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Exit probability for an integrated geometric Brownian motion
Statistics and Probability Letters 79 (2009) 1363–1365
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Exit probability for an integrated geometric Brownian motionI Cloud Makasu ∗ Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa
article
a b s t r a c t
info
Article history: Received 30 June 2008 Received in revised form 13 February 2009 Accepted 14 February 2009 Available online 27 February 2009
In this note, we present an explicit form for the exit probability of an integrated geometric Brownian motion from a given curved domain. Explicit bounds for the exit probability and one possible application are also given, under certain conditions. © 2009 Elsevier B.V. All rights reserved.
1. Statement of the exit probability problem Assume that Mt = (xt , yt ) is a two-dimensional, degenerate diffusion process that evolves as follows: dxt = µyt dt ; dyt = β yt dt +
x(0) = x,
p
β yt dBt ;
y(0) = y,
(1.1)
initially starting in the interior of a curved domain Dθ in the first positive quadrant, given by
Dθ =
(x, y) ∈ R2+ :
y
θ
(log y − 1) < x <
1
θ
y log y ,
(1.2)
where µ, β , and θ are some fixed positive constants, Bt is a standard Brownian motion defined on a probability space (Ω , F, P), and yt and xt are respectively a geometric Brownian motion and its corresponding integrated process at any instant time t. Let
τ = inf{t > 0 : (xt , yt ) 6∈ Dθ |(x, y) ∈ Dθ },
(1.3)
be the first exit time of the process (xt , yt ) from the curved domain Dθ . Denote by
yτ φ(x, y) = Px,y xτ = (log yτ − 1) θ
for x, y ∈ Dθ
(1.4)
the exit probability of the integrated geometric Brownian motion through the upper boundary x = θ (log y − 1). In this note, we present an explicit form and some bounds for the exit probability of the process (1.1) from the curved y domain Dθ first through the upper boundary x = θ (log y − 1) or the lower boundary x = θ1 y log y. Our original motivation for the present note is some recent results on the integral of geometric Brownian motion; see for instance Donati-Martin et al. (2000), Dufresne (2001), Lefebvre (2005) and Schröder (2003). The above cited papers are mainly concerned with explicit forms for the moments of the integrated geometric Brownian motion and its distribution law, among other things. y
I Partial results of this note were obtained when the author was holding a postdoctoral grant PRO12/1003 at the Mathematics Institute, University of Oslo, Norway. ∗ Tel.: +27 15 962 8410; fax: +27 15 962 8648. E-mail address: [email protected].
0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.02.009
1364
C. Makasu / Statistics and Probability Letters 79 (2009) 1363–1365
2. Main results Our main result is stated as follows: Proposition 2.1. Let Mt = (xt , yt ) be a degenerate process given by (1.1) in the interior of a curved domain Dθ , and let θ , β , and µ be fixed positive constants such that β/2 = µθ holds. If exp( θyx ) < y, then for all x, y ∈ Dθ the exit probability φ(x, y) for the process Mt through the upper boundary is explicitly given by
θx
φ(x, y) = log y −
y
,
(2.1)
and
ψ(x, y) = 1 − φ(x, y),
(2.2)
through the lower boundary. Proof. It is a well-known fact that the exit probability (1.4) for the process (1.1) from the domain Dθ solves (see Cox and Miller (1965), pp. 230–231), and is a unique viscosity solution (see Jensen et al. (1988), for instance) of, the interior Dirichlet boundary value problem, 1 2
β y2 φyy (x, y) + β yφy (x, y) + µyφx (x, y) = 0 in Dθ ,
(2.3)
subject to the following boundary conditions:
0 if x = 1 y log y, θ φ(x, y) = 1 if x = y (log y − 1). θ
(2.4)
Now observe that the continuous function
φ(x, y) = log
y e
θx
for x, y ∈ Dθ ,
(2.5)
y
solves the second-order partial differential equation (2.3) subject to the boundary conditions (2.4). For the reason that (2.3) and (2.4) admits a unique solution, then the exit probability, φ(x, y), is given by (2.5). This completes the proof. Remark 2.1. Notice that once φ(x, y) is made explicit, the exit probability ψ(x, y) := Px,y (xτ = θ1 yτ log yτ ) follows immediately from
φ(x, y) + ψ(x, y) = 1 for x, y ∈ Dθ , as given in (2.2). In the next assertion, we have established explicit bounds for the exit probability (1.4), as a consequence of our main result. The assertion is here stated without proof since this follows directly, and using an elementary inequality. Corollary 2.1. Let Mt = (xt , yt ) be a degenerate process given by (1.1) in the interior of the curved domain Dθ , where θ > 0 is y a fixed positive constant. If θ (log y − 1) < x < θ1 y log y, then the exit probability φ(x, y) satisfies the following bound: 2
y − exp( θyx ) y + exp( θyx )
! < φ(x, y) < 1 for x, y ∈ Dθ .
