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Introduction to Stochastic Calculus with Applications

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Introduction to Stochastic Calculus with Applications Chapter 2 Concepts of Probability Theory In this chapter we give fundamental definitions of probabilistic concepts. Since the theory is more transparent in the discrete case, it is presented first. The most important concepts not met in elementary courses are the models for in- formation, its flow and conditional expectation. This is only a brief description, and a more detailed treatment can be found in many books on Probability The- ory, for example, Breiman (1968), Loeve (1978), Durret (1991). Conditional expectation with its properties is central for further development, but some of the material in this chapter may be treated as an appendix. 2.1 Discrete Probability Model A probability model consists of a filtered probability space on which variables of interest are defined. In this section we introduce a discrete probability model by using an example of discrete trading in stock. Filtered Probability Space A filtered probability space consists of: a sample space of elementary events, a field of events, a probability defined on that field, and a filtration of increasing subfields. 21 22 CHAPTER 2. CONCEPTS OF PROBABILITY THEORY Sample Space Consider a single stock with price St at time t =1, 2,...T.DenotebyΩthe set of all possible values of stock during these times. Ω={ω : ω =(S1,S2,...,ST )}. If we assume that the stock price can go up by a factor u and down by afactord, then the relevant information reduces to the knowledge of the movements at each time. Ω={ω : ω =(a1,a2,...,aT )} at = u or d. To model uncertainty about the price in the future, we “list” all possible future prices, and call it possible states of the world. The unknown future is just one of many possible outcomes, called the true state of the world. As time passes more and more information is revealed about the true state of the world. At time t = 1 we know prices S0 and S1. Thus the true state of the world lies in a smaller set, subset of Ω, A ⊂ Ω. After observing S1 we know which prices did not happen at time 1. Therefore we know that the true state of the world is in A and not in Ω \ A = A¯. Fields of Events Define by Ft the information available to investors at time t, which consists of stock prices before and at time t. For example when T =2,att = 0 we have no information about S1 and S2,andF0 = {∅, Ω}, all we know is that a true state of the world is in Ω. Consider the situation at t = 1. Suppose at t =1stockwentupbyu.Then we know that the true state of the world is in A, and not in its complement A¯,where A = {(u, S2),S2 = u or d} = {(u, u), (u, d)}. Thus our information at time t =1is F1 = {∅, Ω,A,A¯}. Note that F0 ⊂F1, since we don’t forget the previous information. At time t investors know which part of Ω contains the true state of the world. Ft is called a field or algebra of sets. F is a field if 1. ∅, Ω ∈F 2. If A ∈F,andB ∈F then A ∪ B ∈F, A ∩ B ∈F, A \ B ∈F. 2.1. DISCRETE PROBABILITY MODEL 23 Example 2.1: (Examples of fields.) It is easy to verify that any of the following is a field of sets. 1. {Ω, ∅} is called the trivial field F0. 2. {∅, Ω,A,A¯} is called the field generated by set A, and denoted by FA. 3. {A : A ⊆ Ω} the field of all the subsets of Ω. It is denoted by 2Ω. A partition of Ω is a collection of exhaustive and mutually exclusive subsets, {D1,...,Dk}, such that Di ∩ Dj = ∅, and Di =Ω. i The field generated by the partition is the collection of all finite unions of Dj’s and their complements. These sets are like the basic building blocks for the field. If Ω is finite then any field is generated by a partition. If one field is included in the other, F1 ⊂F2,thenanysetfromF1 is also in F2. In other words, a set from F1 is either a set or a union of sets from the partition generating F2. This means that the partition that generates F2 has “finer” sets than the ones that generate F1. Filtration A filtration IF is the collection of fields, IF = {F0, F1,...,Ft,...,FT }Ft ⊂Ft+1. IF is used to model a flow of information. As time passes, an observer knows more and more