A Stochastic Paradox for Reflected Brownian Motion?

Transcription

1 Proceedings of the 9th International Symposium on Mathematical Theory of Networks and Systems MTNS 2 9 July, 2 udapest, Hungary A Stochastic Parado for Reflected rownian Motion? Erik I. Verriest Abstract This paper is pedagogical in nature. It is shown that a stochastic integral with respect to a Wiener process does not always yield a martingale. Consequently, ignoring the dw term in the calculation of epectations, as might be done in a first back-of-the-envelope approach, can lead to false results. We develop a model for rownian motion constrained to >, for which the moments can be computed eactly. There is no parado if one solves the problem correctly, however a blind application of the Itô calculus yields to the parado that the variance may become negative. In order to put the student back on track, we close the paper by stating some sufficient conditions that guarantee that the stochastic integral is a true martingale. I. INTRODUCTION The objective of this paper is to point out to the student that when one uses the Itô calculus, one should be careful about the interpretation of the Itô differentials, and handle the process of echanging the order of taking mathematical epectations and differentiation or integration with care. Typically, one of the first eamples the student may encounter is the scalar linear system with additive noise, in Itô form given by the equation d = a + bdw Applying the Itô differential rule, it is derived that d 2 = 2a 2 + b 2 + 2bdw. 2 Taking epectations gives respectively and Ed = ae E d 2 = 2aE 2 + b 2. one may conclude that de de 2 = ae and = 2aE 2 +b 2 leading to the well known mean and variance differential equations. ṁ = am P = 2aP + b 2. At this point, the student may epect that E df = d E f, and proceed accordingly in similar problems. The problem of reflected rownian motion shows otherwise. First we build a suitable model for reflected rownian motion in Section II and show some of its properties regarding the eit behavior and show that the process is well defined in Sections III and IV. We compute carelessly, I should say the variance in V and find that it gets negative: a parado! Then we resolve the parado in Section VI. E.I. Verriest is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 2-2, USA erik.verriest@ece.gatech.edu II. A MODEL FOR REFLECTED ROWNIAN MOTION In this section we build a model to approimate a reflected rownian motion by a process which is mathematically more tractable. It is assumed that the process starts deterministically at >. If t is large, it should behave as a pure rownian motion, but when t approaches zero, a potential function should push it back towards the positive side. Let us choose for this potential. This process obeys the Itô equation d = + dw. This does not quite satisfy the Lipschitz conditions, but something can be done by taking ǫ, as the domain of validity, and using various bounding arguments. For now, I shall neglect that issue, and proceed determining some properties. Taking epectations of the above equation yields de = E. 6 Net, we derive the Itô differential for the process /. d = 2 d d = 2 + dw + = 2 dw. Now, taking epectations yields d = 7 The initial condition is =. Hence, for all t > we get E =. 8 t Consequently, from which de = 9 E t = + t. Intuitively, this seems plausible, the effect of the nonlocal potential gives a drift towards the right increasing. We ll sharpen our intuition about this process by determining the average eit time and eit probabilities in the net sections. ISN

2 E. I. Verriest A Stochastic Parado in a Model for Reflected rownian Motion? III. EXPECTED EXIT TIME Consider the closed interval [β, ], with < β < <. The epected first passage time is given as the solution to the equation [] u + 2 u = with the boundary conditions uβ = u =. We get u + 2u = 2 u + u = 2 u + u βu β uβ = β 2 2. The left hand side contains With uβ =, we get [u] = u + u. u βuβ [βu β + uβ] β = β β 2 β u = β β 2 β + βu β β. From u =, it follows that Hence, and thus finally βu β = 2 + β + β 2 β 2. u = u β = β β β + + β 2 or, letting τ [β,] be the eit time for the process t, starting at =, from the interval [β, ], of course with β < < Eτ [β,] = β + + β Also this seems intuitively clear. If is either β or, eit is instantaneous. Taking the limit for β, Eτ [,] = 2 2. If, the eit time approaches infinity, no matter how small is. This reinforces the intuition that eit at zero is not possible. IV. EXIT PROAILITIES Consider again the closed interval [β, ], with < β < <. The probability of eit at is given by the solution to the equation [] v + 2 v = with the boundary conditions vβ = and v =. We get v + v = v + v = A. for some constant A. The left hand side is [v] = A v = A + a, where a is another constant. With vβ =, we get a = Aβ, and from v = = A + a = A β, it follows that A = β and thus finally It follows that v = β Pr eit at = β. 6 β β. 7 If β, then Pr eit at. Hence the probability of the process crossing over = is zero. Also this seems intuitively plausible. V. VARIANCE PARADOX In this section the variance for this approimation to the reflected rownian motion is computed. Since the process starts deterministically at, the initial variance is zero. Consider now the process 2 t. With the Itô-rule, Taking epectations d 2 = 2d + = 2 + dw + = + 2dw. 8 de 2 =. Integrating, this yields eplicitly E 2 t = 2 + t. We also have the deterministic equation E t 2 = t