(2.6)
3. Application In this section, we shall now consider one possible application of our main result in a problem of optimal stochastic control. Here, we treat a problem of optimally controlling the process (xt , yut ) in the curved domain Dθ with the goal of minimizing a certain expected cost functional, where u is the control variable and θ > 0 is a fixed constant. We assume y that the controller incurs either zero or +∞ terminal costs when the process exits either through x = θ (log y − 1) or x = θ1 y log y, respectively. The motivation in this example follows from Section 4 in Lefebvre (1991), where optimal control of an integrated Brownian motion is singled out.
C. Makasu / Statistics and Probability Letters 79 (2009) 1363–1365
1365
Example 3.1. Let (xt , yut ) be a controlled, degenerate diffusion process in the curved domain Dθ available to a controller through the control variable u, whose goal is to minimize the following expected cost functional:
E
x ,y
τu
"Z 0
1 u2t 2 y2t
# dt + Φ (xτ , yτ ) ,
(3.1)
subject to the controlled process (xt , yut ), given by dxt = µyt dt ; dyt = β yt dt +
x(0) = x,
p
β yt dBt + ut dt ;
y(0) = y,
(3.2)
where u is the control variable in R, and Φ (., .) is a terminal cost of the form
+∞ if x = 1 y log y, θ Φ (x, y) = 0 if x = y (log y − 1). θ Let α = 1/β , for which β > 0. It can be shown (see Whittle (1982, p. 289)) that the optimal control u∗ =
y2 φy (x, y)
for x, y ∈ Dθ
αφ(x, y)
(3.3)
(3.4)
minimizes the expected cost functional (3.1). In this case we can now deduce that the above optimal control problem is associated with the exit probability problem (2.3) and (2.4). Consequently, it now follows from our main result that the optimal control u∗ is explicitly given by u∗ =
y + θx
α log y −
θx
for x, y ∈ Dθ .
(3.5)
y
Remark 3.1. In the above example, notice that the optimal control u∗ → +∞ as x → ( θ1 y log y)− , as one would expect; y otherwise u∗ → y log y/α as x → ( θ (log y − 1))+ . 4. Concluding remarks In this note, we have considered an exit probability problem for the integrated geometric Brownian motion from a prescribed curved domain in the first positive quadrant. Under certain conditions, the exit probability is made explicit. At the end of the note, explicit bounds for the exit probability are also derived, which is followed by one possible application in optimal stochastic control. Acknowledgements We gratefully acknowledge Professors Mario Abundo (Italy) and Mario Lefebvre (Montréal) for their constructive comments and interest in our work, and for sending us references (Lefebvre, 1991, 2005) which motivated the present note. We also thank the anonymous referee and the Associate Editor for their helpful suggestions. References Cox, D.R., Miller, H.D., 1965. The Theory of Stochastic Processes. Methuen, London. Donati-Martin, C., Matsumoto, H., Yor, M., 2000. On positive and negative moments of the integrals of geometric Brownian motions. Statist. Probab. Lett. 49, 45–52. Dufresne, D., 2001. The integral of geometric Brownian motion. Adv. Appl. Probab. 33, 223–241. Jensen, R., Lions, P.L., Souganidis, P.E., 1988. A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations. Proc. Amer. Math. Soc. 102, 975–978. Lefebvre, M., 1991. Forcing a stochastic process to stay in or to leave a given region. Ann. Appl. Probab. 1, 167–172. Lefebvre, M., 2005. First hitting time and place for the integrated geometric Brownian motion. J. Differential Equations Appl. 9 (4), 365–374. Schröder, M., 2003. On the integral of geometric Brownian motion. Adv. Appl. Probab. 35, 159–183. Whittle, P., 1982. Optimization Over Time 1. Wiley, Chichester.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Exit probability for an integrated geometric Brownian motionI Cloud Makasu ∗ Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa
article
a b s t r a c t
info
Article history: Received 30 June 2008 Received in revised form 13 February 2009 Accepted 14 February 2009 Available online 27 February 2009
In this note, we present an explicit form for the exit probability of an integrated geometric Brownian motion from a given curved domain. Explicit bounds for the exit probability and one possible application are also given, under certain conditions. © 2009 Elsevier B.V. All rights reserved.
