A. Useful Tools from Martingale Theory This appendix covers a number of standard results from martingale theory and stochastic calculus which are used in the main body of the text. The concepts are outlined in a very brief fashion. Proofs are often omitted or the reader is referred to the literature. The first section introduces the probabilis­ tic setup of the material used in later sections. See, for example, Williams (1991) for a more detailed introduction to probability. The later sections cover some important results from martingale theory and stochastic calcu­ lus. A more extensive treatment of this material can be found in any text on Brownian motion and stochastic calculus. Most of the material presented in this section is adapted from Karatzas and Shreve (1991) and Revuz and Yor (1994). A.I Probabilistic Foundations Definitions. A measurable space (il,9") consists of a sample space il and a collection of subsets of il called a l1-algebra. A count ably additive mapping J.L : 9" -+ JR + is called a measure on (il, 9") and the triple (il, 9", J.L) is called a measure space. A 9"-measurable function is a mapping h from (il,9") to a state space (S, S) such that for any A E S, h-1(A) E 9". A Borel l1-algebra is a l1-algebra containing all open sets of a topological space. For example, 'B(il) is the Borel l1-algebra of il. The mapping h is called a Borel function if it is 'B(il)­ measurable. A random variable is a measurable function. For example, consider a collection of functions h from a topological space il with l1-algebra 9" to S = JRd and S = 'B(JRd). S = JRd is the Euclidean space and 'B(JRd) is the smallest l1-algebra of Borel sets of the topological space JRd such that each h is measurable. We say that the l1-algebra l1(h) is generated by h. Obviously, for a random variable X, I1(X) E 9". Otherwise the mappings would not be measurable. A process is an family of random variables and is written (Xt)O~t~T or simply X t in shorthand notation. The index t is often interpreted as time defined on the interval [0, T] for T < 00. 224 A. Useful Tools from Martingale Theory For fixed sample points wEn, the function t ~ Xt(w) is called the sample path or trajectory of the process. However, a process can be viewed as a function of two variables (t,w) where t is the (time) index and wEn, i.e. we have the mapping (t, w) ~ X t (w) Vw E n. Therefore, a process is a joint mapping h from lR+ x n onto S. This mapping is measurable if we equip it with the corresponding a-algebrae of Borel sets, i.e., if we have h : (t,w) ~ Xt(w) : (lR+ x n, ~(lR+) ® ~ ~ (lRd, ~(lRd») such that for any A E ~(lRd), h-1(A) E ~(lR+) ®~. A probability space is a measure space with /1(n) = 1 and /1(0) = o. In this case /1 is called a probability measure. It associates subsets of n (events) with a probability. For example, consider a probability space (n,~, P). If, for an event A E ~, peA) = 1, then we say that the event occurs almost surely, abbreviated as P-a.s. or a.s. An event A E ~ with peA) = 0 is called almost impossible. If A ¢ ~, then A is impossible. A probability measure Q is said to be equivalent to P if, for any event A E ~, peA) = 0 {:} Q(A) = O. This means that two probability measures P and Q are equivalent if and only if for any event which is almost impossible under P, it is also almost impossible under Q. If only peA) = 0 =} Q(A) = 0, then we say that Q is absolutely continuous with respect to P. A filtration ~O~t~T = {~o, ... ,~T}, sometimes written (~dO~t~T or ~t in shorthand, is a family of sub-a-algebrae included in ~ which is non-decreasing in the sense that ~s s;;: ~t for s < t. Often, a filtration can be interpreted as representing the accumulation of information over time. A process generates a filtration. For example, consider the filtration ~t = a(XsE[O,tJ) generated by Xt. This is the smallest a-algebra with respect to which Xs is measurable for every s E [0, t]. For technical reasons, we usually assume that it also contains all P-null sets, i.e., if A E ~ and peA) = 0 , then A E ~t. \It E [0, T]. In this case it is called the natural filtration of X. Generally, we implicitly assume a filtration to be of the