Thus, approximation (Xthi , θthi ) is iteratively computed from the initial condition (X0 , θ0 ) using the scheme (3.3)-(3.4). At the grid point, (3.4) computes the jump of θh exactly, conditional on (Xthi , θth− ) = (Xthi , θthi−1 ), if ti is indeed a point i

of the Poisson random measure. Otherwise, the jump term is zero. (The integral in (3.4) entails at most a single evaluation of the function c because p({ti }, dz) is a point mass at the mark z that arrives at ti if it is a jump time.) 1Here h denotes the dependence on the time discretization step h. 4

3.2. Convergence. Proposition 3.1. Suppose, functions a, b, c and λij are defined as in section 2 and the Euler type approximating process {Xth , θth } is defined as in section 3.1. We assume, that (1) functions λij (·) (i, j = 1, . . . N ) are Lipschitz, i.e. |λij (x) − λij (y)| ≤ Cλ |x − y|, for all x, y ∈ Rn ;

(3.5)

(2) for all x, y ∈ Rn and θ, η ∈ M (3.6)

|a(x, θ) − a(y, η)|2 + |b(x, θ) − b(y, η)|2 ≤ Cab (|x − y|2 + 1) for θ 6= η and |a(x, θ) − a(y, η)|2 + |b(x, θ) − b(y, η)|2 ≤ Cab (|x − y|2 )

(3.7)

for θ = η. Then (3.8)

sup E(|Xsh − Xs |2 + |θsh − θs |2 ) ≤ e−λmax T K 2

s≤T

(3.9)

∞ X

h2

−k

k=0

·

(λmax T K)k , k!

sup E(|Xsh − Xs |2 + |θsh − θs |2 ) −→ 0, as h −→ 0,

s≤T

and ∞ ³ X −k (λmax T K)k ´1/2 (3.10) E[sup |Xsh − Xs |] ≤ 2T (T + 4)Cab · e−λmax T K 2 h2 · , k! s≤T k=0

(3.11)

E[sup |Xsh − Xs |] −→ 0, as h −→ 0. s≤T

Here the constant K does not depend on h. To prove the Proposition 3.1 one needs the following lemmas. Let Ft denote the σ algebra generated by all random variables Ws , p((0, s], Umax ), s ≤ t (see section 2). Let L2T denote the space of all Ft -adapted stochastic processes that are square integrable: Z TZ ||f ||L2T = f 2 (t, ω)P(dω)dt < ∞. 0

Ω

To shorten expressions we introduce the following notation: 4

Ei [·] = E[·|N (T ) = i]. Lemma 3.2. Suppose, Wt is independent of p(dt, dz). Then for every f ∈ L2T , Z T Z T (3.12) Ei [( f (t, ω)dWt (ω))2 ] = Ei [ f 2 (t, ω)dt]. 0

0

Proof. First we consider the step processes, and then extend the result to arbitrary processes. Let φ be a bounded step process in L2T : (3.13)

φ(t, ω) =

n−1 X

cj (ω)1[tj ,tj+1 ) (t).

j=0 5

4

By adaptedness, ci in (3.13) is independent of ∆Wj = Wtj+1 − Wtj for i ≤ j. Therefore Z T n−1 X i E [( φ(t)dWt )2 ] = Ei [( cj ∆Wj )2 ] 0

j=0 n−1 X n−1 X

=

Ei [ci cj ∆Wi ∆Wj ]

i=0 j=0 n−1 X

=

X

Ei [c2j ∆Wj2 ] + 2

j=0 n−1 X

=

Ei [ci cj ∆Wi ]E[∆Wj ]

i 0 take s ∈ [tk , tk+1 ), then applying conditions (3.6)-(3.7) and lemma 3.3, and since θs = θtk for any tk ∈ I, and θth , θt ∈ M (unit vectors), we obtain |ah (s, X h , θh ) − a(Xs , θs )|2 + |bh (s, X h , θh ) − b(Xs , θs )|2 = |a(Xthk , θthk ) − a(Xs , θs )|2 + |b(Xthk , θthk ) − b(Xs , θs )|2 ≤ Cab (|Xthk − Xs |2 + |θthk − θs |2 ) ≤ 2Cab (|Xthk − Xtk |2 + |Xtk − Xs |2 + |θthk − θtk |2 ) Z |ch (tk −, X h , θh , z) − c(Xtk , θtk − , z)|2 dz R Z = |c(Xthk , θthk − , z) − c(Xtk , θtk − , z)|2 dz R

