On the quantiles of the Brownian motion and their hitting times.
|
|
- Evelyn Melton
- 6 years ago
- Views:
Transcription
1 On the quantiles of the Brownian motion and their hitting times. Angelos Dassios London School of Economics May 23 Abstract The distribution of the α-quantile of a Brownian motion on an interval [, t] has been obtained motivated by a problem in financial mathematics. In this paper we generalise these results by calculating an explicit expression for the joint density of the α-quantile of a standard Brownian motion, its first and last hitting times and the value of the process at time t. Our results can be easily generalised for a Brownian motion with drift. It is shown that the first and last hitting times follow a transformed arcsine law. 1 Introduction Let X s, s be a real valued stochastic process on a probability space Ω, F, Pr. For < α < 1, define the α quantile of the path of X s, s up to a fixed time t by M X α, t = inf { x : t } 1 X s x ds > αt. 1 The study of the quantiles of various stochastic processes has been recently undertaken as a response to a problem arising in the field of mathematical finance, the pricing of a particular path-dependent financial option; see Miura [7], Akahori [1] and Dassios [3]. This involves calculating quantities such as E h M X α, t, where h x = e x b + or some other appropriate function. This requires obtaining the distribution of X t. In the case where X s, s is a Lévy process having stationary and independent increments the following result was obtained: 1
2 Proposition 1 Let X 1 s and X 2 s be independent copies of X s. Then, MX α, t law sup s αt X = 1 s + inf s 1 αt X 2 s X t X 1 αt + X α t When X s, s is a Brownian motion, we can use this result and obtain an explicit formula for the joint density of M X α, t and X t. This result was first proved for a Brownian motion with drift; see Dassios [3] and Embrechts, Rogers and Yor [5] and for Lévy processes by Dassios [4]. There is also a similar result for discrete time random walks first proved by Wendel [8]. We now let be the first, and L X α, t = inf {s [, t] : X s = M X α, t} K X α, t = sup {s [, t] : X s = M X α, t}, the last time the process hits M X α, t. One can now introduce a barrier element to the financial application by making the option worthless if the quantile is hit too early or too late. For example, this can involve calculating quantities such as E h M X α, t 1 L X α, t > v, K X α, t < u. The first study of these quantities can be found in Chaumont [2]. By using combinatorial arguments he derives of the same type as Proposition 1 that are extensions to Wendel s results in discrete time. In the case where the random walk steps can only take the values +1 or -1, a represenation for the analogues of L X α, t and K X α, t is obtained. Finally he derives a continuous time representation for the triple law of M X α, t, L X α, t and X t, extending Proposition 1 when X t is a Brownian motion. We will adopt a direct approach that seems better suited to obtaining explicit expressions for the densities involved. We will also derive alternative representations and prove a remarkable arc-sine law. For the rest of the paper we assume that X s, s is a standard Brownian motion. We will derive the joint density of M X α, t, L X α, t, K X α, t and X t. If we denote this density by f y, x, u, v, our results can be generalised for a Brownian motion with drift m, using a Cameron- Martin-Girsanov transformation. The corresponding density will be f y, x, u, v exp mx m 2 t/2. Before we obtain the density of M X α, t, L X α, t, K X α, t, X t, we will first show that the law of L X α, t and K X α, t is a transformed arcsine law. 2
