Stochastic Calculus For Finance , By Steven E. Shreve, Springer ...

BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 46, Number 1, January 2009, Pages 165–174
S 0273-0979(08)01217-2
Article electronically published on August 28, 2008
Stochastic calculus for ﬁnance, by Steven E. Shreve, Springer Finance Textbook
Series, in two volumes: Volume I: The binomial asset pricing model, Springer,
New York, 2005, x+187 pages, $34.95, ISBN 13: 978-0387-24968-1 and Volume
II: Continuous-time models, Springer, New York, 2004, x+550 pages, $69.95,
ISBN 0-387-40101-6.
The recent turmoil in ﬁnancial markets has been partly caused by insuﬃcient
attention to rigorous ﬁnancial modeling. Among the causes of this failing is the rel-
ative shortage of mathematically well trained professionals in the ﬁnancial services
industry. Shreve is a co-founder of one of the oldest and most successful masters
degree programs in ﬁnancial engineering, established at Carnegie-Mellon University
in 1994. The lecture notes on which this book was based were tested and honed by
Shreve over many years of teaching in this Computational Finance program. The
result is a remarkable piece of pedagogy and a great service to all entrants to the
ﬁeld.
What follows is a review of Steven Shreve’s masterful two-volume text, Stochastic
Calculus for Finance, which introduces students to stochastic calculus as a tool for
ﬁnancial derivative pricing. I will begin with a brief outline of the nature of the
subject and some of the major historical milestones, and then explain why I believe
that Shreve’s text is the ideal introduction to the topic.
The stochastic integral as a model of trading profits
Michael Harrison, whose role in the development of the subject will come up
shortly, once remarked to me that stochastic calculus has the appearance of having
been expressly designed as a tool for ﬁnancial analysis, so naturally does it ﬁt the
application. Stochastic calculus is now the language of pricing models and risk
management at essentially every major ﬁnancial ﬁrm, and it is the backbone of a
large body of academic research on asset pricing, corporate ﬁnance, and investor
behavior.
The typical stochastic-calculus-based ﬁnancial model describes the random vari-
ation of the market price, say Xt at time t, of some ﬁnancial asset. For proper foun-
dations, one ﬁxes a probability space (Ω, F, P ) (a measure space with P (Ω) = 1),
as well as a ﬁltration {Ft : t ∈ [0, ∞)} of sub-σ-algebras of F that determines the
timing of the revelation of information. The “usual conditions” for a ﬁltration are
laid out, for example, by Protter [19]. One may loosely view Ft as the set of events
(elements of F) whose outcomes are certain to be revealed to investors as true or
false by (or at) time t. For any event A, the probability assigned to A by investors is
P (A). The price process X = {Xt : t ∈ [0, ∞)} is adapted1 to the ﬁltration, mean-
ing that Xt : Ω →
is a random variable whose outcome is revealed to investors
at or before time t.
2000 Mathematics Subject Classiﬁcation. Primary 60-01, 60H10, 60J65, 91B28.
I am grateful for conversations with Julien Hugonnier and Philip Protter, for decades worth of
interesting discussions with Mike Harrison, and also for the patient encouragement of the editor,
Bob Devaney.
1 A process X is adapted if, for all t, Xt is Ft-measurable.
c 2008 American Mathematical Society
Reverts to public domain 28 years from publication
165

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Occasionally, one hears that market eﬃciency implies that the price process X
must be a martingale,2 meaning essentially that the current price Xt is a condi-
tionally unbiased predictor of the price Xu at any future time u. This is a mis-
conception: Investors would generally not take the risk of owning an asset unless
they are compensated with expected returns. Beyond compensation for risk, even
a risk-free asset must oﬀer a return that compensates the investor for tying up cap-
ital. Allowing for nonzero expected price changes, it is therefore natural to treat
the price process X as, loosely speaking, a “martingale plus something,” or, to pick
a precise and natural deﬁnition, a semimartingale.3 The most classical example of
a semimartingale used in ﬁnancial modeling, suggested in 1965 by the economist
Paul Samuelson, is a geometric Brownian motion, which we will get to later.
