Approximating diffusions by piecewise constant parameters
|
|
- Ophelia Dorsey
- 6 years ago
- Views:
Transcription
1 Approximating diffusions by piecewise constant parameters Lothar Breuer Institute of Mathematics Statistics, University of Kent, Canterbury CT2 7NF, UK Abstract We approximate the resolvent of a one-dimensional diffusion process by the resolvent of a level dependent Brownian motion. This is a diffusion process with piecewise constant parameters. The approximating resolvent is determined for the case of a diffusion on an interval with absorbing barriers. A recursive scheme to solve the two-sided exit problem for the approximating process is provided. Key words: diffusion process, resolvent, approximation 2 MSC: 6J6, 6J65 1. Introduction Diffusion processes serve as models for many phenomena in physics, biology economics, to name but a few application fields (see e.g. section 15.2 in [4] for some examples. They further arise as approximations for many Markov processes after rescaling normalising (see [2]. Although many functionals, like first passage probabilities or mean exit times, can be determined explicitly for one-dimensional diffusions, many others still elude an exact solution. There is of course a good reason for this. Diffusion processes arise as solutions to some stochastic differential equations (SDEs, see section II in [5]. Even for one-dimensional time-homogeneous SDEs, their deterministic counterpart, which are ordinary differential equations (ODEs, have explicit solutions only in special cases. Thus a general approach to solve SDEs can only be expected in terms of numerical approximation schemes. The vast majority of existing numerical schemes mimic the methods applied to deterministic differential equations, resulting in the Euler scheme refinements thereof. The basic idea is a discretisation of the time axis the determination Preprint submitted to Stochastic Processes their Applications February 17, 211
2 of a suitable system of difference equations. Many examples of such schemes can be found in [5]. While time discretisation of a one-dimensional ODE yields a finite sequence of numbers as a result, the same method applied to a SDE results in a rom variable for each step in time each fixed initial value. Hence, functionals like transition probabilities can only be obtained by simulating many path realisations under any fixed initial value (cf. the MCMC approach. This of course may quickly amount to high computational costs. The present paper aims to explore possibilities of a numerical approximation that is based on a discretisation of the state space rather than the time axis. We shall consider only one-dimensional diffusion processes, which correspond to solutions of one-dimensional time-homogeneous SDEs. We intend to derive an approximation of the resolvent of a given diffusion process. This preserves to a large degree the distribution of the process allows to approximate functionals like the stationary distribution without the need to recur to costly MCMC methods. The approximation of the resolvent is introduced in the following section. In section 3, the approximating process is introduced in more detail. Section 4 contains the determination of the approximating resolvent. The last section provides a recursive scheme to solve the two-sided exit problem. 2. The approximation Let X = (X t : t denote a one-dimensional diffusion process with a state space E = [l, u] where l < u R. Denote the infinitesimal drift variance functions by µ : E R σ : E R, respectively. Assume that µ σ are Lipschitz continuous. Let T (t, t, denote the transition semi-group of X, acting on a suitable domain D as T (tf(x := E (f(x t X = x for all f D x E. The generator of X is defined as 1 Af := lim (T (εf f ε ε for all f D(A, the domain of A, given by Af(x = σ2 (x f (x µ(xf (x 2 2