3 2 An arcsine law for L X α, t. Let S X t = sup s t {X s} and θ X t = sup {s [, t] : X s = S X t}. We prove the following theorem: Theorem 1 For u >, Pr L X α, t > u = Pr u < θ X t αt + Pr u < θ X t 1 α t 3 and 1 u αt + 1 u 1 α t Pr L X α, t du = π du. 4 u t u Furthermore, K X α, t has the same distribution as t L X α, t. Proof We will first prove that We observe that Pr M X α, t >, L X α, t > u = Pr u < θ X t αt. 5 Pr M X α, t >, L X α, t > u = Pr M X α, t > S X u = t Pr 1 X s S X u ds < αt = t Pr 1 X s X u S X u X u ds < αt. 6 u Let X s = X u + s X u. X s, s is a standard Brownian motion which is independent of X s, s u. We condition on S X u X u = c, and set τ c = inf {s > : X s = c} and X s = X τ c + s c. X s, s is a standard Brownian motion which is independent of both X s, s u and X s, s τ c. We have that t u Pr 1 X s c ds < αt u = αt u Pr τ c dr Pr t u r 1 X s ds < αt u r t u r and since 1 X s ds has the same arcsine law as θ X t u r, this is equal to αt u Pr τ c dr Pr θ X t u r < αt u r = 3
4 αt u Pr τ c dr Pr Pr sup X s > s αt u sup X s > s αt u r sup αt u s t u X s, sup αt u r s t u r X s = sup s αt u X s > c and so 6 is equal to supu s αt X s X u > sup Pr αt s t X s X u, = sup u s αt X s X u > sup s u X s X u Pr u < θ X t αt. Since X s, s is also a standard Brownian motion and M X α, t = M X 1 α, t almost surely, we use X s instead of X s and we get Pr M X α, t <, L X α, t > u = Pr u < θ X t 1 α t. 7 Adding 5 and 7 we get 3, and since θ X t has an arcsine law, 4 follows. To see that K X α, t has the same distribution as L X α, t, set X s = X t s X t. Clearly X s, s t is a standard Brownian motion and we can easily see that M X α, t = M X α, t X t and K X α, t = t L X α, t. We can also extend our result and obtain the joint distribution of M X α, t, L X α, t also of M X α, t X t, t K X α, t. Theorem 2 For b >, and for b <, Pr M X α, t db, L X α, t du = 8 Pr S X t db, θ X t du 1 < u < αt, 9 Pr M X α, t db, L X α, t du = 1 Pr S X t d b, θ X t du 1 < u < 1 α t. 11 Furthermore M X α, t, L X α, t and M X α, t X t, t K X α, t have the same distribution. 4
5 Proof Let b > and u < αt. We then have that Pr M X α, t > b, L X α, t > u = Pr S X u < M X α, t, M X α, t > b = Pr b < S X u < M X α, t + Pr S X u < b < M X α, t. 12 Let τ b = inf {s > : X s = b} and X s = X τ b + s c. X s, s is a standard Brownian motion which is independent of X s, s τ c. Using theorem 1, we have u u Pr b < S X u < M X α, t = t r Pr τ b dr Pr 1 X s S X u r < αt r = Pr τ b dr Pr Furthermore, u M X αt r t r, t r >, L X αt r t r, t r > u r = Pr τ b dr Pr u r < θ X t r < αt r = 13 Pr S X u < b < M X α, t = Pr αt u = Pr u < θ X t < αt, S X u > b. 14 t r Pr τ b dr Pr αt u S X u < b, t 1 X s b ds < αt = 1 X s < αt r Pr τ b dr Pr θ X t r < αt r = Pr u < θ X t < αt, S X u < b, sup u s αt X s > b. 15 Adding 14 and 15 together, we see that 12 is equal to Pr u < θ X t < αt, sup X s > b = Pr u < θ X t < αt, S X t > b u s αt which leads to 9. Since X s, s is also a standard Brownian motion and M X α, t = M X 1 α, t almost surely, we use X s instead of X s and we get that for b <, Pr M X α, t < b, L X α, t > u = Pr u < θ X t 1 α t, S X t > b, 5
