A. Hilbert Space

Recall our use of n-dimensional Euclidean space JRn , the set of n-vectors or n-tuples x = (Xl, ... ,xn) with each Xi E JR. Here one has the ordinary Euclidean length - the norm

- and the inner product (or dot product) of two vectors x and y:

n

(x, y), or X· y, := L XiYi· i=l

This setting is adequate for handling finite-dimensional situations, but not for infinite-dimensional ones. The simplest infinite-dimensional situation contain­ing the above as a special case is the space £2 of square-summable sequences x = (Xl, X2,··· ) with

Because of the Cauchy-Schwarz inequality

if x, y E £2, then 00

(x,y) := LXnYn n=l

is defined, and l(x,y)1 ~ Ilxllllyll

(the series 2:::=1 xnYn is absolutely convergent, that is, 2:::=1 IXnllYnl < 00 - just replace Xn, Yn by IXnl ,IYnl in the above). One may choose to work instead with complex sequences: all the above goes through, with the changes

410 A. Hilbert Space

00

(x,y):= LXnYn. n=l

A Hilbert space is a (possibly - indeed, usually) infinite-dimensional vector space endowed with such an inner product, and also complete (in the sense of metric spaces; all Cauchy sequences converge - see Burkill and Burkill (1970)). For background, we must refer to the excellent textbook treatments of e.g. Young (1988) and Bollobas (1990). Note that we already have a good supply of Hilbert spaces: finite-dimensional ones such as JRn and infinite­dimensional ones such as £2 and L2.

Hilbert spaces closely resemble ordinary Euclidean spaces in many re­spects. In particular, one can take orthogonal complements. If M is a vector subspace of a Hilbert space H which is closed (contains all its limit points), one can form its orthogonal complement

M.l. := {y E H: (x,y) = 0 \Ix EM};

then M.l. is also a closed vector subspace of H, and any z E H is uniquely expressible in the form

z=x+y with xEM,yEM.l.

(and so (x, y) = 0). One then says that H is the direct sum of M and M.l., written H = M EB M.l.. If z = x + y,

IIzll2 = (z, z) = (x + y, x + y) = (x, x) + (x, y) + (y, x) + (y, y) = IIxl12 + 2(x, y) + IIyl12 .

In particular, if (x, y) = 0 (Le. x, y are orthogona{),

This is the (in general) infinite-dimensional version of Pythagoras' theorem. Hilbert spaces are the easiest of infinite-dimensional spaces to work with

because they so closely resemble finite-dimensional, or Euclidean, ones. In particular, one can think geometrically in a Hilbert space, using diagrams as one would for (say) JR2 or JR3; see Appendix C.

Functional analysis is the study of infinite-dimensional spaces, and so Hilbert-space theory forms an important part of it. Excellent textbook treat­ments are available; we particularly recommend Young (1988) for Hilbert space alone, Bollobas (1990) for the more general setting of functional anal­ysis.

B. Projections and Conditional Expectations

Given a Hilbert space (or more generally, an inner product space) V, suppose V is the direct sum of a closed subspace M and its orthogonal complement MJ..:

V=MEBMJ...

In the direct-sum decomposition

of a vector z E V into a sum of x E M and y E M J.., consider the map P : z -+ x. This is called the orthogonal projection of V onto M. It is linear, and idempotent: since the direct-sum decomposition of x = pz is x = x + 0, P(pz) = pz = x, or

p2 =P.

By Pythagoras' theorem,

In particular, IIxl12 = IIPzl12 ::::: Ilz112. That is, application of P decreases the norm of a vector: one says that P has norm IIPII ::::: 1. Conversely, we quote that on an inner product space, these properties - P linear and idempotent with IIPII ::::: 1 - characterise orthogonal projections.

The range R(P) of P, that is, the set of x of the form x = pz for some z, is, as above, the set of vectors invariant under P (that is, the set of z with z = pz); P is called the projection onto its range M = R(P). The orthogonal complement N J.. of the range - the set of z with zero x-component - is the set of z annihilated by P, the kernel (or nullspace) N(P) of P. Thus the direct-sum decomposition for the orthogonal projection Ponto M is

V = M EB MJ.. = R(P) EB N(P).

The situation is symmetric between M and MJ..: write Q := 1- P, with I the identity mapping. Then Q is linear, and as p 2 = P,

Q2 = (I - p)2 = 1- 2P + p2 = I - 2P + P = I - P = Q:

Q2 = Q, and conversely, if Q2 = Q, then p 2 = P. Thus also

412 B. Projections and Conditional Expectations

V=MEBMl.=N(Q)EBR(Q), Q:=I-P.