3 for all x E o =]l, u[, the interior of E. We assume that the boundaries are absorbing. Let Y = (Y t : t denote a diffusion process with the same state space E, but piecewise constant infinitesimal drift variance functions µ : E R σ : E R. Assume that they are given as µ(x := µ i σ(x := σ i for b i1 < x b i i = 1,..., N, where b := l b N := u. Assume further that ( µ(x µ(x 1 2 σ2 (x σ 2 (x < ε sup x E o The boundaries are assumed to be absorbing. Let T (t, t, denote the transition semi-group of Y Ã its generator. The fact that for t > s yields d ds T (t s T (sf = T (t sa T (sf T (t sã T (sf T (tf T (tf = t T (t s(ã A T (sf ds (1 cf. [2], lemma 6.2 of chapter 1. We are interested in the λ-resolvents R λ := e λs T (s ds Rλ := see proposition 2.1 of chapter 1 in [2]. Equation (1 yields R λ f R λ f = = = s= e λs T (s ds t e λt T (t s(ã A T (sf ds dt t e λt T (s(ã A T (t sf ds dt e λs T (s(ã A e λ(ts T (t sf dt ds = R λ (Ã A R λ f t=s Since R λ g e λs T (sg ds 1 λ g 3
4 for any g D, we obtain R λ f R λ f 1 λ (Ã A R λ f in particular for g := R λ f R λ f(x R λ f(x 1 1 λ 2 ( σ2 (x σ 2 (xg (x ( µ(x µ(xg (x g (x g (x ε λ for all x E o. This implies weak convergence of the resolvents. The next two sections are dedicated to determining R λ. 3. The approximating process The approximating process Y = (Y t : t may be called a level dependent Brownian motion with a finite number of thresholds b,..., b N. It is defined by the stochastic differential equation { µ k dt σ k dw t, b k1 < Y t b k, 1 k N 1 dy t = µ N dt σ N dw t, b N1 < Y t < b N where µ i R σ i > for i N, W = (W t : t denotes the stard Wiener process. Define the intervals I k :=]b k1, b k ] for k {1,..., N 1} I N :=]b N1, b N [. We call I k together with the parameters (µ k, σ k the kth regime of Y. If N = 1, i.e. if there is only one regime, we call the process Y homogeneous (in space. It is then of course a Brownian motion on an interval with absorbing boundaries. Define u ± k = u± k (λ := ±µ k µ 2 k 2λσ2 k σ 2 k for k N. Within each regime I k the behaviour of Y equals that of a classical Brownian motion with λ-scale function W (λ k (x := e u k x e u k x. For instance, the exit times τ(l, u := inf{t : Y t / [l, u]} 4
5 from an interval [l, u] I k are given by their Laplace transforms Ψ k (u l x l := E ( e λτ(l,u ; Y τ(l,u = u Y = x = eu k (xl e u k (xl (2 e u k (ul e u k (ul for all x [l, u]. By reflection around the initial value x, we obtain further Ψ k (u l x l := E ( e λτ(l,u ; Y τ(l,u = l Y = x = eu k (ux e u k (ux (3 e u k (ul e u k (ul using the fact that the roles of u k u k are simply reversed for the Brownian motion with parameters (µ k, σk 2. We now wish to determine the exit times of Y from an interval [b k1, b k1 ], given that Y = b k. Abbreviate those by τ(k := inf{t : Y t / [b k1, b k1 ]} (4 for k = 1,..., N 1 write b k := b k b k1 for k = 1,..., N. Theorem 1 For k = 1,..., N 1, the distribution of the exit time τ(k, given that Y = b k, is given by the Laplace transforms E (k := E ( e λτ(k ; Y τ(k = b k1 Y = b k u k1 = u k1 e u k1 b k1 e u k1 b k1 ( u k eu k b k u k eu k b k e u k b k e u k b k u k1 eu E (k := E ( e λτ(k ; Y τ(k = b k1 Y = b k u k = u k e u k b k e u k b k ( u k eu k b k u k eu k b k e u k b k e u k b k for λ. u k1 eu 5 k1 b k1 u k1 eu k1 b k1 e u k1 b k1 e u k1 b k1 k1 b k1 u k1 eu k1 b k1 e u k1 b k1 e u k1 b k1 1 1