6 which leads to 11. To see that t K X α, t, M X α, t X t has the same distribution as L X α, t, M X α, t, set again X s = X t s X t. Clearly X s, s t is a standard Brownian motion and we can easily see that M X α, t = M X α, t X t, and so M X α, t X t = M X α, t and K X α, t = t L X α, t. Remarks 1. The distribution of θ X t, S X t is well known see for example Karatzas and Shreve [6], page 12. From this and theorem 2, we can deduce the density of L X α, t, M X α, t. This is given by b Pr M X α, t db, L X α, t du = π u 3 t u exp b2 2u [1 < u < αt, b > + 1 < u < 1 α t, b < ] dbdu Theorem 2 also leads to an alternative expression for the distribution of M X α, t ; that is for b > and Pr M X α, t db = Pr S X t db, < θ X t < αt, Pr M X α, t db = Pr S X t d b, < θ X t < 1 α t, for b <. 3. Using the argument at the end of the proof, we can generalise the last assertion of the theorem and observe that has the same law as and so we see that and K X α, t, M X α, t X t, X t t L X α, t, M X α, t, X t K X α, t, M X α, t, X t t L X α, t, M X α, t X t, X t, have the same distribution, a fact we will use in the following section. 6
7 3 The joint law of L X α, t, K X α, t, M X α, t, X t. From now on we will denote the density of τ b by k, ; that is for v >, Pr τ b dv = k v, b dv = 2 b exp b2 dv. 17 2πv 3 2v We will also denote the joint density of M v X t, t, X t by g,,, ; that is for < v < t, v Pr M X t, t db, X t da = g b, a, v, t dbda. We can calculate g,,, by using the proposition in the introduction. MX v t, t, X t has the same distribution as S X1 v S X2 t v, X 1 v X 2 t v, where X 1 s, s v and X 2 s, s t v are independent standard Brownian motions. The density of S X t, X t is given by Pr S X t db, X t da = b a 2b a2 exp 1 b, b a dadb 19 2πt 3 2t see Karatzas and Shreve [6], p.95. We observe that since 19 is bounded, g,,, is a bounded density. For our results, we need to calculate g,, v, t. This is the same as the value of the density of M X v t, t, M X v t, t X t at,. From 19 we see that Pr S X t dy, S X t X t dx = 2 2 y + x y + x2 exp 1 y, x dydx 21 2πt 3 2t and it is a simple exercise to verify that 2 y + x exp 2πv 3 g,, v, t = y + x2 2v 2 y + x 2π t v 3 exp y + x2 dxdy 2 t v v t v = t We will now obtain a preliminary result. 7
8 Lemma 1 For any u and v, such that < u < v < t, we have that Pr L X α, t > u, M X α, t db, X t da, K X α, t > v = Pr τ b > u, M X α, t db, X t da, K X α, t > v. 23 Proof Since M X α, t = M X 1 α, t, it suffices to prove 23 for b >. We have to prove that 1 lim δ,ε δε {Pr L X α, t > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v Pr τ b > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v} =. 24 Let X s = X s + u X u. We then have that Pr L X α, t > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v Pr τ b > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v = Pr b < S X u < M X α, t b + δ, X t a, a + ε], K X α, t > v Pr b < S X u < M X α, t b + δ, X t a, a + ε] = b < S X u < b + δ, Pr S X u < M X αt u, t u + X u b + δ,. 25 X t u + X u a, a + ε] Since X s, s t u is independent of X s, s u, and g,,, is bounded, we condition on S X u = y and X u = x and see that there is a constant K, such that y < MX αt u, t u + x b + δ, Pr X Kε b + δ y. t u + x a, a + ε] We therefore conclude that 25 is bounded by KεE b + δ S X u 1 b < S X u < b + δ Kεδ Pr b < S X u < b + δ and by the continuity of the distribution of S X u, we see that the limit in 24 is zero. As a corollary we will obtain the distribution of L X α, t, M X α, t, X t. 8
9 Corollary 1 The law of L X α, t, M X α, t, X t is given by Pr L X α, t du, M X α, t db, X t da = { k b, u g, a b, αt u, t u 1 < u < αt dudbda b > k b, u g, a b, αt, t u 1 < u < αt dudbda b < 26 Proof For b >, since X s + τ b X τ b, s t τ b is independent of X s, s τ b, we have that αt v Pr τ b > v, M X α, t b, b + δ], X t a, a + ε = Pr τ b du Pr M X αt u, t u, δ], X t a b, a b + ε. For b <, we use that M X α, t = M X 1 α, t and so g, b a, 1 α t u, t u = g, a b, αt, t u. We can now obtain the law of L X α, t, K X α, t, M X α, t, X t. Theorem 3 Pr L X α, t du, K X α, t dv, M X α, t db, X t da = 2 b b a dudvdbda exp b2 b a2 π 2 v u 2 u 3 t v 3 2u 2 t v v u 1 α t 1 α t1 u >, u + 1 α t < v < t b >, b > a αt u v αt1 < u < αt < v < t b >, b < a. v u αt αt1 u >, u + αt < v < t b <, b > a 1 α t u v 1 α t1 < u < 1 α t < v < t b <, b < a 27 Proof We start with the case b >, b > a. Using 23, and choosing ε such that a + ε < b, we need to look at Pr τ b r, K X α, t v, M X α, t b, b + δ], X t a, a + ε] = τ Pr b r, M X α, t b, b + δ], = X t a, a + ε], M X α, t sup v s t X s 9