1. Statement of the exit probability problem Assume that Mt = (xt , yt ) is a two-dimensional, degenerate diffusion process that evolves as follows: dxt = µyt dt ; dyt = β yt dt +
x(0) = x,
p
β yt dBt ;
y(0) = y,
(1.1)
initially starting in the interior of a curved domain Dθ in the first positive quadrant, given by
Dθ =
(x, y) ∈ R2+ :
y
θ
(log y − 1) < x <
1
θ
y log y ,
(1.2)
where µ, β , and θ are some fixed positive constants, Bt is a standard Brownian motion defined on a probability space (Ω , F, P), and yt and xt are respectively a geometric Brownian motion and its corresponding integrated process at any instant time t. Let
τ = inf{t > 0 : (xt , yt ) 6∈ Dθ |(x, y) ∈ Dθ },
(1.3)
be the first exit time of the process (xt , yt ) from the curved domain Dθ . Denote by
yτ φ(x, y) = Px,y xτ = (log yτ − 1) θ
for x, y ∈ Dθ
(1.4)
the exit probability of the integrated geometric Brownian motion through the upper boundary x = θ (log y − 1). In this note, we present an explicit form and some bounds for the exit probability of the process (1.1) from the curved y domain Dθ first through the upper boundary x = θ (log y − 1) or the lower boundary x = θ1 y log y. Our original motivation for the present note is some recent results on the integral of geometric Brownian motion; see for instance Donati-Martin et al. (2000), Dufresne (2001), Lefebvre (2005) and Schröder (2003). The above cited papers are mainly concerned with explicit forms for the moments of the integrated geometric Brownian motion and its distribution law, among other things. y
I Partial results of this note were obtained when the author was holding a postdoctoral grant PRO12/1003 at the Mathematics Institute, University of Oslo, Norway. ∗ Tel.: +27 15 962 8410; fax: +27 15 962 8648. E-mail address: [email protected].
0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.02.009
1364
C. Makasu / Statistics and Probability Letters 79 (2009) 1363–1365
2. Main results Our main result is stated as follows: Proposition 2.1. Let Mt = (xt , yt ) be a degenerate process given by (1.1) in the interior of a curved domain Dθ , and let θ , β , and µ be fixed positive constants such that β/2 = µθ holds. If exp( θyx ) < y, then for all x, y ∈ Dθ the exit probability φ(x, y) for the process Mt through the upper boundary is explicitly given by
θx
φ(x, y) = log y −
y
,
(2.1)
and
ψ(x, y) = 1 − φ(x, y),
(2.2)
through the lower boundary. Proof. It is a well-known fact that the exit probability (1.4) for the process (1.1) from the domain Dθ solves (see Cox and Miller (1965), pp. 230–231), and is a unique viscosity solution (see Jensen et al. (1988), for instance) of, the interior Dirichlet boundary value problem, 1 2
β y2 φyy (x, y) + β yφy (x, y) + µyφx (x, y) = 0 in Dθ ,
(2.3)
subject to the following boundary conditions:
0 if x = 1 y log y, θ φ(x, y) = 1 if x = y (log y − 1). θ
(2.4)
Now observe that the continuous function
φ(x, y) = log
y e
θx
for x, y ∈ Dθ ,
(2.5)
y
solves the second-order partial differential equation (2.3) subject to the boundary conditions (2.4). For the reason that (2.3) and (2.4) admits a unique solution, then the exit probability, φ(x, y), is given by (2.5). This completes the proof. Remark 2.1. Notice that once φ(x, y) is made explicit, the exit probability ψ(x, y) := Px,y (xτ = θ1 yτ log yτ ) follows immediately from
φ(x, y) + ψ(x, y) = 1 for x, y ∈ Dθ , as given in (2.2). In the next assertion, we have established explicit bounds for the exit probability (1.4), as a consequence of our main result. The assertion is here stated without proof since this follows directly, and using an elementary inequality. Corollary 2.1. Let Mt = (xt , yt ) be a degenerate process given by (1.1) in the interior of the curved domain Dθ , where θ > 0 is y a fixed positive constant. If θ (log y − 1) < x < θ1 y log y, then the exit probability φ(x, y) satisfies the following bound: 2
y − exp( θyx ) y + exp( θyx )
! < φ(x, y) < 1 for x, y ∈ Dθ .