6 Proof: We begin with the first statement consider E(ε := E ( e λτ(k ; Y τ(k = b k1 Y = b k ε First we assume that the regime changes at b k ε for downward crossings of b k at b k ε for upward crossings. Then we let ε. Summing up over the number of down up crossings of the interval [b k ε, b k ε] before leaving the interval [b k1, b k1 ] at b k1, we obtain E(ε = = n= ( Ψ k1 ( b k1 ε 2εΨ k ( b k ε b k ε n Ψ k1 ( b k1 ε 2ε Ψ k1 ( b k1 ε 2ε 1 Ψ k1 ( b k1 ε 2εΨ k ( b k ε b k ε Equations (2 (3 along with L Hospital s rule now yield the first statement. The proof of the second statement is completely analogous. 4. The resolvent We consider the level dependent Brownian motion as in section 3 seek to find an expression for its resolvent density function r λ (x, y, defined by ( r λ (x, ydy := R λ (x, dy = E e λt 1 {Yt dy} dt Y = x for λ >. Let E(λ denote an independent exponential time of parameter λ. Then R λ (x, dy is simply the probability that Y is located in dy before the exponential time E(λ has expired. We separate any path of Y from x to y within the time E(λ into three parts. First, if x / {b 1,..., b N }, denote the probabilities of hitting the next upper/lower grid point before E(λ by p ± 1 (x. Let r(x denote the regime of x, i.e. r(x = k if x I k, k {1,..., N}. With k := r(x b k := b k b k1 the values p ± 1 (x are given by cf. (2 (3. p ± 1 (x = Ψ ± k ( b k x b k1 (5 6
7 The second part of the path consists of movements among the grid points b,..., b N before E(γ. To this aim, define the square matrix P = (p ij i,j N of dimension N 1 by 1, j = i {, N} E (i, j = i 1, i {1,..., N 1} p ij := E (i, j = i 1, i {1,..., N 1} else Theorem 2 Define the hitting times T (x := min{t : Y t = x} for all x E. Further define the matrix M = (m ij i,j N by M := (I P 1. Then for all i, j N. m ij = E ( e λt (b j Y = b i = P(T (bj < E(λ Y = b i Proof: We first observe that E (k = P ( τ(k < E(λ, Y τ(k = b k1 Y = b k E (k = P ( τ(k < E(λ, Y τ(k = b k1 Y = b k Thus P is sub-stochastic with the kth row sum e kp 1 = 1 P(τ(k > E(λ Y = b k < 1 for k = 2,..., N 1, where e k denotes the kth canonical row base vector 1 the column vector with all entries being 1. Due to the memoryless property of the exponential distribution, the matrix M = (I P 1 = n= P n contains the probabilities m ij of hitting state b j before E(λ, given that Y = b i. The last part is a movement from a grid point to y before E(λ without hitting another grid point in between. Thus we are interested in the occupation measures ( τ(k l k (ydy := E where b k1 < y < b k1. e λt I {Yt dy} dt Y = b k 7
8 Theorem 3 Define further m k (y = euk1 (b k1y e u k1 (b k1y e u k1 b k1 e u k1 b k1 eu k (yb k1 e u k (yb k1 e u k b k e u k b k I {y>bk } I {y<bk } h k Then = u k eu k b k u e u k b k e u k b k for b k1 < y < b k1. k eu k b k, h k1 = u k1 euk1 b k1 u k1 b k1 eu k1 e u k1 b k1 e u k1 b k1 l k (y = m k(y h k h k1 Proof: The density function l k (y is the product of the occupation density at b k before min{τ(k, E(λ} the occupation density at y before returning to b k before min{τ(k, E(λ}. Thus we write l k (y = d k m k (y. The first factor, the occupation density at b k, is obtained as cf. [3], section 2.4, where d k := lim ε 2ε(1 h k1 (εh k (ε1 h k1 (ε = Ψ k1 ( b k1 ε 2ε h k (ε = Ψ k ( b k ε b k ε Since lim ε h k1 (ε = lim ε h k (ε = 1, we can write ( d lim 2ε(1 h k1 (εh k (ε1 = 2 ε dε h k1 (εh k (ε 1 ε= where d dε h k1 (ε = 2 u k1 euk1 b k1 u k1 b k1 eu k1 ε= e u k1 b k1 e u k1 b k1 d dε h k (ε = 2 u k eu k b k u ε= e u k b k e u k b k 8 k eu k b k (6 (7