10 r Pr τ b du Pr r Pr τ b du Pr M X αt u, t u, δ], X t u a b, a b + ε], M X αt u, t u sup v u s t u X s M X αt u, t u, δ], X t u a b, a b + ε], K X αt u, t u v u Using the last remark of the previous section, we then see that M X αt u, t u, δ], Pr X t u a b, a b + ε], = K X αt u, t u v u Pr M X αt u, t u X t u, δ], X t u a b, a b + ε], L X αt u, t u t v From the previous theorem we see that the density of = L X αt u, t u, M X αt u, t u X t u, X t u at t v,, a b is k b a, t v g,, v u 1 α t, v u 1 < t v < αt u. Combining this with 28 we get that L X α, t, K X α, t, M X α, t, X t has a continuous density at u, v, b, a that is given by k b, u k b a, t v g,, v u 1 α t, v u 3 1 u >, u + 1 α t < v < t. 31 We now look at the case b >, b < a. Using 23, and choosing δ such that b + δ < a, we need to look at Pr τ b r, K X α, t > v, M X α, t b, b + δ], X t a, a + ε] = τ Pr b r, M X α, t b, b + δ], = X t a, a + ε], M X α, t < inf v s t X s r Pr τ b du Pr M X αt u, t u, δ], X t u a b, a b + ε], M X αt u, t u < inf v u s t u X s = 1
11 r Pr τ b du Pr M X αt u, t u, δ], X t u a b, a b + ε], K X αt u, t u < v u Using 29 and the previous theorem we see that the density of. 32 L X αt u, t u, M X αt u, t u X t u, X t u at t v,, a b is k b a, t v g,, αt u, v u 1 αt < v. Combining this with 28 we get that L X α, t, K X α, t, M X α, t, X t has a continuous density at u, v, b, a that is given by k b, u k b a, t v g,, αt u, v u 1 < u < αt < v < t. 33 Substituting 17 and 22 into 31 and 33, we get the first two legs of 27. Considering X s, s t and observing that M X α, t = M X 1 α, t, L X α, t = L X 1 α, t and K X α, t = K X 1 α, t yields the rest of 27. References [1] Akahori, J., 1995, Some formulae for a new type of path-dependent option, Ann. Appl. Prob., 5, [2] Chaumont, L., 1999, A path transformation and its applications to fluctuation theory, J. London Math. Soc. 2, 59, no 2, [3] Dassios, A., 1995, The distribution of the quantiles of a Brownian motion with drift and the pricing of related path-dependent options, Ann. Appl. Prob., 5, [4] Dassios, A., 1996, Sample quantiles of stochastic processes with stationary and independent increments, Ann. Appl. Prob., 6, [5] Embrechts P.,Rogers, L.C.G., and Yor, M., 1995, A proof of Dassios representation of the α-quantile of Brownian motion with drift, Ann. Appl. Prob., 5,
12 [6] Karatzas I. and S.E. Shreve, 1988, Brownian motion and Stochastic Calculus, Springer-Verlag. [7] Miura, R., 1992, A note on a look-back option based on order statistics, Hitosubashi Journal of Commerce and Management, 27, [8] Wendel, J.G., 196, Order statistics of partial sums, Ann. of Math. Stat., 31,
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family
More informationON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER
ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER GERARDO HERNANDEZ-DEL-VALLE arxiv:1209.2411v1 [math.pr] 10 Sep 2012 Abstract. This work deals with first hitting time densities of Ito processes whose
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 7 9/25/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 7 9/5/013 The Reflection Principle. The Distribution of the Maximum. Brownian motion with drift Content. 1. Quick intro to stopping times.