3 Proceedings of the 9th International Symposium on Mathematical Theory of Networks and Systems MTNS 2 9 July, 2 udapest, Hungary The variance is obtained from Var t = E 2 t E t 2 = 2 + t t 2 = t t For t > 2, this variance is negative, a parado! What went wrong? VI. RESOLUTION OF THE PARADOX Consider the d-dimensional rownian motion, where the coordinates of the sample paths are independent standard one-dimensional rownian motions. It is known [2, p.9] that if rt = t, then Pr { rt dr R s } = gt s, rs, rdr, where gt, a, b = t ep a2 + b 2 ab ab d 2 I d 2 b d t and I d/2 is the modified essel function. The probability is conditioned on the filtration R s = {rτ τ s}. This gt, a, b is also the fundamental solution of u t = 2 with boundary condition 2 u b 2 + d b lim b d u b b =. u b, The corresponding Itô equation for the process rt is drt = d + dwt. 2rt Note that for d =, the three-dimensional rownian motion, this gives drt = + dwt, 2 rt which is precisely the equation we were investigating. Equation 2 and its connection with the essel process provides now a different avenue to resolve the parado described in section V. Consider thus the three-dimensional rownian motion M, starting at the point with coordinates,,, where = r. Let rt be the distance of the M from the origin. The stochastics of rt, the distance of the rownian particle from the origin, is identical to the -D dynamics on the halfline for the particle in a / potential. ut this -dimensional problem is easily solved by elementary probability []. The easiest way is to obtain first the density of r 2. From the properties of -D rownian motion, we know that t is normally distributed with mean r and variance t standard case. Likewise, both yt and zt are normally distributed with mean zero and variance t. These three processes are independent. Hence ut = y 2 t+z 2 t has a χ-square distribution with two degrees of freedom. The density is f ut u = ep u Hu, 2 where H is the Heaviside step function. The random variable v = 2 t has density With f vt v = 2 f v + f v Hv. 22 v f t = 2πt ep 2, 2 this gives ep v r 2 f vt v = Hv. 2 v 2πt + erf r Independence of ut and vt results in the convolution formula for the density of their sum r 2 t = ut + vt., f r 2 tρ = = ρ f vt ρ sf ut sds ep ρ r 2 2 /2 r πt ep ρ + r 2 + Hρ. 2 With this density, the epected value of r 2 t is readily computed E r 2 t = ρf r2 tρdρ = r 2 + t. 26 Likewise, we obtain the epected value of rt by E rt = = r π ρfr 2 tρdρ r 2 r ep + + πr 2 + terf r. 27 Finally, the inverse distance, /rt has average value E = f r rt ρ 2 tρdρ = r erf. 28 r Unless t =, this is strictly less than /r. Figure r displays E rt and it can be seen that the naive prediction equality to only holds approimately for small values of t and. It is now easily verified that for the process we considered in Section II, we get indeed, de = E,