9 Thus d k = ( 1. h k k1 h Note that dk > for λ >, since u ± k < for all k N. The second factor in l k (y is the occupation density at y before returning to b k before min{τ(k, E(λ}. Denote this by m k (y. Exploiting the Siegmund duality (see [1], we consider the time-reversed process Y r. This is defined as a Brownian motion with parameters (µ k, σ k in the kth regime. Define u ±,r k := µ k µ 2 k 2λσ2 k σ 2 k = u k (8 for k = 1,..., N. Then where m k (y = { Ψ,r k1 ( b k1 y b k, y > b k Ψ,r k ( b k y b k1, y < b k (9,r Ψ,r k1 ( b k1 y b k = euk1 (b k1y e u,r k1 (b k1y e u,r k1 b k1 e u,r k1 b k1 Ψ,r k ( b k y b k1 = eu,r k (yb k1 e u,r e u,r k k (yb k1 b k e u,r k b k Now the second equality in (8 yields the statement. If r(x = r(y, then it is possible to reach y from x before E(λ without hitting any threshold b k, k N. Theorem 4 Define m(x, y = eu k (bky e u k (b ky e u k (bkx e I u k (b {y>x} eu k (ybk1 e u k (yb k1 kx e u k (xbk1 e I {y<x}, u k (xb k1 h (x := u k eu k (bkx u k eu k (b kx, e u k (bkx e u k (b kx h (x := u k eu k (xbk1 u k eu k (xb k1 e u k (xbk1 e u k (xb k1 9
10 Then ( τ(bk1,b k p (x, ydy := E e λt m(x, y I {Yt dy} dt Y = x = h (x h (x dy for r(x = r(y = k. Proof: This is shown by the same arguments as theorem 3. We can determine p (x, y as the product of d(x m(x, y, where d(x is the occupation density at x before min{τ(b k1, b k, E(λ} m(x, y is the occupation density at y before returning to x before min{τ(b k1, b k, E(λ}. The first factor is obtained as d(x = lim ε 2ε(1h (x, εh (x, ε 1, where h (x, ε = Ψ k (b k x ε 2ε h (x, ε = Ψ k (x b k1 ε x b k1 ε Since lim ε h (x, ε = lim ε h (x, ε = 1, we can write ( d 1 lim 2ε(1 h (x, εh (x, ε 1 = 2 ε dε h (x, εh (x, ε ε= where d dε h (x, ε = 2 u k eu k (bkx u k eu k (b kx = 2h (x ε= e u k (bkx e u k (b kx d dε h (x, ε = 2 u k eu k (xbk1 u k eu k (xb k1 = 2h (x ε= e u k (xbk1 e u k (xb k1 This yields d(x = (h (x h (x 1. The second factor m(x, y is given in terms of the time-reversed process Y r, namely { Ψ,r k (b k x y x, y > x m(x, y = Ψ,r k (x b k1 y b k1, y < x Now the formulas (2, (3, the second equality in (8 yield the statement. 1
11 Assembling the above results, the resolvent density r(x, y can now be given explicitly as N1 N1 r(x, y = p 1 (x m r(x1,k l k (y p 1 (x m r(x,k l k (y for y / {b,..., b N } k=2 I {r(x=r(y} p (x, y k=2 r(x, y = p 1 (xm r(x,i p 1 (xm r(x1,i for y = b i i {,..., N}. The ingredients are given in (5 theorems Exit problems We wish to derive Laplace transforms for the exit times τ(l, u := inf{t : Y t / [l, u]} from an interval [l, u] ]b, b N [ in the form of E (l, a, u := E ( e λτ(l,u ; Y τ(l,u = u Y = a where l < a < u λ. The derivations for are analogous. E (l, a, u := E ( e λτ(l,u ; Y τ(l,u = l Y = a 5.1. The two-sided exit problem for two regimes We first restrict our considerations to the case N = 2, i.e. one threshold b 1 dividing two regimes I 1 =]b, b 1 ] I 2 =]b 1, b 2 [. There are some simple cases, namely E ( e λτ(l,u ; Y τ(l,u = u Y = a { Ψ 1 (u l a l, l < a < u < b 1 = Ψ 2 (u l a l, b 1 < l < a < u For the other cases we obtain, by path continuity, E ( e λτ(l,u ; Y τ(l,u = u Y = a = Ψ 1 (bl al E ( e λτ(l,u ; Y τ(l,u = u Y = b 1 11