A trading strategy θ determines the quantity θt(ω) of the asset held in each state
ω ∈ Ω and at each time t. For a model of a well-functioning market, it is crucial to
rule out trading strategies that are based on advance knowledge of price changes,
for there would otherwise be arbitrages, meaning trading strategies with unlimited
proﬁts at no risk. The natural corresponding measurability restriction is that θ is
a predictable process.4 Given a price process X and a trading strategy θ satisfying
t
technical conditions, the total ﬁnancial gain
θ
s
u dXu between any times s and
t ≥ s is deﬁned as a stochastic integral.5
An elemental type of trading strategy is a “buy-and-hold” strategy θ, which
initiates a position immediately after some stopping time T and closes it at some
later stopping time U . For a position size θ that is FT -measurable,6 the trading
strategy θ is deﬁned by θt = 1{T < t ≤ U} θ. The total gain from trade for this buy-
U
and-hold strategy is naturally
θ
0
t dXt = θ(XU − XT ), the position size multiplied
by the interim price change. The gain from trade for a general stochastic trading
strategy can be deﬁned as the total gain of an approximating portfolio of buy-
and-hold strategies, in a particular limiting sense.7 For the cases most commonly
encountered in ﬁnancial applications, based on Brownian motion, Shreve gives a
very clear explanation of this limit in his second volume, Continuous-Time Models,
Sections 4.2–4.3.
A typical ﬁnancial model allows for n diﬀerent securities, with price processes
X1, . . . , Xn. An investor can choose an associated n-dimensional trading strategy
θ = (θ1, . . . , θn) from some allowable set Θ, determining the total gain-from-trade
process
n
θt dXt ≡
θit dXit.
i=1
2 A martingale is an integrable adapted process M whose conditional expected change
E(Mu − Mt | Ft) is zero whenever u ≥ t.
3 A semimartingale is deﬁned as the sum M + A of local martingale M , a slight relaxation of a
martingale, and an adapted process A whose sample paths have ﬁnite variation on each bounded
time interval.
4 A predictable process is a map from Ω × [0, ∞) to
that is measurable with respect to the
σ-algebra generated by left-continuous adapted processes.
5 For general settings, minimal restrictions on the trading strategy θ and the price process X
for
θt dXt to be a well-deﬁned stochastic integral can be reviewed in [19].
6 For a stopping time T , the σ-algebra FT has a special deﬁnition. See [19].
7 Again, see [19] for the case of semimartingale X.

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167
In addition to incorporating technical restrictions under which these stochastic
integrals are well deﬁned, the allowable set Θ can enforce budget limits, credit
constraints, short-sales limitations, or various other natural investment restrictions.
Signiﬁcant strands of research literature address the following two classes of
problems:
• Given some “utility” functional U on the space of potential gains from
t
trade, solve the optimization problem sup
U
θ
θ∈Θ
0
s dXs
. The utility
functional U can encode preferences regarding risk, intertemporal substi-
tution, and the timing of information about trading gains, among other
properties.
• Apply the laws of supply and demand to characterize the price processes
of the available ﬁnancial securities. A minimal restriction on the behavior
of prices is the absence of arbitrages; demand and supply could never be
matched in the presence of an arbitrage.
Both volumes of Shreve’s text focus on arbitrage-free asset pricing, perhaps
because most of the students for whom he has written the book are aiming for
business careers in ﬁnance. The theory of optimal investment, while a signiﬁcant
subject area in academia, has achieved much less traction in business practice.
In an earlier book, Mathematical Finance, Shreve and his frequent collaborator
Ioannis Karatzas provide a detailed treatment of mathematical models of optimal
investment.
The ﬁeld of ﬁnance is replete with many other applications of stochastic calculus,
such as the ﬁnancial policies of corporations, the design of new securities, and risk
management, which usually involves control or characterization of the left tail of
t
the probability distribution of the gain from trade
θ
0
s dXs.
The beginnings of stochastic calculus
Even as early as 1900, Louis Bachelier had introduced Brownian motion as a
ﬁnancial price process. In 1905, Albert Einstein, unaware of Bachelier’s prior work,
suggested the name “Brownian motion” and characterized its essential properties.