4 E. I. Verriest A Stochastic Parado in a Model for Reflected rownian Motion? r~ 2 t~ t~ Fig.. The process r rt Fig. 2. Variance of rt but which is not zero! de = ep 2, 2πt Finally, the correct epression of the variance follows from Varrt = r 2 t rt 2. Figure 2 displays the variance for different values of as function of time. For, the variance has slope 8/π, whereas for we get slope as epected standard rownian motion. The curves in the figure correspond to =., 2,,,, 2 and. The figure below shows the average position as function of time together with the -sigma bounds for the uncertainty σ = t for initialization at =, and 2. VII. FURTHER THOUGHTS The origin of the problem is that in the differential Itô equation we get correctly E d =. However, this does not imply that de =. The integrated form is: E = t E t dws s 2. More detailed arguments, invoking stopping times, show that /mint, τ k approaches /t a.e. as k +, however the convergence is neither monotone nor dominated 2 2 t~ Fig.. Evolution of mean and uncertainty -σ bounds so the only thing one can do to pass to the limit in the integral is to use Fatou s Lemma to get E t = E lim k + mint, τ k lim inf E = /. k + mint, τ k So one gets only inequality instead of the equality E /t = /. Hence interchanging epectation and integration is not implied, as only Fatou s lemma can be 678

5 Proceedings of the 9th International Symposium on Mathematical Theory of Networks and Systems MTNS 2 9 July, 2 udapest, Hungary used to provide the bound E. 2 t Also we note that does not satisfy a Lipshitz condition since y cannot be bounded by C y for any constant C. Hence this process may not be an Itô diffusion. We note that the three dimensional rownian motion is trivially an Itô diffusion. Applying the Itô-rule to the map φ : w = w w 2 w w = w 2 + w2 2 + w2 is dubious, since φ is not C 2 at the origin. Formal application gives i= d w = w + i= w i dw i w. w Now, idw i w has the same finite-dimensional distributions as a one dimensional rownian motion dw [, p. ]. Since [w, w 2, w ] never hits the origin, is an Itô diffusion. It can be shown that the process uw = w is a positive supermartingale [, p.]. If {T n} is an increasing sequence of stopping times, with T n, then the process uwt T n is a martingale, implying that uwt is a local martingale. Since E uwt as t, and E uw =, it follows that uwt cannot be a martingale. VIII. CONCLUSIONS We have illustrated by way of eample, that it is worthwhile to check the intricate details and conditions involved to guarantee that the differential rule and interchanging integrals integration and epectation are applicable. We have shown that carelessness leads to the paradoical situation of resulting in a process with a negative variance. The parado is resolved by developing properties of the three dimensional rownian motion, and its related essel process. For this approimation of the reflected rownian motion by a repulsive potential, eit probabilities and epected eit times from an interval were calculated. Sufficient conditions were presented under which the result of the Itô integral is a true martingale. Acknowledgement The author is indebted to Prof. Andrzej Swiech Math, GT, for valuable discussions on the parado epressed in section V and providing the impetus to unravel the parado. In addition, the reviewers comments, which helped to improve this note, are gratefully acknowledged. REFERENCES [] Z. Schuss, Theory and Application of Stochastic Differential Equations Wiley 98. [2] K. Itô and H.P. McKean, Jr. Diffusion Processes and their Sample Paths. Academic Press, 96. [] A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw-Hill, 96 []. /Oksendal, Stochastic Differential Equations, Fourth Ed. Springer- Verlag 99. [] P. Protter, Stochatic Integration and Differential Equations, Springer- Verlag 99. We recall the following result []. Let w be a Wiener process rownian motion. Define F t to be the σ-algebra generated by the random variables ws, for s t. Let ft, ω : [, Ω R be such that i t, ω ft, ω is F-measurable, where denotes the σ-algebra on [, ii ft, ω is F t -adapted iii E t s fτ, ω2 dτ < then the integral t fτ, ωdwt, ω in the Itô-sense is a s martingale. In fact the conditions can be relaed [, p.]: Condition ii may be replaced by ii : There eists an increasing family of σ-algebras H t, such that wt is a martingale with respect to H t and ft is H t -adapted. Condition iii can be relaed to: iii { t } Pr fs, ω 2 ds < for all t =. 679