COMPUTING THE EXIT-TIME FOR A FINITE ... - Semantic Scholar
COMPUTING THE EXIT-TIME FOR A FINITE-RANGE SYMMETRIC JUMP PROCESS NATHANIAL BURCH∗ AND R. B. LEHOUCQ†
16 October 2014 Abstract. This paper investigates the exit-time for a broad class of symmetric finite-range jump processes via the corresponding master equation, a nonlocal diffusion equation suitably constrained. In direct analogy to the classical diffusion equation with a homogeneous Dirichlet boundary condition, the nonlocal diffusion equation is augmented with a homogeneous volume constraint. The volume-constrained master equation provides an efficient alternative over Monte Carlo simulation for computing an important statistic of the process. Several numerical examples are given. Key words. nonlocal diffusion, jump process, random walks, anomalous diffusion, volumeconstraints, exit-time, first-passage AMS subject classifications. 26A33, 34A08, 34B10, 35A15, 35L65, 35B40, 45A05, 45K05, 60G22, 76R51
1. Introduction. Statistics of a symmetric jump Markov process, e.g., a moment, that characterize the process are often desired. These processes arise in various application areas where non-classical diffusion prevails; see [19] and [9, 14] for books on biological and financial applications, respectively. The classical Brownian motion model is not well-suited in these applications due to its continuous sample paths. In contrast, the processes of interest in this paper exhibit jumps so that the sample paths have discontinuities but are of finite-range, i.e., the distance of a jump is bounded. Moreover, the mean square displacement of a diffusing particle undergoing a jump process is distinct from that of Brownian motion making such jump processes viable models for non-classical diffusion. In particular, such diffusions are expedient models when the process sample-path is discontinuous because nearly instantaneous price volatility or species migration is suggested by the length and time scales over which the data is collected. Process statistics are easily computed given the density, typically unavailable. Instead, a common practice is to simulate the process multiple times and produce an estimate of the density used to approximate the desired statistic. When the particle sample path is further constrained, to be of finite-range or to occur over a bounded domain Ω, the simulations are more involved. Such a constraint arises in computing the exit-time distribution. The master equation for such a process, i.e., the deterministic equation that governs the time evolution of the density, suitably restricted to a bounded domain via a volume-constraint, therefore provides a powerful alternative for computing the exit-time. The volume-constraint specifies the density over some region of positive volume and is the nonlocal analogue of a boundary condition used for the exit-time associated with Brownian motion. In fact, with probability one, a particle originating in a domain does not touch the boundary upon exiting the domain; see, e.g., [15]. Hence augmenting the master equation with a boundary ∗ North
Carolina State University, Raleigh, North Carolina 27695 National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM 87185–1320; rblehou@ sandia.gov. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under contract DE-AC04-94AL85000. Supported in part by the U.S. Department of Energy grant number FWP-09-014290 through the Office of Advanced Scientific Computing Research, DOE Office of Science. † Sandia
1
2
NATHANIAL BURCH AND R. B. LEHOUCQ
condition does not, in general, suffice for the determination of the exit-time. The paper [5] provides a formal overview of the exit-time problem for Markov jump process and its many distinctions with the Brownian motion. Related work is exclusively related to L´evy processes with a non-zero Brownian component or killing term; see e.g., [12, 14]. In contrast, this paper investigates in more mathematical and numerical detail the exit-time, or first-passage, for a broad class of symmetric finite-range jump processes via the corresponding volume-constrained master equation. Such an equation enables the restriction of the particle sample path to a bounded domain, is compatible with Monte Carlo simulations and results in a well-posed master equation, i.e., there is a unique solution that depends continuously upon the initial condition and volume constraint. The recent analytic results given in [11] establish that the master equation is well-posed for two general classes of finite-range jumprates we consider in this paper. These analytic results then imply that the mean and variance of the exit-time are finite and that the probability of remaining in the domain decreases to zero as time increases; see §3.2. We also present a numerical technique in §4 for approximating the density and compare to