(2.6)
3. Application In this section, we shall now consider one possible application of our main result in a problem of optimal stochastic control. Here, we treat a problem of optimally controlling the process (xt , yut ) in the curved domain Dθ with the goal of minimizing a certain expected cost functional, where u is the control variable and θ > 0 is a fixed constant. We assume y that the controller incurs either zero or +∞ terminal costs when the process exits either through x = θ (log y − 1) or x = θ1 y log y, respectively. The motivation in this example follows from Section 4 in Lefebvre (1991), where optimal control of an integrated Brownian motion is singled out.
C. Makasu / Statistics and Probability Letters 79 (2009) 1363–1365
1365
Example 3.1. Let (xt , yut ) be a controlled, degenerate diffusion process in the curved domain Dθ available to a controller through the control variable u, whose goal is to minimize the following expected cost functional:
E
x ,y
τu
"Z 0
1 u2t 2 y2t
# dt + Φ (xτ , yτ ) ,
(3.1)
subject to the controlled process (xt , yut ), given by dxt = µyt dt ; dyt = β yt dt +
x(0) = x,
p
β yt dBt + ut dt ;
y(0) = y,
(3.2)
where u is the control variable in R, and Φ (., .) is a terminal cost of the form
+∞ if x = 1 y log y, θ Φ (x, y) = 0 if x = y (log y − 1). θ Let α = 1/β , for which β > 0. It can be shown (see Whittle (1982, p. 289)) that the optimal control u∗ =
y2 φy (x, y)
for x, y ∈ Dθ
αφ(x, y)
(3.3)
(3.4)
minimizes the expected cost functional (3.1). In this case we can now deduce that the above optimal control problem is associated with the exit probability problem (2.3) and (2.4). Consequently, it now follows from our main result that the optimal control u∗ is explicitly given by u∗ =
y + θx
α log y −
θx
for x, y ∈ Dθ .
(3.5)
y
Remark 3.1. In the above example, notice that the optimal control u∗ → +∞ as x → ( θ1 y log y)− , as one would expect; y otherwise u∗ → y log y/α as x → ( θ (log y − 1))+ . 4. Concluding remarks In this note, we have considered an exit probability problem for the integrated geometric Brownian motion from a prescribed curved domain in the first positive quadrant. Under certain conditions, the exit probability is made explicit. At the end of the note, explicit bounds for the exit probability are also derived, which is followed by one possible application in optimal stochastic control. Acknowledgements We gratefully acknowledge Professors Mario Abundo (Italy) and Mario Lefebvre (Montréal) for their constructive comments and interest in our work, and for sending us references (Lefebvre, 1991, 2005) which motivated the present note. We also thank the anonymous referee and the Associate Editor for their helpful suggestions. References Cox, D.R., Miller, H.D., 1965. The Theory of Stochastic Processes. Methuen, London. Donati-Martin, C., Matsumoto, H., Yor, M., 2000. On positive and negative moments of the integrals of geometric Brownian motions. Statist. Probab. Lett. 49, 45–52. Dufresne, D., 2001. The integral of geometric Brownian motion. Adv. Appl. Probab. 33, 223–241. Jensen, R., Lions, P.L., Souganidis, P.E., 1988. A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations. Proc. Amer. Math. Soc. 102, 975–978. Lefebvre, M., 1991. Forcing a stochastic process to stay in or to leave a given region. Ann. Appl. Probab. 1, 167–172. Lefebvre, M., 2005. First hitting time and place for the integrated geometric Brownian motion. J. Differential Equations Appl. 9 (4), 365–374. Schröder, M., 2003. On the integral of geometric Brownian motion. Adv. Appl. Probab. 35, 159–183. Whittle, P., 1982. Optimization Over Time 1. Wiley, Chichester.