(On nonoverlapping time intervals, the increments of a Brownian motion are in-
dependent and normally distributed with means and variances that are each in
ﬁxed proportions to the lengths of the time intervals.) Norbert Wiener showed the
existence of such a process in 1923. Paul L´
evy and Andrei Kolmogorov provided
successively fuller developments of the Brownian model.
In 1944, Kiyoshi Itˆ
o laid the foundations for stochastic calculus with his model
of a stochastic process X that solves a stochastic diﬀerential equation of the form
t
t
(1)
Xt = X0 +
µ(Xs) ds +
σ(Xs) dBs,
0
0
where B is a standard Brownian motion,8 and where µ and σ are functions from
to
satisfying some technical condition. (A Lipschitz condition suﬃces.) As a
8 A standard Brownian motion B is a Brownian motion whose initial condition B0 is zero with
probability one, and whose increments have zero expectation and have a variance equal to the
length of the time interval.

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generalization of (1), Itˆ
o then characterized a class of processes of the form
t
t
(2)
Xt = X0 +
Hs ds +
Vs dBs,
0
0
for adapted processes H and V satisfying suitable technical conditions. Such a
process is now called an “Itˆ
o process,” a special case of what later became known
as a semimartingale. Itˆ
o’s next important achievement, in 1951, was to establish
that, for any smooth function f :
→ ,
t
1
t
(3)
f (Xt) = f (X0) +
f (Xs)Hs + f (Xs)V 2
ds +
f (X
s
s)Vs dBs.
0
2
0
Itˆ
o’s formula (3), now named for him, has important generalizations. In his notes
to Chapter 4 of Volume II, Shreve relates that Vincent Doeblin,9 a French soldier
in World War II, independently discovered essentially the same result in 1940. This
only came to light in May 2000. As a result, throughout his book Shreve has called
(3) the “Itˆ
o–Doeblin formula.”
Jarrow and Protter in [14] oﬀer a colorful and much more complete history of
these developments up to the time of modern ﬁnancial theory.
The modern era of financial asset pricing
In 1969, Robert C. Merton introduced stochastic calculus to ﬁnance, indeed
to the broader ﬁeld of economics, beginning an amazing decade of developments,
most famously the formula of Black and Scholes in [2] for the price of an option
that conveys the right to buy an asset at some future time T at a ﬁxed price of K.
If the underlying asset has the price process X, the eﬀective payoﬀ of the option at
time T is max(XT − K, 0).
Black and Scholes chose a setting of constant interest rates, and followed Samuel-
son’s lead by taking X to be geometric Brownian motion, the solution of a stochastic
diﬀerential equation of the simple form
(4)
dXt = µXt dt + σXt dBt,
for constants µ and σ. The constant µ represents the expected rate of return on
the asset, while the constant σ is known as the “volatility.”
Black and Scholes originally based their option pricing formula on a restrictive
theory of demand and supply in market equilibrium. Robert Merton provided a
revolutionary alternative derivation of the formula by pointing out that the option
is actually redundant: The option payoﬀ max(XT − K, 0) can be replicated by
trading the underlying asset, borrowing and lending over time as needed. Merton’s
replicating trading strategy has an initial market value that must be the price of
the option. (If not, for example, if the option price is higher, one could arbitrage by
selling the option, investing in the option replication strategy at a lower cost than
the price of the option, and using the proceeds of the replication strategy at time
T to cover the payoﬀ required on the short option position.) The initial cost of the
replication strategy is indeed given by the same Black–Scholes formula! In their
path-breaking journal article, Black and Scholes gave both their original market-
equilibrium-based justiﬁcation, as well as Merton’s more general arbitrage-based
argument.
9 Shreve refers to “W. Doeblin,” but as pointed out in [14], Wolfgang Doeblin changed his name
to Vincent Doeblin.

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169
Although Black and Scholes initially had diﬃculty getting their paper published,
the option pricing model was eventually the basis of a Nobel prize awarded to
Robert Merton and Myron Scholes in 1997. Sadly, Fischer Black had not survived
to receive the prize.