estimates computed from Monte Carlo simulations of the process; comparisons are made in §5 demonstrating that the numerical solution of a volume-constrained master equation robustly approximates the density estimate. Enforcing a volume-constraint for the master equation is then seen to be symbiotic with Monte-carlo realization of the jump process. We also establish the equivalence between the smoothing of the jump, or nonlocal, operator and the activity of the jump process in §3.1. By smoothing we mean that the steady-state density is rendered smoother than the data. Perhaps not well appreciated is that a finite activity process is diffusive but any discontinuity in the initial condition persists for all finite time; in contrast, an infinite activity process removes any discontinuities in the initial condition in finite time. We relate the index of the underlying fractional Sobolev space to the activity, and hence smoothing of the steady-state operator. This provides a strictly deterministic characterization associated with sample paths of jump processes. Volume-constraints also allow the consideration of disconnected domains, which appear naturally in many applications but are not easily handled. An example is species migration or animal foraging across waterways. We present a striking example for computing the exit-time over an unconnected domain in §5.3. This is in stark contrast with Brownian motion where the sample path remains in connected domains and is absorbed when encountering the boundary. 2. Exit-time and the cumulative distribution function. Let Xt denote a symmetric jump Markov process that originates in a bounded domain Ω ⊂ Rd . The symmetry of the process is induced by the symmetry of the jump-rate density γ, i.e., γ(x, y) = γ(y, x), associated with the jump-rate to x from y (or y from x). The process starts at a point x0 ∈ Ω, i.e., X0 = x0 , that is sampled at random from the density u0 . Such processes are Markov and have discontinuous sample paths due to the jumps governed by γ that are of finite-range when the maximum jump is bounded independent of x. The Markov property states that the probability of the process located at z depends only on the most recent location of the process rather than the entire time history, see, e.g., [2, 14] for a mathematical definition. Let the random variable T := inf{t : Xt ∈ / Ω}
EXIT-TIME FOR A SYMMETRIC JUMP PROCESS
3
and Px0 (T > t) = P Xt ∈ Ω | X0 = x0
denote the exit-time and the probability of the process remaining in Ω, respectively, through a time t > 0 conditioned on X0 = x0 ∈ Ω. The density u of particles that has not yet exited evolves according to the volume-constrained nonlocal diffusion problem ut = Lu, on Ω, (2.1) u(·, t) = 0, on Rd \ Ω, u(·, 0) = u0 (·), on Ω, R where the initial density u0 is nonnegative and satisfies Ω u0 (x) dx = 1 and Z Lu(x, t) := u(y, t) − u(x, t) γ(x, y) dy. Rd
Whether the nonlocal diffusion problem (2.1) is well-posed depends upon the kernel γ; two examples of compactly supported kernels leading to a well-posed volumeconstrained problems are given in §3. These two classes of kernels also grant that P(T > t) → 0 as t increases and that the first two moments are finite; see §3.2. Since the jump-rate is symmetric in its variables, L is self-adjoint so that the equation ut = Lu does also evolve the law of the process; see [5] for a formal treatment considering the case of a nonsymmetric jump-rate. The volume-constraint (2.1)2 on the density u is the nonlocal analogue of a Dirichlet boundary condition and corresponds to realizations of the process having exited the domain. Specifying the value of the density on the volume Rd \ Ω is necessary because the process sample-path is discontinuous and may “jump” a finite distance away from Ω and not touch the boundary. This explains why simply prescribing the density u on the boundary ∂Ω does not suffice to render the system (2.1) well-posed, i.e., a unique solution exists and depends continuously upon the initial condition. In contrast, if Xt is a Brownian, or Wiener, process, then the sample-path is continuous and must exit Ω through the boundary. The mean exit-time is given by Z ∞ Ex0 (T ) = − t dPx0 (T > t) 0
R see, e.g., Gardiner [13, pp.136–138] or [16, p.600]. If we assume that t Ω u(x, t) dx → 0 as t increases, then integration by parts leads to Z Z Z ∞ Z ∞ d u(x, t) dx dt = u(x, t) dt dx. (2.2a) Ex0 (T ) = − t dt Ω Ω 0 0 R In a similar fashion, when we assume that t2 Ω u(x, t) dx → 0 as t increases, the second moment of the exit-time is given by Z ∞ Z Z ∞ 2 2 Ex0 (T ) = − t dPx0 (T > t) = 2 t u(x, t) dt dx. (2.2b) 0
Ω
0
The variance of the exit-time follows from combining (2.2a) and (2.2b). For the remainder of our study, omission of the subscript x0 for P and E signifies that the initial density u0 does not induce a Dirac delta measure.