detailed information, that is, finer and finer partitions of Ω. In the example of the price of stock, IF describes how the information about prices is revealed to investors. Ω Example 2.2: IF={F0, FA, 2 }, is an example of filtration. Stochastic Processes If Ω is a finite sample space, then a function X defined on Ω attaches numerical values to each ω ∈ Ω. Since Ω is finite, X takes only finitely many values xi, i =1,...,k. If a field of events F is specified, then any set in it is called a measurable set. If F =2Ω, then any subset of Ω is measurable. A function X on Ω is called F-measurable or a random variable on (Ω, F) if all the sets {X = xi}, i =1,...,k,aremembersofF. This means that if we have the information described by F, that is, we know which event in F has occurred, then we know which value of X has occurred. Note that if F =2Ω, then any function on Ω is a random variable. 24 CHAPTER 2. CONCEPTS OF PROBABILITY THEORY Example 2.3: Consider the model for trading in stock, t =1, 2, where at each time the stock can go up by the factor u or down by the factor d. Ω={ω1 =(u, u),ω2 =(u, d),ω3 =(d, u),ω4 =(d, d)}.TakeA = {ω1,ω2},whichis Ω the event that at t = 1 the stock goes up. F1 = {∅, Ω,A,A¯},andF2 =2 contains all 16 subsets of Ω. Consider the following functions on Ω. X(ω1)=X(ω2)=1.5, X(ω3)=X(ω4)=0.5. X is a random variable on F1. Indeed, the set {ω : X(ω)= 2 1.5} = {ω1,ω2} = A ∈F1.Also{ω : X(ω)=0.5} = A¯ ∈F1.IfY (ω1)=(1.5) , 2 Y (ω2)=0.75, Y (ω3)=0.75, and Y (ω4)=0.5 ,thenY is not a random variable on F1,itisnotF1-measurable. Indeed, {ω : Y (ω)=0.75} = {ω2,ω3} ∈F/ 1. Y is F2-measurable. Definition 2.1 A stochastic process is a collection of random variables {X(t)}. For any fixed t, t =0, 1,...,T, X(t) is a random variable on (Ω, FT ). A stochastic process is called adapted to filtration IF if for all t =0, 1,...,T, X(t) is a random variable on Ft, that is, if X(t) is Ft-measurable. Example 2.4: (Example 2.3 continued.) X1 = X, X2 = Y is a stochastic process adapted to IF={F1, F2}.Thisprocess represents stock prices at time t under the assumption that the stock can appreciate or depreciate by 50% in a unit of time. Field Generated by a Random Variable Let (Ω, 2Ω) be a sample space with the field of all events, and X be a random variable with values xi, i =1, 2,...k.Considersets Ai = {ω : X(ω)=xi}⊆Ω. These sets form a partition of Ω, and the field generated by this partition is called the field generated by X. It is the smallest field that contains all the sets of the form Ai = {X = xi} and it is denoted by FX or σ(X). The field generated by X represents the information we can extract about the true state ω by observing X. Example 2.5: (Example 2.3 continued.) {ω : X(ω)=1.5} = {ω1,ω2} = A, {ω : X(ω)=0.5} = {ω3,ω4} = A¯. FX = F1 = {∅, Ω,A,A¯}. Filtration Generated by a Stochastic Process Given (Ω, F) and a stochastic process {X(t)} let Ft = σ({Xs, 0 ≤ s ≤ t}) be the field generated by random variables Xs, s =0,...,t.Itisallthe information available from the observation of the process up to time t. Clearly, Ft ⊆Ft+1, so that these fields form a filtration. This filtration is called the natural filtration of the process {X(t)}. 2.1. DISCRETE PROBABILITY MODEL 25 If A ∈Ft then by observing the process from 0 to t we know at time t whether the true state of the world is in A or not. We illustrate this on our financial example. Example 2.6: Take T = 3, and assume that at each trading time the stock can go up by the factor u or down by d. uuu duu B¯ uud dud Ω= udu ddu B udd ddd A A¯ Look at the sets generated by information about S1. This is a partition of Ω, {A, A¯}. Together with the empty set and the whole set, this is the field F1. Sets generated by information about S2 are B and B¯. Thus the sets formed by knowledge of S1 and S2 is the partition of Ω, consisting of all intersections of the above sets. Together with the empty set and the whole set this is the field F2. Clearly any set in F1 is also in F2, for example A =(A∩B)∪(A∩B¯). Similarly if we add information about Ω S3 we obtain all the elementary sets, ω’s and hence all subsets of Ω, F3 =2 .In particular we will know the true state of the world when T =3.
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