In particular, when V is finite-dimensional, the content of the above re­duces to linear algebra (there is no need to assume M closed, as closure is automatic, so analysis is not needed). For a textbook treatment in this context, see Halmos (1958).

The above use of orthogonal projection, Pythagoras' theorem etc. under­lies the theory of the familiar linear model of statistics: normally distributed errors, least squares, regression, analysis of variance etc. For such a geometric treatment of the linear model, see e.g. Rawlings (1988), Chapter 6.

Conditional Expectations and Projections

We confine ourselves here to the L2 theory. Take the vector space L2 = L2(a, F, JP) of square-integrable random

variables X on a probability space (a, F, JP). This is a Hilbert space under the norm

and inner product (X, Y) := JE(XY).

The space L2 is complete, by the Riesz-Fischer theorem (see e.g. BolloMs (1990)).

If M is a vector subspace of L2 which is closed (equivalently: complete), given any X E L2 one can 'drop a perpendicular' from X to M, obtaining Y E M with

IIX - YII = inf {IIX - WII : W E M}

and (X - Y, Z) = 0 for all Z E M

(see e.g. Williams (1991), §6.11). Then the map X -+ Y is the orthogonal projection onto M: it is linear, idempotent and of norm at most 1.

Suppose now that Q is a sub-a-field of F and M is the L2-space of Q­measurable functions. Then

X --+ JE(XIQ)

is the orthogonal projection of X onto M = L2(a,Q,JP). It gives the best predictor (in the least-squares sense) of X given Q (that is, it minimises the mean-square error of all predictors of X given the information represented by Q): see e.g. Williams (1991), §9.4. The idempotence of this conditional expectation operator follows from the iterated conditional expectation oper­ation of §2.5. This idempotence is also suggested by the above interpretation: forming our best estimate given available information should give the same result done once as done twice.

B. Projections and Conditional Expectations 413

This picture of conditional expectation as projection is powerful - partly because, in the L2-setting, it allows us to think and argue geometrically. The excellent text Neveu (1975) on martingales is based on this viewpoint.

c. The Separating Hyperplane Theorem

In a vector space V, if x and y are vectors, the set of linear combinations ax + f3y, with scalars a, f3 2:: 0 with sum a + f3 = 1, represents geometrically the line segment joining x to y. Each such linear combination AX + (1- A)Y, with 0 ~ A ~ 1, is called a convex combination of x and y. A set C in V is called convex if, for all pairs x and y of points in C, all convex combinations of x and y are also in C.

If V has dimension nand U is a subspace of dimension n - 1, U is said to have codimension 1. If U is a subspace,

x + U := {x + u : U E U}

is called the translate of U by the vector x. A hyperplane in V is a translate of a subspace of co dimension 1. Such a

hyperplane is always representable in the form

H = [I, a] := {x: I(x) = a},

for some scalar a and linear functional I: that is, a map I : V -t IR with

I(x + y) = I(x) + I(y) (x,y E V), f(>.x) = >.f(x) (x E V, >. E IR).

Such an I is of the form

I(x) = /IXI + ... + Inxn;

then I = (/I, ... , In) defines a vector I in V, and the hyperplane H = [I, a] consists of those vectors x in V whose projections onto I have magnitude a.

The hyperplane [I, a] bounds the set A c V if

I(x) 2:: a 'Ix E V or I(x) ~ a 'Ix E V;

the hyperplane [I, a] separates the sets A, B c V if

I(x) 2:: a 'Ix E A and I(x) ~ a 'Ix E B,

or the same inequalities with A < B (or 2::,~) interchanged. The following result is crucial for many purposes, both in mathematics

and in economics and finance.

416 C. The Separating Hyperplane Theorem

Theorem C.0.1 (Separating Hyperplane Theorem). If A, B are two non-empty disjoint convex sets in a vector space V, they can be separated by a hyperplane.

For proof and background, see e.g. Valentine (1964), Part II and Bott (1942). The restriction to finite dimension is not in fact necessary: the re­sult is true as stated even if V has infinite dimension (for proof, see e.g. Bollobas (1990), Chapter 3). In this form, the result is closely linked to the Hahn-Banach theorem, the cornerstone of functional analysis. Again, Bol­lobas (1990) is a fine introduction.

Remark. When using a book on functional analysis, it is usually a good idea to look out for the results whose proof depends on the Hahn-Banach theorem: these are generally the key results, and the hard ones. The same is true in mathematical economics or finance of the separating hyperplane theorem.