12 for l < a < b 1 < u, E ( e λτ(l,u ; Y τ(l,u = u Y = a = Ψ 2 (u b 1 a b 1 Ψ 2 (u b 1 a b 1 E ( e λτ(l,u ; Y τ(l,u = u Y = b 1 for l < b 1 < a < u. Thus it suffices to determine E (l, b 1, u. The same arguments as in theorem 1 yield with k = 1 E (l, b k, u = u k1 u k1 e u k1 (ub k e u k1 (ub k ( u k eu k (b kl u k eu k (b kl E (l, b k, u = for λ. e u k (b kl e u k (b kl u k u k e u k (b kl e u k (b kl ( u k eu k (b kl u k eu k (b kl e u k (b kl e u k (b kl u k1 eu u k1 eu k1 (ub k u k1 eu k1 (ub k e u k1 (ub k e u k1 (ub k k1 (ub k u k1 eu k1 (ub k e u k1 (ub k e u k1 (ub k The two-sided exit problem for N > 2 regimes For h := min{n 1 : b n > l}, the matrix E (l, b h, b h1 has been determined in the previous subsection. Define k := max{n 1 : b n < u}. If k = h, then E (l, a, u is given by the previous results. Thus assume that k > h 1. We obtain by path continuity where E (l, a, u = E (l, a, b k E (l, b k, u E (l, b k, u = E (b k1, b k, u E (b k1, b k, ue (l, b k1, b k E (l, b k, u This yields E (l, b k, u = ( I E (b k1, b k, ue (l, b k1, b k 1 E (b k1, b k, u Since the matrices E (b k1, b k, u E (b k1, b k, u have been determined in the previous subsection, this provides a recursion scheme for E (l, a, u. 12
13 References [1] J. Cox U. Rösler. A duality relation for entrance exit laws for Markov processes. Stochastic Processes their Applications, 16: , [2] S. N. Ethier T. G. Kurtz. Markov processes. Characterization convergence. New York etc.: John Wiley & Sons, [3] K. Ito H. P. McKean. Difffusion Processes their Sample Paths. Springer, [4] S. Karlin H. M. Taylor. A second course in stochastic processes. New York etc.: Academic Press, [5] P. E. Kloeden E. Platen. Numerical Solution of Stochastic Differential Equations. Springer,
Threshold dividend strategies for a Markov-additive risk model
European Actuarial Journal manuscript No. will be inserted by the editor Threshold dividend strategies for a Markov-additive risk model Lothar Breuer Received: date / Accepted: date Abstract We consider
More informationReflected Brownian Motion
Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide
More informationChapter 3: Markov Processes First hitting times
Chapter 3: Markov Processes First hitting times L. Breuer University of Kent, UK November 3, 2010 Example: M/M/c/c queue Arrivals: Poisson process with rate λ > 0 Example: M/M/c/c queue Arrivals: Poisson
More informationIntroduction to Random Diffusions
Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales
More informationc 2003 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 24, No. 5, pp. 1809 1822 c 2003 Society for Industrial and Applied Mathematics EXPONENTIAL TIMESTEPPING WITH BOUNDARY TEST FOR STOCHASTIC DIFFERENTIAL EQUATIONS KALVIS M. JANSONS
More informationTyler Hofmeister. University of Calgary Mathematical and Computational Finance Laboratory
JUMP PROCESSES GENERALIZING STOCHASTIC INTEGRALS WITH JUMPS Tyler Hofmeister University of Calgary Mathematical and Computational Finance Laboratory Overview 1. General Method 2. Poisson Processes 3. Diffusion
More informationBrownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion
Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of
More informationProperties of an infinite dimensional EDS system : the Muller s ratchet
Properties of an infinite dimensional EDS system : the Muller s ratchet LATP June 5, 2011 A ratchet source : wikipedia Plan 1 Introduction : The model of Haigh 2 3 Hypothesis (Biological) : The population
More informationStochastic Areas and Applications in Risk Theory
Stochastic Areas and Applications in Risk Theory July 16th, 214 Zhenyu Cui Department of Mathematics Brooklyn College, City University of New York Zhenyu Cui 49th Actuarial Research Conference 1 Outline
More informationNumerical Integration of SDEs: A Short Tutorial
Numerical Integration of SDEs: A Short Tutorial Thomas Schaffter January 19, 010 1 Introduction 1.1 Itô and Stratonovich SDEs 1-dimensional stochastic differentiable equation (SDE) is given by [6, 7] dx
More informationfor all f satisfying E[ f(x) ] <.
. Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if
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 informationA numerical method for solving uncertain differential equations
Journal of Intelligent & Fuzzy Systems 25 (213 825 832 DOI:1.3233/IFS-12688 IOS Press 825 A numerical method for solving uncertain differential equations Kai Yao a and Xiaowei Chen b, a Department of Mathematical
More informationOptimal Sojourn Time Control within an Interval 1
Optimal Sojourn Time Control within an Interval Jianghai Hu and Shankar Sastry Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, CA 97-77 {jianghai,sastry}@eecs.berkeley.edu
More informationOn the submartingale / supermartingale property of diffusions in natural scale
On the submartingale / supermartingale property of diffusions in natural scale Alexander Gushchin Mikhail Urusov Mihail Zervos November 13, 214 Abstract Kotani 5 has characterised the martingale property
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
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 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 informationLAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC
LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic
More informationLecture 10: Semi-Markov Type Processes
Lecture 1: Semi-Markov Type Processes 1. Semi-Markov processes (SMP) 1.1 Definition of SMP 1.2 Transition probabilities for SMP 1.3 Hitting times and semi-markov renewal equations 2. Processes with semi-markov
More informationStochastic differential equation models in biology Susanne Ditlevsen
Stochastic differential equation models in biology Susanne Ditlevsen Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential
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 informationSTATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, = (2n 1)(2n 3) 3 1.
STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, 26 Problem Normal Moments (A) Use the Itô formula and Brownian scaling to check that the even moments of the normal distribution
More informationPartial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula
Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Group 4: Bertan Yilmaz, Richard Oti-Aboagye and Di Liu May, 15 Chapter 1 Proving Dynkin s formula
More informationSimulation Method for Solving Stochastic Differential Equations with Constant Diffusion Coefficients
Journal of mathematics and computer Science 8 (2014) 28-32 Simulation Method for Solving Stochastic Differential Equations with Constant Diffusion Coefficients Behrouz Fathi Vajargah Department of statistics,
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 informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationWeak convergence and large deviation theory
First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov
More informationPoisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation
Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant
More informationElliptic Operators with Unbounded Coefficients
Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli Motivation Consider the Stochastic Differential
More informationBranching Processes II: Convergence of critical branching to Feller s CSB
Chapter 4 Branching Processes II: Convergence of critical branching to Feller s CSB Figure 4.1: Feller 4.1 Birth and Death Processes 4.1.1 Linear birth and death processes Branching processes can be studied
More informationFeller Processes and Semigroups
Stat25B: Probability Theory (Spring 23) Lecture: 27 Feller Processes and Semigroups Lecturer: Rui Dong Scribe: Rui Dong ruidong@stat.berkeley.edu For convenience, we can have a look at the list of materials
More informationDynamical systems with Gaussian and Levy noise: analytical and stochastic approaches
Dynamical systems with Gaussian and Levy noise: analytical and stochastic approaches Noise is often considered as some disturbing component of the system. In particular physical situations, noise becomes
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 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 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 informationIntroduction to the Numerical Solution of SDEs Selected topics
0 / 23 Introduction to the Numerical Solution of SDEs Selected topics Andreas Rößler Summer School on Numerical Methods for Stochastic Differential Equations at Vienna University of Technology, Austria
More informationON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME
ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems
More informationc 2002 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 4 No. pp. 507 5 c 00 Society for Industrial and Applied Mathematics WEAK SECOND ORDER CONDITIONS FOR STOCHASTIC RUNGE KUTTA METHODS A. TOCINO AND J. VIGO-AGUIAR Abstract. A general
More informationSecond order weak Runge Kutta type methods for Itô equations
Mathematics and Computers in Simulation 57 001 9 34 Second order weak Runge Kutta type methods for Itô equations Vigirdas Mackevičius, Jurgis Navikas Department of Mathematics and Informatics, Vilnius
More informationSummary of Stochastic Processes
Summary of Stochastic Processes Kui Tang May 213 Based on Lawler s Introduction to Stochastic Processes, second edition, and course slides from Prof. Hongzhong Zhang. Contents 1 Difference/tial Equations