More informationBrownian Motion and Stochastic Calculus
ETHZ, Spring 17 D-MATH Prof Dr Martin Larsson Coordinator A Sepúlveda Brownian Motion and Stochastic Calculus Exercise sheet 6 Please hand in your solutions during exercise class or in your assistant s
More informationRichard F. Bass Krzysztof Burdzy University of Washington
ON DOMAIN MONOTONICITY OF THE NEUMANN HEAT KERNEL Richard F. Bass Krzysztof Burdzy University of Washington Abstract. Some examples are given of convex domains for which domain monotonicity of the Neumann
More informationA MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT
A MODEL FOR HE LONG-ERM OPIMAL CAPACIY LEVEL OF AN INVESMEN PROJEC ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider an investment project that produces a single commodity. he project s operation yields
More informationPotentials of stable processes
Potentials of stable processes A. E. Kyprianou A. R. Watson 5th December 23 arxiv:32.222v [math.pr] 4 Dec 23 Abstract. For a stable process, we give an explicit formula for the potential measure of the
More informationThe Cameron-Martin-Girsanov (CMG) Theorem
The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version,
More informationInformation and Credit Risk
Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information
More informationLecture 22 Girsanov s Theorem
Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n
More informationOn Optimal Stopping Problems with Power Function of Lévy Processes
On Optimal Stopping Problems with Power Function of Lévy Processes Budhi Arta Surya Department of Mathematics University of Utrecht 31 August 2006 This talk is based on the joint paper with A.E. Kyprianou:
More informationA Barrier Version of the Russian Option
A Barrier Version of the Russian Option L. A. Shepp, A. N. Shiryaev, A. Sulem Rutgers University; shepp@stat.rutgers.edu Steklov Mathematical Institute; shiryaev@mi.ras.ru INRIA- Rocquencourt; agnes.sulem@inria.fr
More informationRegular Variation and Extreme Events for Stochastic Processes
1 Regular Variation and Extreme Events for Stochastic Processes FILIP LINDSKOG Royal Institute of Technology, Stockholm 2005 based on joint work with Henrik Hult www.math.kth.se/ lindskog 2 Extremes for
More informationOn Reflecting Brownian Motion with Drift
Proc. Symp. Stoch. Syst. Osaka, 25), ISCIE Kyoto, 26, 1-5) On Reflecting Brownian Motion with Drift Goran Peskir This version: 12 June 26 First version: 1 September 25 Research Report No. 3, 25, Probability
More informationSquared Bessel Process with Delay
Southern Illinois University Carbondale OpenSIUC Articles and Preprints Department of Mathematics 216 Squared Bessel Process with Delay Harry Randolph Hughes Southern Illinois University Carbondale, hrhughes@siu.edu
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationarxiv: v1 [math.pr] 11 Jan 2013
Last-Hitting Times and Williams Decomposition of the Bessel Process of Dimension 3 at its Ultimate Minimum arxiv:131.2527v1 [math.pr] 11 Jan 213 F. Thomas Bruss and Marc Yor Université Libre de Bruxelles
More informationThe Uniform Integrability of Martingales. On a Question by Alexander Cherny
The Uniform Integrability of Martingales. On a Question by Alexander Cherny Johannes Ruf Department of Mathematics University College London May 1, 2015 Abstract Let X be a progressively measurable, almost
More informationThe exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: the Cramér case
The exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: the Cramér case Konstantin Borovkov and Zbigniew Palmowski Abstract For a multivariate Lévy process satisfying
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationAN EXTREME-VALUE ANALYSIS OF THE LIL FOR BROWNIAN MOTION 1. INTRODUCTION
AN EXTREME-VALUE ANALYSIS OF THE LIL FOR BROWNIAN MOTION DAVAR KHOSHNEVISAN, DAVID A. LEVIN, AND ZHAN SHI ABSTRACT. We present an extreme-value analysis of the classical law of the iterated logarithm LIL
More informationPredicting the Time of the Ultimate Maximum for Brownian Motion with Drift
Proc. Math. Control Theory Finance Lisbon 27, Springer, 28, 95-112 Research Report No. 4, 27, Probab. Statist. Group Manchester 16 pp Predicting the Time of the Ultimate Maximum for Brownian Motion with
More informationShort-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility
Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility José Enrique Figueroa-López 1 1 Department of Statistics Purdue University Statistics, Jump Processes,
More informationPath transformations of first passage bridges
Path transformations of first passage idges Jean Bertoin 1),Loïc Chaumont 2) and Jim Pitman 3) Technical Report No. 643 Department of Statistics University of California 367 Evans Hall # 3860 Berkeley,
More informationWhite noise generalization of the Clark-Ocone formula under change of measure
White noise generalization of the Clark-Ocone formula under change of measure Yeliz Yolcu Okur Supervisor: Prof. Bernt Øksendal Co-Advisor: Ass. Prof. Giulia Di Nunno Centre of Mathematics for Applications
More information9 Brownian Motion: Construction
9 Brownian Motion: Construction 9.1 Definition and Heuristics The central limit theorem states that the standard Gaussian distribution arises as the weak limit of the rescaled partial sums S n / p n of
More informationFinite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims
Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims Tao Jiang Abstract
More informationSkew Brownian Motion and Applications in Fluid Dispersion