4
NATHANIAL BURCH AND R. B. LEHOUCQ
Define the functions Z ∞ u ¯(x) := u(x, t) dt
and u ¯(2) (x) :=
0
Z
∞
t u(x, t) dt . 0
By the volume-constrained nonlocal diffusion equation (2.1), we have Z ∞ ut (x, t) dt = −u0 (x), x ∈ Ω, L¯ u(x) = 0 Z ∞ L¯ u(2) (x) = t ut (x, t) dt = −¯ u(x), x ∈ Ω, 0
where the last equality used integration by parts and we assumed that t u(x, t) → 0 as t increases. The first and second moments of the exit-time are then given by Z Z E(T ) = u ¯(x) dx, and E(T 2 ) = u ¯(2) (x) dx. (2.3) Ω
Ω
The volume-constrained problem (2.1) grants that Z Z d u(x, t) dx = Lu(x, t) dx dt Ω ZΩ Z = u(y, t) − u(x, t) γ(x, y) dy dx Ω Rd \Ω Z Z =− u(x, t)γ(x, y) dy dx, Ω
(2.4)
Rd \Ω
where the antisymmetry of the integrand and the volume-constraint (2.1)2 were used for the second and third equalities, respectively. Integration with respect to time and the initial condition results in the exit-time distribution function F (t) := P(T ≤ t) Z =1− u(x, t) dx Ω Z tZ Z = u(x, s)γ(x, y) dy dx ds . 0
Ω
(2.5)
Rd \Ω
In words, equation (2.5) is the probability that the particle remains in Ω at time t. In contrast, (2.4)2 states that the rate of change of the probability that the particle remains in Ω is given by the (nonlocal) flux from Ω into Rd \ Ω; see [11] for a justification. The probabilistic interpretation of (2.4) is that the particle is located either Ω or Rd \ Ω. This is in stark contrast to the exit-time problem for Brownian motion where the particle exits to the boundary of Ω. The discussion following (2.1) explains the rate of change of P(T > t) decreases to zero so that F (t) → 1 as t increases. Hence there is a relationship between the flux, and the probability of the process, exiting Ω, respectively. 3. Finite-range processes. In this section, we consider in §3.1 two cases of finite-range jump kernels or, equivalently, jump-rate densities that bound the maximum jump. The two forms depend upon whether the kernel is integrable, i.e., R γ(x, y) dy is finite valued on Ω or not. The paper [11, Thm 5.1] establishes the d R
5
EXIT-TIME FOR A SYMMETRIC JUMP PROCESS
well-posedness of (2.1) for a broad class of kernels in both cases, i.e., given a suitable initial density u0 the solution is unique and depends continuously upon the data. We then justify in §3.2 the comments following (2.1) that these two classes of kernels also grant that P(T > t) → 0 as t increases and that the first two moments are finite. We then interpret our results in §3.3 with respect to jump diffusions. 3.1. Two finite-range kernels. Let 1Ω denote the indicator function on the set Ω. In the first case, the kernel satisfies c∗ c∗ 1 ≤ γ(x, y) ≤ 1|x−y|
16 October 2014 Abstract. This paper investigates the exit-time for a broad class of symmetric finite-range jump processes via the corresponding master equation, a nonlocal diffusion equation suitably constrained. In direct analogy to the classical diffusion equation with a homogeneous Dirichlet boundary condition, the nonlocal diffusion equation is augmented with a homogeneous volume constraint. The volume-constrained master equation provides an efficient alternative over Monte Carlo simulation for computing an important statistic of the process. Several numerical examples are given. Key words. nonlocal diffusion, jump process, random walks, anomalous diffusion, volumeconstraints, exit-time, first-passage AMS subject classifications. 26A33, 34A08, 34B10, 35A15, 35L65, 35B40, 45A05, 45K05, 60G22, 76R51
1. Introduction. Statistics of a symmetric jump Markov process, e.g., a moment, that characterize the process are often desired. These processes arise in various application areas where non-classical diffusion prevails; see [19] and [9, 14] for books on biological and financial applications, respectively. The classical Brownian motion model is not well-suited in these applications due to its continuous sample paths. In contrast, the processes of interest in this paper exhibit jumps so that the sample paths have discontinuities but are of finite-range, i.e., the distance of a jump is bounded. Moreover, the mean square displacement of a diffusing particle undergoing a jump process is distinct from that of Brownian motion making such jump processes viable models for non-classical diffusion. In particular, such diffusions are expedient models when the process sample-path is discontinuous because nearly instantaneous price volatility or species migration is suggested by the length and time scales over which the data is collected. Process statistics are easily computed given the density, typically unavailable. Instead, a common practice is to simulate the process multiple times and produce an estimate of the density used to approximate the desired statistic. When the particle sample path is further constrained, to be of finite-range or to occur over a bounded domain Ω, the simulations are more involved. Such a constraint arises in computing the exit-time distribution. The master equation for such a process, i.e., the deterministic equation that governs the time evolution of the density, suitably restricted to a bounded domain via a volume-constraint, therefore provides a powerful alternative for computing the exit-time. The volume-constraint specifies the density over some region of positive volume and is the nonlocal analogue of a boundary condition used for the exit-time associated with Brownian motion. In fact, with probability one, a particle originating in a domain does not touch the boundary upon exiting the domain; see, e.g., [15]. Hence augmenting the master equation with a boundary ∗ North