Once “the formula” was digested and researchers recognized the power of sto-
chastic calculus for analyzing business and theoretical problems, the range and
depth of applications expanded rapidly. Among the most important early applica-
tions, beyond option pricing, were dynamic models of the term structure of interest
rates, beginning with those of Vasicek in [24] and Cox, Ingersoll, and Ross in [3].
As these developments unfolded, Cox and Ross in [4] noticed that, without loss
of generality for the purpose of pricing a large class of derivative securities such
as options, one could invariably get the correct result by pretending that all of
the securities have an expected rate of return that is equal to the current risk-free
interest rate. This was initially surprising, for the actual mean rate of return of a
ﬁnancial asset is sensitive to its risk; assets that impose adverse risk on investors
demand higher mean rates of return.
The notion of “risk-neutral” pricing was given abstract foundations by Harri-
son and Kreps in [10], who formalized (under conditions) the near equivalence of
the absence of arbitrage with the existence of some new “risk-neutral” probability
measure Q under which all expected rates of return are indeed equal to the current
risk-free rate. In particular, this allows one to price any security as the Q-expected
sum of its future discounted cash ﬂows, a dramatic practical simpliﬁcation that was
almost immediately adopted in a wide range of applications in the ﬁnancial services
industry. This new “risk-neutral” pricing method augmented, and in many cases
largely displaced, the partial-diﬀerential-equation (PDE) approach to derivative
pricing that had been used by Black and Scholes in [2].
As an illustration, suppose that the risk-free rate is a constant r and that a stock
has an Itˆ
o price process X of the form (2). We are interested in the price of an
option that conveys the right to buy the stock at some time T at a price of K. By
the deﬁnition of a risk-neutral probability measure Q, the initial price of the option
must be
(5)
EQ e−rT max(XT − K, 0) ,
where EQ denotes expectation with respect to the measure Q. For the geometric
Brownian model (4) of X, Harrison and Kreps applied Girsanov’s Theorem, laid out
in Chapter 5 of Shreve’s second volume, to show that log XT is normally distributed
under Q with mean (r + σ2/2)T and variance σ2T . The explicit calculation of (5) is
then routine, and of course matches the Black–Scholes option pricing formula that
was originally derived by solving a PDE.
Harrison and Pliska in [11] pushed on, placing the theory of trading gains
squarely on the more general foundation of stochastic integration with respect
to semimartingales. Only much later did Delbaen and Schachermayer in [6] ﬁ-
nally establish essentially minimal conditions connecting the notions of risk-neutral
probability measures with the absence of arbitrage. These theoretical developments
are not covered in Shreve’s text; they are accessible in full generality only to more
advanced researchers.
Meanwhile, application after application, in both academic research and industry
practice, was built on the familiar template of arbitrage-free pricing and risk-neutral
probability measures.

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Volume I. The Binomial Asset Pricing Model
Shreve’s wonderful two-volume treatment of the topic can be viewed as two self-
contained books. Throughout the ﬁrst volume, The Binomial Asset Pricing Model,
the price of an asset changes only at integer dates, and, conditional on its current
level, the next price change has only two possible directions, “up” and “down.”
In this “binomial” framework, measure-theoretic technicalities are easily avoided,
making this an ideal setting for an introduction to the modeling of derivative prices.
Under the assumption that X is a Markov process under the risk-neutral prob-
ability measure Q, the analysis is dramatically simpliﬁed, as follows. Suppose that
a derivative security pays g(XT ) at time T , for some payoﬀ function g. For our
previous example of an option, g(x) = max(x − K, 0). The deﬁnition of a risk-
neutral probability measure implies that the initial price of the derivative security
is EQ(e−rT g(XT )).
In order to compute the initial derivative price in the binomial setting, suppose
that at a given time t and current price level x, the underlying price jumps up by
the factor Ux,t with risk-neutral probability px,t, and jumps down by the factor
Dx,t with probability 1 − px,t. Then the associated derivative price f (x, t) must
satisfy the recursion
(6)
f (x, t) = e−rt (px,tf (xUx,t, t + 1) + (1 − px,t)f (xDx,t, t + 1)) .