Skew Brownian Motion and Applications in Fluid Dispersion Ed Waymire Department of Mathematics Oregon State University Corvallis, OR 97331 *Based on joint work with Thilanka Appuhamillage, Vrushali Bokil,
More informationStochastic Calculus for Finance II - some Solutions to Chapter VII
Stochastic Calculus for Finance II - some Solutions to Chapter VII Matthias hul Last Update: June 9, 25 Exercise 7 Black-Scholes-Merton Equation for the up-and-out Call) i) We have ii) We first compute
More informationBrownian Motion. Chapter Definition of Brownian motion
Chapter 5 Brownian Motion Brownian motion originated as a model proposed by Robert Brown in 1828 for the phenomenon of continual swarming motion of pollen grains suspended in water. In 1900, Bachelier
More informationCitation Osaka Journal of Mathematics. 41(4)
TitleA non quasi-invariance of the Brown Authors Sadasue, Gaku Citation Osaka Journal of Mathematics. 414 Issue 4-1 Date Text Version publisher URL http://hdl.handle.net/1194/1174 DOI Rights Osaka University
More informationChange detection problems in branching processes
Change detection problems in branching processes Outline of Ph.D. thesis by Tamás T. Szabó Thesis advisor: Professor Gyula Pap Doctoral School of Mathematics and Computer Science Bolyai Institute, University
More informationRepresentations of Gaussian measures that are equivalent to Wiener measure
Representations of Gaussian measures that are equivalent to Wiener measure Patrick Cheridito Departement für Mathematik, ETHZ, 89 Zürich, Switzerland. E-mail: dito@math.ethz.ch Summary. We summarize results
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationShifting processes with cyclically exchangeable increments at random
Shifting processes with cyclically exchangeable increments at random Gerónimo URIBE BRAVO (Collaboration with Loïc CHAUMONT) Instituto de Matemáticas UNAM Universidad Nacional Autónoma de México www.matem.unam.mx/geronimo
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More informationLimit theorems for multipower variation in the presence of jumps
Limit theorems for multipower variation in the presence of jumps Ole E. Barndorff-Nielsen Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8 Aarhus C, Denmark oebn@imf.au.dk
More informationSome Aspects of Universal Portfolio
1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical
More informationMalliavin Calculus in Finance
Malliavin Calculus in Finance Peter K. Friz 1 Greeks and the logarithmic derivative trick Model an underlying assent by a Markov process with values in R m with dynamics described by the SDE dx t = b(x
More informationThe Subexponential Product Convolution of Two Weibull-type Distributions
The Subexponential Product Convolution of Two Weibull-type Distributions Yan Liu School of Mathematics and Statistics Wuhan University Wuhan, Hubei 4372, P.R. China E-mail: yanliu@whu.edu.cn Qihe Tang
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationGARCH processes continuous counterparts (Part 2)
GARCH processes continuous counterparts (Part 2) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/
More informationLECTURE 2: LOCAL TIME FOR BROWNIAN MOTION
LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in
More informationSupplement A: Construction and properties of a reected diusion
Supplement A: Construction and properties of a reected diusion Jakub Chorowski 1 Construction Assumption 1. For given constants < d < D let the pair (σ, b) Θ, where Θ : Θ(d, D) {(σ, b) C 1 (, 1]) C 1 (,
More informationC.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series
C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationBrownian Motion and lorentzian manifolds
The case of Jürgen Angst Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg École d été de Probabilités de Saint-Flour juillet 2008, Saint-Flour 1 Construction of the diffusion,
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationLarge Deviations for Perturbed Reflected Diffusion Processes
Large Deviations for Perturbed Reflected Diffusion Processes Lijun Bo & Tusheng Zhang First version: 31 January 28 Research Report No. 4, 28, Probability and Statistics Group School of Mathematics, The
More information1 Independent increments
Tel Aviv University, 2008 Brownian motion 1 1 Independent increments 1a Three convolution semigroups........... 1 1b Independent increments.............. 2 1c Continuous time................... 3 1d Bad
More informationStochastic Calculus. Kevin Sinclair. August 2, 2016
Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed
More informationSome Tools From Stochastic Analysis
W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click
More informationIntegral representations in models with long memory
Integral representations in models with long memory Georgiy Shevchenko, Yuliya Mishura, Esko Valkeila, Lauri Viitasaari, Taras Shalaiko Taras Shevchenko National University of Kyiv 29 September 215, Ulm
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationBayesian quickest detection problems for some diffusion processes
Bayesian quickest detection problems for some diffusion processes Pavel V. Gapeev Albert N. Shiryaev We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process
More informationLAN property for sde s with additive fractional noise and continuous time observation
LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday,
More informationLOCATION OF THE PATH SUPREMUM FOR SELF-SIMILAR PROCESSES WITH STATIONARY INCREMENTS. Yi Shen
LOCATION OF THE PATH SUPREMUM FOR SELF-SIMILAR PROCESSES WITH STATIONARY INCREMENTS Yi Shen Department of Statistics and Actuarial Science, University of Waterloo. Waterloo, ON N2L 3G1, Canada. Abstract.