natural kind. A filtration to which the null sets are added is called augmented. A filtered probability space is a probability space equipped with a fil­ tration and is often written (n,~, (~dO~t~T' P), (fl,~, (~tE[O,TJ)' P), or (fl,~, (~t)O~t~T' P). Consider a filtered probability space. If, for an event A E ~, peA) = 1 for all t E [0, T], then we say that A is given almost everywhere. In shorthand notation we write A P-a.e. or a.e. Let a filtered probability space (fl,~, (~dO<t<T' P) be given. A process (Xt)O~t~T is called adapted (to the filtration :ftfif it is ~rmeasurable. We also say that the process is ~t-adapted. A process X t is called predictable if it is ~t--measurable for continuous processes and ~t_l-measurable for dis­ crete processes. A predictable process is ~t--adapted (~t_l-adapted) and ~radapted. For notational simplicity, we sometimes write ~s, s < t, for ~t-. The difference between adapted and predictable processes is mostly relevant in the case of discrete processes. A.3 Martingales 225 A.2 Process Classes Definition A.2.1. Let £00 be the class of all adapted processes. 0 is a pro­ cess. Define the following spaces: £1 = {o E £00: loT 10ti dt < 00 a.s.}, £2 = { 0 E £00 : loT O;dt < 00 a.s.}, 1{2 = { 0 E £2 : E (loT O;dt) < 00 } , £1,1 = {o E £00 : loTI loT2 10(s, t)ldsdt < 00 a.s.} , £2,1 = {o E £1,1 : loT O(s, u)2ds < 00 a.s., f (f lo(s.U)ldU), ds < = as} We write 0 E £1(f.?,3',P) or simply 0 E £1 if it is clear from context on which probability space the process is defined. In most instances in this text, integrable (£1) or square-integrable (£2) processes are used. A.3 Martingales Definition A.3.1. A process Mt adapted to 3't and satisfying E [lMt I] < 00, 'it E [0, T] is called a P-submartingale if and a P -supermartingale if M t is a martingale if it is both a submartingale and a supermartingale, i.e., if Ep[MTI3'd = M t . Remark A. 3.1. Informally, we call the process M t a local martingale if it does not satisfy the technical condition E(IMtl) < 00, Vt E [0, T]. 226 A. Useful Tools from Martingale Theory Theorem A.3.1 (Dooh-Meyer). Let X be a right-continuous submartin­ gale. Under some technical conditions, X has the decomposition °:s t < 00, such that M t is right-continuous martingale and At an increasing process. Proof. Cf. Karatzas and Shreve (1991), p. 25. Definition A.3.2. For a martingale M t E £2, the quadratic variation pro­ cess is (M, M)t = At, where At is the increasing process of the Doob-Meyer decomposition of X = M2. Quadratic variation is also known as cross­ variation. Remark A.3.2. M is a unique process such that M2 - (M, M) is a martingale. (M, M) is often abbreviated (M)=(M, M). Definition A.3.3. Similar to the previous definition, for M and N two different continuous martingales, there exists a unique continuous increas­ ing process (M, N) = ~((M + N) - (M - N) vanishing at zero such that M N - (M, N) is a martingale. Remark A. 3. 3. This definition immediately follows from the previous defini­ tion. Consider two different martingales M and N. By the previous definition, the process (M +N)2_(M + N) is a martingale, but so is (M _N)2_(M - N) and their difference 4MN - (M + N) + (M - N). Therefore, the cross­ variation process is (M, N) = ~((M + N) - (M - N). If M and N are independent martingales, then (M, N) = 0. Definition A.3A. A continuous semimartingale is a continuous adapted process X which has the decomposition X = Xo + M + A where M is a con­ tinuous local martingale and A a non-decreasing, continuous, adapted process of finite variation. Xo is a :Fo-measurable random variable. Theorem A.3.2 (Martingale Representation). Let M be a continuous local martingale. For any adapted local martingale X, there is a predictable process <p such that X t = Xo + lot <Ps dMs· Proof. Cf. Revuz and Yor (1994), Chapter IV. Corollary A.3.1. Let W t be a JRd-valued Brownian motion. If X t is a con­ tinuous martingale in (£2)n with X O = 0, then there exists a unique pre­ dictable process <p in (1t2)nxd such that X t = Xo + lot <Ps . dWs. Proof. Cf. Karatzas and Shreve (1991), Section 3.D. Remark A.3.4. For X t is a local martingale, it is sufficient that <Pt E (£2)d (cf.