More informationOn a class of stochastic differential equations in a financial network model
1 On a class of stochastic differential equations in a financial network model Tomoyuki Ichiba Department of Statistics & Applied Probability, Center for Financial Mathematics and Actuarial Research, University
More informationEULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS
Qiao, H. Osaka J. Math. 51 (14), 47 66 EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS HUIJIE QIAO (Received May 6, 11, revised May 1, 1) Abstract In this paper we show
More informationHarmonic Functions and Brownian motion
Harmonic Functions and Brownian motion Steven P. Lalley April 25, 211 1 Dynkin s Formula Denote by W t = (W 1 t, W 2 t,..., W d t ) a standard d dimensional Wiener process on (Ω, F, P ), and let F = (F
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationSimulation of diffusion. processes with discontinuous coefficients. Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan
Simulation of diffusion. processes with discontinuous coefficients Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan From collaborations with Pierre Étoré and Miguel Martinez . Divergence
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
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 informationDiscretization of SDEs: Euler Methods and Beyond
Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo
More informationProblem Max. Possible Points Total
MA 262 Exam 1 Fall 2011 Instructor: Raphael Hora Name: Max Possible Student ID#: 1234567890 1. No books or notes are allowed. 2. You CAN NOT USE calculators or any electronic devices. 3. Show all work
More informationTMS165/MSA350 Stochastic Calculus, Lecture on Applications
TMS165/MSA35 Stochastic Calculus, Lecture on Applications In this lecture we demonstrate how statistical methods such as the maximum likelihood method likelihood ratio estimation can be applied to the
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 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 information1 Types of stochastic models
1 Types of stochastic models Models so far discussed are all deterministic, meaning that, if the present state were perfectly known, it would be possible to predict exactly all future states. We have seen
More informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More informationUniversal examples. Chapter The Bernoulli process
Chapter 1 Universal examples 1.1 The Bernoulli process First description: Bernoulli random variables Y i for i = 1, 2, 3,... independent with P [Y i = 1] = p and P [Y i = ] = 1 p. Second description: Binomial
More informationSolving the Poisson Disorder Problem
Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, Springer-Verlag, 22, (295-32) Research Report No. 49, 2, Dept. Theoret. Statist. Aarhus Solving the Poisson Disorder Problem
More informationHomework #6 : final examination Due on March 22nd : individual work
Université de ennes Année 28-29 Master 2ème Mathématiques Modèles stochastiques continus ou à sauts Homework #6 : final examination Due on March 22nd : individual work Exercise Warm-up : behaviour of characteristic
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationExact Simulation of Multivariate Itô Diffusions
Exact Simulation of Multivariate Itô Diffusions Jose Blanchet Joint work with Fan Zhang Columbia and Stanford July 7, 2017 Jose Blanchet (Columbia/Stanford) Exact Simulation of Diffusions July 7, 2017
More information{σ x >t}p x. (σ x >t)=e at.
3.11. EXERCISES 121 3.11 Exercises Exercise 3.1 Consider the Ornstein Uhlenbeck process in example 3.1.7(B). Show that the defined process is a Markov process which converges in distribution to an N(0,σ
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 informationOn the quantiles of the Brownian motion and their hitting times.
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]
More informationControlled Diffusions and Hamilton-Jacobi Bellman Equations
Controlled Diffusions and Hamilton-Jacobi Bellman Equations Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 2014 Emo Todorov (UW) AMATH/CSE 579, Winter
More informationPH.D. PRELIMINARY EXAMINATION MATHEMATICS
UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 207 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem
More informationA Short Introduction to Diffusion Processes and Ito Calculus
A Short Introduction to Diffusion Processes and Ito Calculus Cédric Archambeau University College, London Center for Computational Statistics and Machine Learning c.archambeau@cs.ucl.ac.uk January 24,
More informationAn adaptive numerical scheme for fractional differential equations with explosions
An adaptive numerical scheme for fractional differential equations with explosions Johanna Garzón Departamento de Matemáticas, Universidad Nacional de Colombia Seminario de procesos estocásticos Jointly
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 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 informationHandbook of Stochastic Methods
C. W. Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Third Edition With 30 Figures Springer Contents 1. A Historical Introduction 1 1.1 Motivation I 1.2 Some Historical
More informationIntroduction to Diffusion Processes.