≤ Cc (|Xthk − Xtk | + |θthk − − θtk − |2 ). 4

Denote C = max(2Cab , Cc ). Taking conditional expectation we obtain (3.17) Ei [|ah (s, X h , θh ) − a(Xs , θs )|2 + |bh (s, X h , θh ) − b(Xs , θs )|2 ] ≤ C(Ei [|Xthk − Xtk |2 ] + Ei [|Xtk − Xs |2 ] + Ei [|θthk − θtk |2 ]), Z (3.18) Ei [ |ch (tk −, X h , θh , z) − c(Xtk , θtk − , z)|2 dz] R

≤ C(Ei [|Xthk − Xtk |] + Ei [|θthk− − θtk− |2 ]). Denote

(3.19)

 4  ϕhi (t) =      4 ψi (h) =    4    γih (t) =

sup Ei [|Xuh − Xu |2 ],

t ∈ [0, T ],

u≤t

sup Ei [|Xs − Xu |2 ], h > 0, |s−u|≤h

sup Ei [|θuh − θu |2 ],

t ∈ [0, T ].

u≤t

Then, using (3.19) and the inequality (3.17) we have Ei [|ah (s, X h , θh ) − a(Xs , θs )|2 + |bh (s, X h , θh ) − b(Xs , θs )|2 ] ≤ 2C(ϕhi (s) + ψi (h) + γih (s)), 9

s ∈ [0, T ], h > 0. Thus, for t < τk+1 , k < i (i.e. less then (k + 1)-th jump time) ϕhi (t) = sup Ei [|Xsh − Xs |2 ] s≤t

Z s Z s h h h (bh (u, X h , θh ) − b(Xu , θu ))dWu )2 ] = sup E [( (a (u, X , θ ) − a(Xu , θu ))du + i

s≤t

0

0

Z

s

≤ sup Ei [2T s≤t

0

Z

t

≤ K1 Z

Z

0 t

≤ K1 0

s

(ah (u, X h , θh ) − a(Xu , θu ))2 du +

(bh (u, X h , θh ) − b(Xu , θu ))2 du]

0

(ϕhi (u) + ψi (h) + γih (u))du ϕhi (u)du + K1 T (ψi (h) + γih (τk ))

here K1 = max(1, 2T, 2C). By Gronwall’s lemma: ϕhi (t) ≤ K1 T (ψi (h) + γih (τk ))eK1 t ≤ K1 T (ψi (h) + γih (τk ))eK1 T , for t < τk+1 . Note, that Z

Z s a(Xv , θv )dv + b(Xv , θv )dWv |2 ] u u Z s Z s i 2 ≤ 2(s − u) E [|a(Xv , θv )| ]dv + 2 Ei [|b(Xv , θv )|2 ]dv

Ei [|Xs − Xu |2 ] = Ei [|

s

u

≤ K2 (s − u),

u

0 ≤ u ≤ s,

and thus ψi (h) =

sup Ei [|Xs − Xu |2 ] ≤ K2 h, h → 0. |s−u|≤h

From here (3.20)

ϕhi (t) ≤ K1 T (K2 h + γih (τk ))eK1 T ,

h for t < τk+1 , 1 ≤ k < i, and using the fact that (Xsh − Xs ) = (Xs− − Xs− ), (i.e. continuity from the left) we get

(3.21)

ϕhi (τk+1 ) ≤ K1 T (K2 h + γih (τk ))eK1 T , 1 ≤ k < i.