Given the boundary condition f (x, T ) = g(x), this allows an easy calculation of the
initial derivative price EQ(e−rT g(XT )).
For the trivial case of a derivative that is a claim to the underlying asset itself,
the price is f (x, t) = x for all x and t, so we can apply (6) in this case and calculate
that
Ux,t − Dx,t
(7)
px,t =
.
er − Dx,t
(If U > D, we must have Ux,t > er > Dx,t, for otherwise an arbitrage arises.) Thus,
the model is completely determined by the interest rate r and the jump factors Ux,t
and Dx,t.
If the jump factors Ux,t and Dx,t do not depend on x or t, then by virtue of
(7), neither does px,t, so the logarithm of the price process X is, under Q, the
sum of independent and identically distributed Bernoulli trials. This is a popular
special case in Shreve’s book and in ﬁnancial industry practice during the 1970s and
1980s. After the Black–Scholes formula appeared, Bill Sharpe (better known for his
Nobel-prize-winning Capital Asset Pricing Model) noticed that the continuous-time
option pricing model has this simple “binomial” analogue.
Cox and Ross in [5] developed the binomial model in a general multi-period
setting and showed how the Black–Scholes formula arises as the limit of the option
price in the binomial model as the number of trading periods per unit of time goes
to inﬁnity, and the jump factors U and D are normalized to give a ﬁxed return
variance per unit of time. This limit can be viewed as a consequence of Donsker’s
Theorem (recorded, for example, in [9]), by which the distribution of a Brownian
motion such as log X can be approximated as a normalized sum of Bernoulli trials.
After laying out the basics, Shreve provides two extensive applications of the
binomial model: American options and interest-rate derivatives. The nomenclature
“American,” due to Paul Samuelson, refers to the contractual freedom oﬀered to
the investor to collect the derivative payoﬀ g(Xτ ) at any stopping time τ before the

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expiration date T . It can be shown, using the theory of optimal stopping, that the
price of the American security must, in the absence of arbitrage, be what Merton
in [17] called the “rational price,” given by
sup EQ e−rτ g(Xτ ) .
τ ≤T
Curiously, Shreve does not show precisely why this pricing formula is required by
the absence of arbitrage, but he does show something tantamount to this. Using
a novel pedagogical approach based on the reﬂection principle, Shreve goes on to
treat “perpetual” American options, those with no upper bound on the exercise
stopping time τ .
Elementary by design, Shreve’s ﬁrst volume is a lovely introduction to ﬁnancial
modeling, and could be taught to masters or even undergraduate students as a
short, say month-long, course.
Volume II. Continuous-Time Models
Shreve’s much longer second volume, Continuous-Time Models, is a self-
contained introduction to stochastic calculus and its applications to ﬁnancial mod-
eling. In my view, there is no better introductory treatment of the topic. Shreve’s
facility with the subject at a deep mathematical level, combined with his pedagog-
ical talent, allowed him to make this treatment both rigorous and easily accessible
to good masters-level students. Apparently no eﬀort has been spared to get it
right and to make it understandable. Shreve uses measure theory to explain and
reinforce concepts, as opposed to some introductory treatments that attempt to
avoid measure theory and invariably get into diﬃculties in providing intuitive, not
to mention correct, results.
Volume II begins with a general measure-theoretic introduction to probability
theory, a routine two-chapter summary. Chapter 3 is devoted to the deﬁnition and
properties of Brownian motion. Getting to the heart of the matter, Chapter 4
introduces stochastic calculus and, in a lovely stroke of exposition, uses the Black–
Scholes option pricing model to explain how the theory works.
Chapter 4 ends with the Brownian bridge, a stochastic process whose paths
between 0 and t have the probability distribution of a Brownian motion conditional
on a given ending point at time t. Shreve provides two alternative constructions
of the Brownian bridge: as a particular Gaussian process and as the solution of
a stochastic diﬀerential equation. I had never seen the Brownian bridge so nicely
explained from several viewpoints. Although the Brownian bridge does have some
applications in ﬁnance, none are developed in Shreve’s book. Perhaps this section
was originally written for some other book?