More informationThe strictly 1/2-stable example
The strictly 1/2-stable example 1 Direct approach: building a Lévy pure jump process on R Bert Fristedt provided key mathematical facts for this example. A pure jump Lévy process X is a Lévy process such
More informationThe absolute continuity relationship: a fi» fi =exp fif t (X t a t) X s ds W a j Ft () is well-known (see, e.g. Yor [4], Chapter ). It also holds with
A Clarification about Hitting Times Densities for OrnsteinUhlenbeck Processes Anja Göing-Jaeschke Λ Marc Yor y Let (U t ;t ) be an OrnsteinUhlenbeck process with parameter >, starting from a R, that is
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationNested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model
Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model Xiaowei Chen International Business School, Nankai University, Tianjin 371, China School of Finance, Nankai
More informationLOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES
LOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES ADRIAN D. BANNER INTECH One Palmer Square Princeton, NJ 8542, USA adrian@enhanced.com RAOUF GHOMRASNI Fakultät II, Institut für Mathematik Sekr. MA 7-5,
More informationProf. Erhan Bayraktar (University of Michigan)
September 17, 2012 KAP 414 2:15 PM- 3:15 PM Prof. (University of Michigan) Abstract: We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled
More informationRandomization in the First Hitting Time Problem
Randomization in the First Hitting Time Problem Ken Jacson Alexander Kreinin Wanhe Zhang February 1, 29 Abstract In this paper we consider the following inverse problem for the first hitting time distribution:
More informationBernstein-gamma functions and exponential functionals of Lévy processes
Bernstein-gamma functions and exponential functionals of Lévy processes M. Savov 1 joint work with P. Patie 2 FCPNLO 216, Bilbao November 216 1 Marie Sklodowska Curie Individual Fellowship at IMI, BAS,
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationProving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control
Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control Erhan Bayraktar University of Michigan joint work with Virginia R. Young, University of Michigan K αρλoβασi,
More informationarxiv: v1 [math.pr] 24 Sep 2018
A short note on Anticipative portfolio optimization B. D Auria a,b,1,, J.-A. Salmerón a,1 a Dpto. Estadística, Universidad Carlos III de Madrid. Avda. de la Universidad 3, 8911, Leganés (Madrid Spain b
More information2012 NCTS Workshop on Dynamical Systems
Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ gentz 2012 NCTS Workshop on Dynamical Systems National Center for Theoretical Sciences, National Tsing-Hua University Hsinchu,
More informationMaximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments
Austrian Journal of Statistics April 27, Volume 46, 67 78. AJS http://www.ajs.or.at/ doi:.773/ajs.v46i3-4.672 Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments Yuliya
More informationObstacle problems for nonlocal operators
Obstacle problems for nonlocal operators Camelia Pop School of Mathematics, University of Minnesota Fractional PDEs: Theory, Algorithms and Applications ICERM June 19, 2018 Outline Motivation Optimal regularity
More informationPITMAN S 2M X THEOREM FOR SKIP-FREE RANDOM WALKS WITH MARKOVIAN INCREMENTS
Elect. Comm. in Probab. 6 (2001) 73 77 ELECTRONIC COMMUNICATIONS in PROBABILITY PITMAN S 2M X THEOREM FOR SKIP-FREE RANDOM WALKS WITH MARKOVIAN INCREMENTS B.M. HAMBLY Mathematical Institute, University
More informationOther properties of M M 1
Other properties of M M 1 Přemysl Bejda premyslbejda@gmail.com 2012 Contents 1 Reflected Lévy Process 2 Time dependent properties of M M 1 3 Waiting times and queue disciplines in M M 1 Contents 1 Reflected
More informationOPTIMAL STOPPING OF A BROWNIAN BRIDGE
OPTIMAL STOPPING OF A BROWNIAN BRIDGE ERIK EKSTRÖM AND HENRIK WANNTORP Abstract. We study several optimal stopping problems in which the gains process is a Brownian bridge or a functional of a Brownian
More informationA NEW PROOF OF THE WIENER HOPF FACTORIZATION VIA BASU S THEOREM
J. Appl. Prob. 49, 876 882 (2012 Printed in England Applied Probability Trust 2012 A NEW PROOF OF THE WIENER HOPF FACTORIZATION VIA BASU S THEOREM BRIAN FRALIX and COLIN GALLAGHER, Clemson University Abstract
More informationThe Codimension of the Zeros of a Stable Process in Random Scenery
