An Introduction to Stochastic Processes in Continuous Time

1 An Introduction toStochastic Processes inContinuous time :the non-Jip-and-Janneke-language approachFlora Spieksmaadaptation of the text byHarry van Zantento be used at your own expenseMay 5, 2016 Contents1 Stochastic Introduction .. Finite-dimensional distributions .. Kolmogorov s continuity criterion .. Gaussian Processes .. Non-differentiability of the Brownian sample paths .. Filtrations and stopping times .. Exercises .. 252 Definition and examples .. Discrete- time martingales .. transforms .. decomposition .. theorems .. sampling theorems .. of Large numbers .. Continuous - time martingales .. in Continuous time .. theorems .. sampling .. Applications to Brownian motion .. variation.

2 Inequality .. law of the iterated logarithm .. of hitting times .. Poisson process and the PASTA property .. Exercises .. 623 Markov Basic definitions: a mystification? .. Existence of a canonical version .. Strong Markov property .. Markov property .. on optional times: Markov property and strong Markovproperty .. strong Markov property for right- Continuous canonicalMarkov Processes .. Applications to Brownian Motion .. principle .. limit result .. time distribution .. a random variable in Brownian motion .. Exercises .. 974 Generator of a Markov process with countable state The generator .. Bounded rates: supxqx< .. Construction of Markov Processes with given generatorQ.

3 Unbounded rates .. Exercises .. 1085 Feller-Dynkin Semi-groups .. The generator determines the semi-group: the Hille-Yosida theorem .. Feller-Dynkin transition functions .. of the generator .. of the generator and alternative computation .. Killed Feller-Dynkin Processes .. Regularisation of Feller-Dynkin Processes .. of canonical, cadlag version .. filtration and strong Markov property .. Feller diffusions .. Exercises .. 138 Chapter 1 Stochastic IntroductionLoosely speaking, a Stochastic process is a phenomenon that can be thought of as evolvingin time in a random manner. Common examples are the location of a particle in a physicalsystem, the price of stock in a financial market, interest rates, mobile phone networks, internettraffic, basic example is the erratic movement of pollen grains suspended in water, so-calledBrownian motion.

4 This motion was named after the English botanist R. Brown, who firstobserved it in 1827. The movement of pollen grain is thought to be due to the impacts ofwater molecules that surround it. Einstein was the first to develop a model for studying theerratic movement of pollen grains in in an article in 1926. We will give a sketch of how thismodel was derived. It is more heuristically than mathematically basic assumptions for this model (in dimension 1) are the following:1)the motion is , in a time -interval [t,t+ ], small,2)particle movements in two non-overlapping time intervals of length are mutually inde-pendent;3)the relative proportion of particles experiencing a displacement of size between and +d is approximately ( ) with the probability ofsomedisplacement is 1: ( )d = 1; theaveragedisplacement is 0: ( )d = 0; the variation in displacement is linear in the length of the time interval: 2 ( )d =D , whereD 0 is called thediffusion byf(x,t) the density of particles at positionx, at timet.

5 Under differentiabilityassumptions, we get by a first order Taylor expansion thatf(x,t+ ) f(x,t) + f t(x,t).12 CHAPTER 1. Stochastic PROCESSESOn the other hand, by a second order expansionf(x,t+ ) = f(x ,t) ( )d [f(x,t) f x(x,t) +12 2 2f x2(x,t)] ( )d f(x,t) +12D 2f x2(x,t).Equating gives rise to the heat equation in one dimension: f t=12D 2f x2,which has the solutionf(x,t) =#particles 4 Dt e x2 (x,t) is the density of aN(0,4Dt)-distributed random variable multiplied by the numberof remark. In section we will see that under these assumptions paths of pollengrain through liquid are non-differentiable. However, from physics we know that the velocityof a particle is the derivative (to time ) of its location. Hence pollen grain paths must bedifferentiable. We have a conflict between the properties of the physical model and themathematical model.