Carolina State University, Raleigh, North Carolina 27695 National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM 87185–1320; rblehou@ sandia.gov. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under contract DE-AC04-94AL85000. Supported in part by the U.S. Department of Energy grant number FWP-09-014290 through the Office of Advanced Scientific Computing Research, DOE Office of Science. † Sandia
1
2
NATHANIAL BURCH AND R. B. LEHOUCQ
condition does not, in general, suffice for the determination of the exit-time. The paper [5] provides a formal overview of the exit-time problem for Markov jump process and its many distinctions with the Brownian motion. Related work is exclusively related to L´evy processes with a non-zero Brownian component or killing term; see e.g., [12, 14]. In contrast, this paper investigates in more mathematical and numerical detail the exit-time, or first-passage, for a broad class of symmetric finite-range jump processes via the corresponding volume-constrained master equation. Such an equation enables the restriction of the particle sample path to a bounded domain, is compatible with Monte Carlo simulations and results in a well-posed master equation, i.e., there is a unique solution that depends continuously upon the initial condition and volume constraint. The recent analytic results given in [11] establish that the master equation is well-posed for two general classes of finite-range jumprates we consider in this paper. These analytic results then imply that the mean and variance of the exit-time are finite and that the probability of remaining in the domain decreases to zero as time increases; see §3.2. We also present a numerical technique in §4 for approximating the density and compare to estimates computed from Monte Carlo simulations of the process; comparisons are made in §5 demonstrating that the numerical solution of a volume-constrained master equation robustly approximates the density estimate. Enforcing a volume-constraint for the master equation is then seen to be symbiotic with Monte-carlo realization of the jump process. We also establish the equivalence between the smoothing of the jump, or nonlocal, operator and the activity of the jump process in §3.1. By smoothing we mean that the steady-state density is rendered smoother than the data. Perhaps not well appreciated is that a finite activity process is diffusive but any discontinuity in the initial condition persists for all finite time; in contrast, an infinite activity process removes any discontinuities in the initial condition in finite time. We relate the index of the underlying fractional Sobolev space to the activity, and hence smoothing of the steady-state operator. This provides a strictly deterministic characterization associated with sample paths of jump processes. Volume-constraints also allow the consideration of disconnected domains, which appear naturally in many applications but are not easily handled. An example is species migration or animal foraging across waterways. We present a striking example for computing the exit-time over an unconnected domain in §5.3. This is in stark contrast with Brownian motion where the sample path remains in connected domains and is absorbed when encountering the boundary. 2. Exit-time and the cumulative distribution function. Let Xt denote a symmetric jump Markov process that originates in a bounded domain Ω ⊂ Rd . The symmetry of the process is induced by the symmetry of the jump-rate density γ, i.e., γ(x, y) = γ(y, x), associated with the jump-rate to x from y (or y from x). The process starts at a point x0 ∈ Ω, i.e., X0 = x0 , that is sampled at random from the density u0 . Such processes are Markov and have discontinuous sample paths due to the jumps governed by γ that are of finite-range when the maximum jump is bounded independent of x. The Markov property states that the probability of the process located at z depends only on the most recent location of the process rather than the entire time history, see, e.g., [2, 14] for a mathematical definition. Let the random variable T := inf{t : Xt ∈ / Ω}
EXIT-TIME FOR A SYMMETRIC JUMP PROCESS
3
and Px0 (T > t) = P Xt ∈ Ω | X0 = x0