Introduction to Diffusion Processes. Arka P. Ghosh Department of Statistics Iowa State University Ames, IA 511-121 apghosh@iastate.edu (515) 294-7851. February 1, 21 Abstract In this section we describe
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 informationON THE OPTIMAL STOPPING PROBLEM FOR ONE DIMENSIONAL DIFFUSIONS
ON THE OPTIMAL STOPPING PROBLEM FOR ONE DIMENSIONAL DIFFUSIONS SAVAS DAYANIK Department of Operations Research and Financial Engineering and the Bendheim Center for Finance Princeton University, Princeton,
More informationFunctional central limit theorem for super α-stable processes
874 Science in China Ser. A Mathematics 24 Vol. 47 No. 6 874 881 Functional central it theorem for super α-stable processes HONG Wenming Department of Mathematics, Beijing Normal University, Beijing 1875,
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 informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 12, 215 Averaging and homogenization workshop, Luminy. Fast-slow systems
More informationStochastic Integration.
Chapter Stochastic Integration..1 Brownian Motion as a Martingale P is the Wiener measure on (Ω, B) where Ω = C, T B is the Borel σ-field on Ω. In addition we denote by B t the σ-field generated by x(s)
More informationA new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009
A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance
More informationNumerical methods for solving stochastic differential equations
Mathematical Communications 4(1999), 251-256 251 Numerical methods for solving stochastic differential equations Rózsa Horváth Bokor Abstract. This paper provides an introduction to stochastic calculus
More informationMarkov decision processes and interval Markov chains: exploiting the connection
Markov decision processes and interval Markov chains: exploiting the connection Mingmei Teo Supervisors: Prof. Nigel Bean, Dr Joshua Ross University of Adelaide July 10, 2013 Intervals and interval arithmetic
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 informationLecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
More informationhal , version 1-13 Mar 2013
AALYSIS OF THE MOTE-CARLO ERROR I A HYBRID SEMI-LAGRAGIA SCHEME CHARLES-EDOUARD BRÉHIER AD ERWA FAOU Abstract. We consider Monte-Carlo discretizations of partial differential equations based on a combination
More informationOptimal transportation and optimal control in a finite horizon framework
Optimal transportation and optimal control in a finite horizon framework Guillaume Carlier and Aimé Lachapelle Université Paris-Dauphine, CEREMADE July 2008 1 MOTIVATIONS - A commitment problem (1) Micro
More informationExponential functionals of Lévy processes
Exponential functionals of Lévy processes Víctor Rivero Centro de Investigación en Matemáticas, México. 1/ 28 Outline of the talk Introduction Exponential functionals of spectrally positive Lévy processes
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 informationMarkov processes and queueing networks
Inria September 22, 2015 Outline Poisson processes Markov jump processes Some queueing networks The Poisson distribution (Siméon-Denis Poisson, 1781-1840) { } e λ λ n n! As prevalent as Gaussian distribution
More information1. Stochastic Process
HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 216 STOCHASTIC CALCULUS BASICS SANG YOON (TIM) LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real
More informationStochastic differential equations in neuroscience
Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) http://www.univ-orleans.fr/mapmo/membres/berglund/ Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans
More informationAdaptive dynamics in an individual-based, multi-resource chemostat model
Adaptive dynamics in an individual-based, multi-resource chemostat model Nicolas Champagnat (INRIA Nancy) Pierre-Emmanuel Jabin (Univ. Maryland) Sylvie Méléard (Ecole Polytechnique) 6th European Congress
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 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 information9.2 Branching random walk and branching Brownian motions
168 CHAPTER 9. SPATIALLY STRUCTURED MODELS 9.2 Branching random walk and branching Brownian motions Branching random walks and branching diffusions have a long history. A general theory of branching Markov
More informationAffine Processes. Econometric specifications. Eduardo Rossi. University of Pavia. March 17, 2009
Affine Processes Econometric specifications Eduardo Rossi University of Pavia March 17, 2009 Eduardo Rossi (University of Pavia) Affine Processes March 17, 2009 1 / 40 Outline 1 Affine Processes 2 Affine
More informationA Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1
Chapter 3 A Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1 Abstract We establish a change of variable
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationOn the First Hitting Times to Boundary of the Reflected O-U Process with Two-Sided Barriers
Int. J. Contemp. Math. Sciences, Vol. 8, 213, no. 5, 235-242 HIKARI Ltd, www.m-hikari.com On the First Hitting Times to Boundary of the Reflected O-U Process with Two-Sided Barriers Lidong Zhang 1 School
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More information