The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More informationMean-square Stability Analysis of an Extended Euler-Maruyama Method for a System of Stochastic Differential Equations
Mean-square Stability Analysis of an Extended Euler-Maruyama Method for a System of Stochastic Differential Equations Ram Sharan Adhikari Assistant Professor Of Mathematics Rogers State University Mathematical
More informationRuin probabilities of the Parisian type for small claims
Ruin probabilities of the Parisian type for small claims Angelos Dassios, Shanle Wu October 6, 28 Abstract In this paper, we extend the concept of ruin in risk theory to the Parisian type of ruin. For
More informationON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary
More informationMAXIMAL COUPLING OF EUCLIDEAN BROWNIAN MOTIONS
MAXIMAL COUPLING OF EUCLIDEAN BOWNIAN MOTIONS ELTON P. HSU AND KAL-THEODO STUM ABSTACT. We prove that the mirror coupling is the unique maximal Markovian coupling of two Euclidean Brownian motions starting
More informationA connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand
A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand Carl Mueller 1 and Zhixin Wu Abstract We give a new representation of fractional
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationLecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
More informationThomas Knispel Leibniz Universität Hannover
Optimal long term investment under model ambiguity Optimal long term investment under model ambiguity homas Knispel Leibniz Universität Hannover knispel@stochastik.uni-hannover.de AnStAp0 Vienna, July
More informationMultivariate Generalized Ornstein-Uhlenbeck Processes
Multivariate Generalized Ornstein-Uhlenbeck Processes Anita Behme TU München Alexander Lindner TU Braunschweig 7th International Conference on Lévy Processes: Theory and Applications Wroclaw, July 15 19,
More informationStochastic optimal control with rough paths
Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction
More informationPseudo-stopping times and the hypothesis (H)
Pseudo-stopping times and the hypothesis (H) Anna Aksamit Laboratoire de Mathématiques et Modélisation d'évry (LaMME) UMR CNRS 80712, Université d'évry Val d'essonne 91037 Évry Cedex, France Libo Li Department
More informationSubordinated Brownian Motion: Last Time the Process Reaches its Supremum
Sankhyā : The Indian Journal of Statistics 25, Volume 77-A, Part, pp. 46-64 c 24, Indian Statistical Institute Subordinated Brownian Motion: Last Time the Process Reaches its Supremum Stergios B. Fotopoulos,
More informationarxiv: v2 [math.pr] 21 Jul 2016
The value of foresight Philip Ernst, L.C.G. Rogers, and Quan Zhou October 17, 2018 arxiv:1601.05872v2 [math.pr] 21 Jul 2016 We dedicate this work to our colleague, mentor, and friend, Professor Larry Shepp
More informationOPTIMAL TRANSPORTATION PLANS AND CONVERGENCE IN DISTRIBUTION
OPTIMAL TRANSPORTATION PLANS AND CONVERGENCE IN DISTRIBUTION J.A. Cuesta-Albertos 1, C. Matrán 2 and A. Tuero-Díaz 1 1 Departamento de Matemáticas, Estadística y Computación. Universidad de Cantabria.
More informationStochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier.
Ito 8-646-8 Calculus I Geneviève Gauthier HEC Montréal Riemann Ito The Ito The theories of stochastic and stochastic di erential equations have initially been developed by Kiyosi Ito around 194 (one of
More informationOPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS
APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationWalsh Diffusions. Andrey Sarantsev. March 27, University of California, Santa Barbara. Andrey Sarantsev University of Washington, Seattle 1 / 1
Walsh Diffusions Andrey Sarantsev University of California, Santa Barbara March 27, 2017 Andrey Sarantsev University of Washington, Seattle 1 / 1 Walsh Brownian Motion on R d Spinning measure µ: probability
More informationAn Uncertain Control Model with Application to. Production-Inventory System
An Uncertain Control Model with Application to Production-Inventory System Kai Yao 1, Zhongfeng Qin 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Economics
More informationExercises in stochastic analysis
Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with
More informationStochastic Processes III/ Wahrscheinlichkeitstheorie IV. Lecture Notes
BMS Advanced Course Stochastic Processes III/ Wahrscheinlichkeitstheorie IV Michael Scheutzow Lecture Notes Technische Universität Berlin Wintersemester 218/19 preliminary version November 28th 218 Contents
More information