6 What is wrong with the assumptions? Already in 1926 editor R. F urthdoubted the validity of the independence assumption (2). Recent investigation seems to haveconfirmed this motion will be one of our objects of study during this course. We will now turnto a mathematical a set and (E,E) a measurable space. Astochastic processindexedbyT, with values in (E,E), is a collectionX= (Xt)t Tof measurable maps from a (joint)probability space ( ,F,P) to (E,E).Xtis called arandom elementas a generalisation of theconcept of a random variable (where (E,E) = (R,B)). The space (E,E) is called the statespace of the BN 1 The indextis a time parameter, and we view the index setTas the set of all observationinstants of the process. In these notes we will usually haveT=Z+={0,1,..}orT=R+= [0, ) (orTis a sub-interval of one these sets).]

7 In the former case, we say that time isdiscrete, in the latter that time iscontinuous. Clearly a discrete- time process can always beviewed as a Continuous - time process that is constant on time -intervals [n,n+ 1).The state space (E,E) will generally be a Euclidian spaceRd, endowed with its Borel -algebraB(Rd). IfEis the state space of the process, we call the every fixed observation instantt T, the Stochastic processXgives us anE-valuedrandom elementXton ( ,F,P). We can also fix and consider the mapt Xt( ) onT. These maps are called thetrajectoriesorsample pathsof the process. The sample INTRODUCTION3are functions fromTtoEand so they are elements of the function spaceET. Hence, we canview the processXas anET-valued random often, the sample paths belong to a nice subset of this space, the continuousor right- Continuous functions, alternatively called the path space.]

8 For instance, a discrete- time process viewed as the Continuous - time process described earlier, is a process with right- Continuous sample we need to put an appropriate -algebra on the path spaceET. For consistencypurposes it is convenient that the marginal distribution ofXtbe a probability measure onthe path space. This is achieved by ensuring that the projectionx xt, wheret T,is measurable. The -algebraET, described in BN 2, is the minimal -algebra with BN 2We will next introduce the formal requirements for the Stochastic Processes that are calledBrownian motion and Poisson process respectively. First, we introduce Processes with inde-pendent a separable Banach space, andEthe Borel- -algebra of subsetsofE. LetT= [0, ] R+. LetX={Xt}t Tbe an (E,E)-valued Stochastic process, definedon an underlying probability space ( ,F,P).

9 I)Xis called a process with independent increments, if (Xt Xs) and (Xu,u s), areindependent for alls t .ii)Xis called a process with stationary, independent increments, if, in addition,Xt Xsd=Xt s X0, fors t .The mathematical model of the physical Brownian motion is a Stochastic process that isdefined as Stochastic processW= (Wt)t 0is called a (standard)Brownian mo-tionorWiener process, ifi)W0= 0, ;ii)Wis a Stochastic process with stationary, independent increments;iii)Wt Wsd=N(0,t s);iv)almost all sample paths are these notes we will abbreviate Brownian motion asBM. Property (i) tells thatstandardBMstarts at 0. A Stochastic process with property (iv) is called acontinuousprocess. Similarly, a Stochastic process is said to beright-continuousif almost all ofits sample paths are right- Continuous functions.

10 Finally, the acronymcadlag(continu `adroite, limites `a gauche) is used for Processes with right- Continuous sample paths havingfinite left-hand limits at every time with Brownian motion we will discuss another fundamental process: the Pois-son 1. Stochastic PROCESSESD efinition real-valued Stochastic processN= (Nt)t 0is called a Poisson process ifi)Nis a counting process, )Nttakes only values in (Z+,2Z+),t 0;b)t7 Ntis increasing, Nt,t )(no two occurrences can occur simultaneously) lims tNs lims tNs+1, for allt )N0= 0, ;iii)Nis a Stochastic process with stationary, independent :so far we do not know yet whether aBMprocess and a Poisson process exist at all!The Poisson process can be constructed quite easily and we will do so first before delving intomore complex of the Poisson processThe construction of a Poisson process is simplerthan the construction of Brownian motion.