Denote q(dt, dz) = p(dt, dz) − dtdz. Now we will derive the similar recurrent formula for γih (τk+1 ): 10

γih (τk+1 ) = sup Ei [|θsh − θs |2 ] s≤τk+1

h¯ Z = sup E ¯ i

s≤τk+1

s

Z

s

Z

0

h¯ Z = sup Ei ¯ s≤τk+1

Umax

0

i

¯2 i (ch (u−, X h , θh , z) − c(Xu , θu− , z))p(du, dz)¯ (ch (u−, X h , θh , z) − c(Xu , θu− , z))q(du, dz)

Umax sZ

Z + 0 hZ s Z

¯2 i (ch (u−, X h , θh , z) − c(Xu , θu− , z))dudz)¯

Umax

≤ sup 2E s≤τk+1

0

Umax

s≤τk+1

0

Umax

|ch (u−, X h , θh , z) − c(Xu , θu− , z)|2 dudz

Z sZ i +( (ch (u−, X h , θh , z) − c(Xu , θu− , z))dudz)2 0 Umax hZ s Z ≤ sup 2Ei |ch (u−, X h , θh , z) − c(Xu , θu− , z)|2 dudz

(3.22)

Z sZ i + T λmax |ch (u−, X h , θh , z) − c(Xu , θu− , z)|2 dudz 0 Umax Z τk+1 h ≤ 2C(1 + T λmax ) Ei [|Xuh − Xu | + |θu− − θu− |2 ]du 0 ¢ ¡q ≤ 2CT (1 + T λmax ) ϕhi (τk+1 ) + γih (τk ) , for 1 ≤ k < i. 4

Assume τ0 = 0. Then γih (τ0 ) = 0 and ϕhi (τ0 ) = 0. Define 4

K3 = 2eK1 T K1 K2 CT (1 + T λmax ). Then, using the recurrent formulas (3.20), (3.21) and (3.22) for ϕhi (τk+1 ) and γih (τk+1 ) we obtain: √ γih (τ1 ) ≤ 2K32 h

ϕhi (τ1 ) ≤ 2K32 h √ ϕhi (τ2 ) ≤ 4K33 h ... ϕhi (τk ) ≤ K3 (2K3 )k h2 ...

1

γih (τ2 ) ≤ 4K33 h 4 ... 1−k

ϕhi (T ) ≤ K3 (2K3 )i+1 h2

γih (τk ) ≤ K3 (2K3 )k h2 ...

−i

γih (T ) ≤ K3 (2K3 )i h2

−k

−i

Denote K = 2K3 . From the above estimates follows that sup Ei [|Xsh − Xs |2 + |θsh − θs |2 ] ≤ ϕhi (T ) + γih (T )

s≤T

≤ K3 (2K3 )i+1 h2

−i

+ K3 (2K3 )i h2

−i

−i

≤ K i+2 h2 , h −→ 0. ¤ 11

Proof of Proposition 3.1. Using the results of lemma 3.4 we have sup E[|Xsh − Xs |2 + |θsh − θs |2 ] = sup

s≤T

s≤T

≤

∞ X k=0

∞ X

Ek [|Xsh − Xs |2 + |θsh − θs |2 ] · P(N (T ) = k)

k=0

(sup Ek [|Xsh − Xs |2 + |θsh − θs |2 ]) · P(N (T ) = k) s≤T

≤ e−λmax T K 2

∞ X

h2

−k

·

k=0

(λmax T K)k . k!

Denote 4

Sm (h) =

m X

h2

−k

·

k=0

(λmax T K)k , k!

and 4

S(h) = lim Sm (h) = m→∞

Since

∞ X

h2

k=0

−k

·

(λmax T K)k . k!

¯ ¯ ¯ 2−k (λmax T K)k ¯ (λmax T K)k ¯h ¯≤ , h ∈ [0, 1] · ¯ ¯ k! k!

and ∞ X (λmax T K)k k=0

k!