denote the exit-time and the probability of the process remaining in Ω, respectively, through a time t > 0 conditioned on X0 = x0 ∈ Ω. The density u of particles that has not yet exited evolves according to the volume-constrained nonlocal diffusion problem ut = Lu, on Ω, (2.1) u(·, t) = 0, on Rd \ Ω, u(·, 0) = u0 (·), on Ω, R where the initial density u0 is nonnegative and satisfies Ω u0 (x) dx = 1 and Z Lu(x, t) := u(y, t) − u(x, t) γ(x, y) dy. Rd
Whether the nonlocal diffusion problem (2.1) is well-posed depends upon the kernel γ; two examples of compactly supported kernels leading to a well-posed volumeconstrained problems are given in §3. These two classes of kernels also grant that P(T > t) → 0 as t increases and that the first two moments are finite; see §3.2. Since the jump-rate is symmetric in its variables, L is self-adjoint so that the equation ut = Lu does also evolve the law of the process; see [5] for a formal treatment considering the case of a nonsymmetric jump-rate. The volume-constraint (2.1)2 on the density u is the nonlocal analogue of a Dirichlet boundary condition and corresponds to realizations of the process having exited the domain. Specifying the value of the density on the volume Rd \ Ω is necessary because the process sample-path is discontinuous and may “jump” a finite distance away from Ω and not touch the boundary. This explains why simply prescribing the density u on the boundary ∂Ω does not suffice to render the system (2.1) well-posed, i.e., a unique solution exists and depends continuously upon the initial condition. In contrast, if Xt is a Brownian, or Wiener, process, then the sample-path is continuous and must exit Ω through the boundary. The mean exit-time is given by Z ∞ Ex0 (T ) = − t dPx0 (T > t) 0
R see, e.g., Gardiner [13, pp.136–138] or [16, p.600]. If we assume that t Ω u(x, t) dx → 0 as t increases, then integration by parts leads to Z Z Z ∞ Z ∞ d u(x, t) dx dt = u(x, t) dt dx. (2.2a) Ex0 (T ) = − t dt Ω Ω 0 0 R In a similar fashion, when we assume that t2 Ω u(x, t) dx → 0 as t increases, the second moment of the exit-time is given by Z ∞ Z Z ∞ 2 2 Ex0 (T ) = − t dPx0 (T > t) = 2 t u(x, t) dt dx. (2.2b) 0
Ω
0
The variance of the exit-time follows from combining (2.2a) and (2.2b). For the remainder of our study, omission of the subscript x0 for P and E signifies that the initial density u0 does not induce a Dirac delta measure.
4
NATHANIAL BURCH AND R. B. LEHOUCQ
Define the functions Z ∞ u ¯(x) := u(x, t) dt
and u ¯(2) (x) :=
0
Z
∞
t u(x, t) dt . 0
By the volume-constrained nonlocal diffusion equation (2.1), we have Z ∞ ut (x, t) dt = −u0 (x), x ∈ Ω, L¯ u(x) = 0 Z ∞ L¯ u(2) (x) = t ut (x, t) dt = −¯ u(x), x ∈ Ω, 0
where the last equality used integration by parts and we assumed that t u(x, t) → 0 as t increases. The first and second moments of the exit-time are then given by Z Z E(T ) = u ¯(x) dx, and E(T 2 ) = u ¯(2) (x) dx. (2.3) Ω
Ω
The volume-constrained problem (2.1) grants that Z Z d u(x, t) dx = Lu(x, t) dx dt Ω ZΩ Z = u(y, t) − u(x, t) γ(x, y) dy dx Ω Rd \Ω Z Z =− u(x, t)γ(x, y) dy dx, Ω
(2.4)
Rd \Ω
where the antisymmetry of the integrand and the volume-constraint (2.1)2 were used for the second and third equalities, respectively. Integration with respect to time and the initial condition results in the exit-time distribution function F (t) := P(T ≤ t) Z =1− u(x, t) dx Ω Z tZ Z = u(x, s)γ(x, y) dy dx ds . 0
Ω
(2.5)
Rd \Ω
In words, equation (2.5) is the probability that the particle remains in Ω at time t. In contrast, (2.4)2 states that the rate of change of the probability that the particle remains in Ω is given by the (nonlocal) flux from Ω into Rd \ Ω; see [11] for a justification. The probabilistic interpretation of (2.4) is that the particle is located either Ω or Rd \ Ω. This is in stark contrast to the exit-time problem for Brownian motion where the particle exits to the boundary of Ω. The discussion following (2.1) explains the rate of change of P(T > t) decreases to zero so that F (t) → 1 as t increases. Hence there is a relationship between the flux, and the probability of the process, exiting Ω, respectively. 3. Finite-range processes. In this section, we consider in §3.1 two cases of finite-range jump kernels or, equivalently, jump-rate densities that bound the maximum jump. The two forms depend upon whether the kernel is integrable, i.e., R γ(x, y) dy is finite valued on Ω or not. The paper [11, Thm 5.1] establishes the d R
5
EXIT-TIME FOR A SYMMETRIC JUMP PROCESS
well-posedness of (2.1) for a broad class of kernels in both cases, i.e., given a suitable initial density u0 the solution is unique and depends continuously upon the data. We then justify in §3.2 the comments following (2.1) that these two classes of kernels also grant that P(T > t) → 0 as t increases and that the first two moments are finite. We then interpret our results in §3.3 with respect to jump diffusions. 3.1. Two finite-range kernels. Let 1Ω denote the indicator function on the set Ω. In the first case, the kernel satisfies c∗ c∗ 1 ≤ γ(x, y) ≤ 1|x−y|