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Index

o-admissible, 234 u-algebra, 31 - stopping time, 84

affine term structure, 340 algebra, 31 almost everywhere, 33 almost surely, 33 American put, 141, 258 arbitrage, 1, 15 - free, 106 - opportunity, 19, 106, 232 - price, 115 - pricing technique, 8, 9, 328 - strategy, 106 arbitrageur, 6 ARCH (autoregressive conditional

heteroscedasticity), 317 Asian option - Geman-Yor method, 261 - Rogers-Shi method, 261

barrier option - Asian, 266 - down-and-out call, 264 - forward start, 266 - knockout discount, 265 - moving boundary, 266 - outside, 266 basket default swap, 400 Bayes formula, 225, 239 binomial model, 121 Black formula - caplets, 363 - swaption, 366 Black's futures option formula, 283 Black-Derman-Toy model, 341 Black-Scholes - complete, 248 - European call price, 133 - formula, 44, 251, 294 - hedging, 115, 252, 279

- martingale measure, 243 - model, 196, 243, 270 - partial differential equation, 253, 255 - risk-neutral valuation, 250 - stochastic calculus, 198 - volatility, 314 Borel u-algebra, 31 Brownian motion, 160 - geometric, 197 - martingale characterization, 171, 215 - quadratic variation, 169

call - European - - convergence of CRR price, 133 -- Cox-Ross-Rubinstein price, 126 cap, 354 - Black's model, 355 caplet, 354, 362 central limit theorem - functional form, 223 Collateralized Debt Obligations, 404 complete market, 22 completeness theorem, 116, 238 conditional expectation - iteration, 50 conditional Jensen formula, 48 conditional mean formula, 48 conditional probability, 44 confluent hypergeometric function, 263 contingent claim, 2, 105, 116, 230, 248,

277 - attainable, 236 convergence - almost surely, 51 - in pth mean, 52 - in distribution, 52 - in probability, 52 - mean square, 52 - weak, 53, 221 convolution, 53 copula, 403

434 Index

cost process, 311 coupon, 328 coupon bonds, 328 Cox-Ross-Rubinstein model, 121 credit default swap, 399 credit migration, 376 currency, 5 currency option, 286

default correlation, 376 default probability, 376 derivative - Radon-Nikodym, 43 derivative securities, 2 diffusion, 159, 243 - constant elasticity of variance, 137 distribution

tn, 67 Bernoulli, 40 binomial, 41 bivariate normal, 46 elliptically contoured, 66 generalized inverse Gaussian, 68 hyperbolic, 42, 68 multinormal, 66 normal, 41, 56 Poisson, 42, 57

distribution function, 38 Doob Decomposition, 93 Doob-Meyer-decomposition, 170 dynamic completeness, 272 dynamic portfolio, 102

early-exercise - decomposition, 258 - premium, 260 elasticity coefficient, 255 equivalent martingale measure, 233 Esscher measure, 292 expectation, 39 expectation hypothesis, 346, 348 expiry, 3

Follmer-Schweizer decomposition, 308 factor modelling, 401 factorization formula, 293 Feynman-Kac formula, 201, 251, 339 filtration, 75, 76, 153 - Brownian, 199, 243 financial market model, 101, 229 finite market approximation, 271 finite-dimensional distributions, 153 first-passage time, 225, 264 first-to-default swap, 400

Flesaker-Hughston-model, 370 formula

currency option, 286 Geman-Yor, 263 Levy-Khintchine, 64, 179 risk-neutral pricing, 119, 120 Stirling's, 60

forward,2 - contract, 3 - price, 3 forward LIBOR measure, 357 forward rate, 330 - instantaneous, 331 free lunch, 19 function - characteristic, 55 - finite variation, 37

indicator, 34 Lebesgue-integrable, 35 measurable, 34 simple, 34

futures, 2, 282

gains process, 102, 230 GARCH (generalised autoregressive

conditional heteroscedasticity), 317 Gaussian process, 158 Girsanov pair, 199 Greeks, 254

delta, 254 gamma, 254 rho, 254 theta, 254 vega, 254, 255

Heath-J arrow-Morton - drift condition, 345 - model, 343 hedge - perfect, 119 hedgers, 6 hedging - mean-variance, 18, 307 - risk-minimizing, 18, 311 hedging strategy - CRR model, 127, 128 Hilbert space, 308, 410 hyperplane, 415

implied volatility, 314 independence, 40 index, 5 infinite divisible, 69

inner product, 409 integral - Lebesgue-Stieltjes, 36 - Legesgue, 34 - Riemann, 36 interest rate, 4 intrinsic value, 259 invariance principle, 223 Ito - calculus, 187 - lemma, 195 Ito formula - basic, 194 - for Ito process, 195 - for semi-martingales, 212 - multidimensional, 195 Ito process, 193