= eλmax T K < ∞,

then, by Weierstrass M-Test, S(h) < ∞ and the convergence is uniform on [0, 1], furthermore, function S(h) is continuous on [0, 1]. Thus lim S(h) = lim lim Sm (h) = lim lim Sm (h) = 0. h→0 m→∞

h→0

m→∞ h→0

Hence, we have proven that (3.23)

sup E[|Xsh − Xs |2 + |θsh − θs |2 ] ≤ e−λmax T K 2

s≤T

∞ X

h2

−k

k=0

·

(λmax T K)k , k!

and sup E[|Xsh − Xs |2 + |θsh − θs |2 ] −→ 0, as h −→ 0.

s≤T

Using Jensen’s inequality we obtain (3.24)

³ ´1/2 E[sup |Xsh − Xs |] ≤ E[sup |Xsh − Xs |2 ] . s≤T

s≤T

12

Next, using Doob’s maximal martingale inequality, conditions (3.6)-(3.7) and (3.23) we show that ¯Z s h ¯ h 2 E[sup |Xs − Xs | ] = E sup ¯ (ah (u, X h , θh ) − a(Xu , θu ))du s≤T

s≤T

Z

0 s

¯2 i ¯ (bh (u, X h , θh ) − b(Xu , θu ))dWu ¯ 0 h ³ Z s |ah (u, X h , θh ) − a(Xu , θu )|2 du ≤ 2E sup T +

s≤T

0

¯Z s ¯2 ´i ¯ ¯ +¯ (bh (u, X h , θh ) − b(Xu , θu ))dWu ¯ 0 h Z T ≤ 2E T |ah (u, X h , θh ) − a(Xu , θu )|2 du 0

¯Z ¯ + 4¯

T

¯2 i ¯ (bh (u, X h , θh ) − b(Xu , θu ))dWu ¯

0

hZ ≤ 2(T + 4)Cab E

T

0

(|Xuh − Xu |2 + |θuh − θu |2 )du

i

≤ 2T (T + 4)Cab sup E[|Xsh − Xs |2 + |θsh − θs |2 ] s≤T

≤ 2T (T + 4)Cab · e−λmax T K 2

∞ X

h2

k=0

−k

·

(λmax T K)k . k!

Hence ∞ ³ X −k (λmax T K)k ´1/2 E[sup |Xsh − Xs |] ≤ 2T (T + 4)Cab · e−λmax T K 2 h2 · , k! s≤T k=0

and

E[sup |Xsh − Xs |] −→ 0, as h −→ 0. s≤T

¤ 4. Approximation of first passage times Our aim is to show that first passage times of discretized switching diffusion converge in distribution to first passage times of original process, i.e. P (τ ≤ T ) − P (τ h ≤ T ) −→ 0 as the maximal discretization step tends to zero. We follow the approach of [5] and [6]. We are interested in computing P (τ ≤ T ) = 1 − E[1{T 0 : xt ∈ D} is a first passage time to a given closed connected set D. So, we have to evaluate E[1{T 0} , ∂Dc = {x ∈ Rn : F (x) = 0} for some globally Lipschitz function F . Provided that the condition (C) below is satisfied ¯ c ) = 0, (C): P (∃t ∈ [0, T ] xt ∈ / Dc ; ∀t ∈ [0, T ] xt ∈ D 13

then we have lim |P (τ ≤ T ) − P (τ h ≤ T )| = 0,

N →+∞

Remark 4.2. If D is of class C 2 with a compact boundary, the existence of such a function F holds. Remark 4.3. Condition (C) rules out the pathological situation where the paths may reach ∂D without leaving Dc . If this condition is not satisfied, then the approximation may not converge to the exact solution. Remark 4.4. If a and b are bounded and Dc is of class C 3 with a compact boundary, then an uniform ellipticity condition on the diffusion implies condition (C).2 Proof. Obviously, we have |P (τ ≤ T ) − P (τ h ≤ T )| = |P (T < τ h ) − P (T < τ )| ≤ E[|1{T