Levy process, 178, 294 Laplace transform, 263, 266 law of large numbers - weak, 58 Lebesgue measure, 31 LIBOR dynamics, 358, 361 LIBOR rate, 357 Lipschitz condition, 203, 274 local martingale, 209 localization, 192 lookback option - call, 267 - partial, 269 - put, 267 Levy exponent, 64

market - complete, 116, 236 - incomplete, 289, 295, 315 market price of risk, 245, 247, 338 Markov chain, 78, 96 Markov process, 78, 158 - strong, 158 martingale, 78, 155 - local, 192 - representation, 118 - square-integrable, 94 - transform, 80 martingale measure, 21, 108 - forward risk-neutral, 241, 346 - minimal, 316 - risk-neutral, 120, 246 martingale modelling, 335 martingale representation, 308 martingale transform lemma, 81

maximal utility, 290 maximum likelihood, 257 mean reversion, 342 measurable space, 31 measure, 32

Index 435

- absolutely continuous, 43 - equivalent, 43 measure space, 32 Merton model, 379, 380 method of images, 265 Monte-Carlo, 266 multinomial models, 148

Newton-Raphson iteration, 256 no free lunch with vanishing risk, 235 norm, 409 Novikov condition, 199 numeraire, 21, 101, 230, 239 numeraire invariance theorem, 231

optimal capital structure, 389 optimal stopping problem, 91 option, 2 - American, 2, 138, 258 - Asian, 3, 260 - barrier, 3, 263 -- discrete, 143 - binary, 269 - call, 2 - European, 2 - exotic, 5 - fair price (Davis), 290 - futures call, 283 - lookback, 3, 266 -- discrete, 145 - put, 2 Ornstein-Uhlenbeck process, 205 orthogonal complement, 410 orthogonal projection, 411

partition, 37 point process, 175 Poisson process, 175 portfolio, 9 portfolio credit risk, 400 - asset-based, 401 - loss distribution, 402 predictable, 80, 192, 209 previsible, 209 price - arbitrage, 119 pricing kernel, 368 probability - measure, 33

436 Index

- space, 33 probability space, 38 - filtered, 76 process - Bessel, 261 - Bessel-squared, 262 - maximum of BM, 264 - minimum of BM, 264 projection, 308 put-call parity, 143

quadratic covariation, 157 quadratic variation, 168, 211

random variable, 38 - expectation, 39 - variance, 39 reduced-form model, 391 - valuation, 395 reflection principle, 264 representation property, 238 Riccati equation, 340 Riesz decomposition, 259 risk - intrinsic, 295 - remaining, 295, 311 risk management, 273, 279 risk premium, 338 risk-neutral valuation, 115 236 250

273, 335 " ,

sample space, 37 self-decomposability, 65 semi-martingale, 186, 209 - characteristics, 218 - good sequence of, 223 separating hyperplane Theorem, 19 Snell envelope, 89, 259 speculator, 6 spot LIB OR measure, 361 spot rate, 331 spreads, 382 state-price vector, 20, 276 stochastic basis, 76, 153 stochastic differential equations - strong solution, 204 - weak solution, 204 stochastic exponential, 197, 198, 216 stochastic integral - quadratic variation of, 192 stochastic process, 77, 153 - adapted, 77, 153 - cadlag, 154 - Poisson, 42

- progressively measurable, 243 - RCLL, 154 stochastic volatility, 219 stock, 4 stock price - jump, 136 stopping time, 82, 155 strategy - replicating, 112 strike price, 3 structural model - Black-Cox, 384 - Leland, 389 - Merton, 380 - stochastic interest rate, 388 structure-preserving property, 270 submartingale, 79, 155 supermartingale, 79, 155 swap, 2, 4, 353, 363 swaption, 365

tail dependence, 403 term structure equation, 337, 339 theorem - central limit, 59 - Doob's Martingale convergence, 95 - Feynman-Kac, 202 - fundamental theorem of asset pricing,

119,235 - Girsanov, 199 - local limit, 60 - monotone convergence, 35 - Optional Sampling, 85 - Poisson limit, 60, 61 - portmanteau, 222 - representation of Brownian martin-

gales, 200 - Stopping time principle, 83 trading strategy, 18, 102 - admissible, 235 - mean-self-financing, 312 - replicating, 236 - self-financing, 230 - tame, 233

uniform integrability, 156 usual conditions (for a stochastic basis)

153 ' utility function, 290

value process, 102, 230 Vasicek model, 340 volatility, 196

historic, 257 implied, 256 non-parametric estimation, 258

- stochastic, 257 volatility matrix, 243 volatility smile, 314

Index 437

weak law of large numbers, 52 wealth process, 102

yield-to-maturity, 329

zero-coupon bond, 329, 330