Chapter 5 completes the underpinnings of the basic theory by covering the
Harrison–Kreps theory of risk-neutral probabilities (what they call “equivalent mar-
tingale measures”), allowing an often painless route to the calculation of derivative
prices.
For example, consider again the price of a derivative paying g(XT ) at T , where
the underlying asset price process X is the Ito process (2). An equivalent martingale
measure is a probability measure Q that is equivalent to the reference measure P
(in the sense that P and Q assign zero probability to the same set of events), and
under which e−rtXt is a martingale. Indeed, a deﬁning property of Q is that any
asset price, discounted by e−rt, is a Q-martingale. So, the initial derivative price

172
BOOK REVIEWS
is EQ(e−rT g(XT )). For a well-deﬁned derivative price, it is now only a question of
ensuring that such a measure Q exists and deducing the probability distribution of
XT under Q. In Shreve’s Brownian setting, following the Harrison–Kreps approach,
this is accomplished with Girsanov’s Theorem.10
Chapter 6 connects the foundational theory with Markov processes and par-
tial diﬀerential equations. If X is a Markov process under Q solving a stochastic
diﬀerential equation of the type (1), then the function f deﬁned by f (Xt, t) =
EQ[e−rT g(XT ) | Xt] can be viewed as the solution of a particular (backward Kol-
mogorov) parabolic partial diﬀerential equation. This PDE approach is now lit-
tle used in mainstream ﬁnancial engineering. Typically, derivative prices, as risk-
neutral expectations, are computed by Monte Carlo simulation. PDE methods are
used principally for initial conceptual model designs. This is in part because of the
restrictiveness of the Markov setting, and partly because essentially all models in
use at a bank or trading ﬁrm must be integrated for risk management and other
purposes within a Monte Carlo setting.
Most of the remainder of the book is taken up with a range of applications:
Exotic Options (Chapter 7), American Derivative Securities (Chapter 8), Change
of Num´
eraire (Chapter 9), and Term-Structure Models (Chapter 10). To this point,
aside from a few gems like the Brownian bridge section, the topic coverage and
overall structure are quite conventional; one would expect as much in a textbook
designed for future practitioners. What marks the book as unusual in comparison
with many other available texts covering this topic is the care applied to detail,
rigor, and exposition.
The ﬁnal chapter extends from the Brownian model to allow processes with
jumps, for example Poisson processes. This extension has become crucial with the
advent of a deep market for credit derivatives, which allows investors to hedge
against, or speculate on, ﬁnancial default. As default frequently occurs with little
or no warning, it would be diﬃcult to parsimoniously model the price of a credit
derivative with the continuous paths of an Itˆ
o process. Unfortunately for readers,
although Shreve has worked on default risk in some of his own research, his textbook
does not deal with the subject. (Perhaps he can be convinced to extend his coverage
to credit derivatives in a second edition!)
Each chapter of Stochastic Calculus for Finance ends with notes to the literature
and instructive exercises. I am pleased that Shreve has not succumbed to the usual
plea that a textbook should include solutions to the exercises. Rather, he nicely
calibrates the diﬃculty of his exercises and lards them with hints, so that the
student can learn by solving problems. In my experience, this is likely to reinforce
the concepts and techniques more deeply than merely learning by reading.
For years, I have proﬁted from reading Steven Shreve’s papers and books and
have been captivated by his remarkably lucid research presentations. It is easy to
see why he is widely admired for bringing clarity to the many subject areas that
he has touched over his career, ranging well beyond ﬁnancial modeling. Not only
do I recommend these two volumes as the place to start one’s education in asset
10 The Radon–Nikod´
ym derivative dQ has a density process ξ, deﬁned by ξ
| F
dP
t = E( dQ
dP
t). The
martingale representation theorem, also found in Shreve’s Chapter 5, implies that dξt = −ξtηt dBt,
for a process η that plays a special role in pricing models. Girsanov’s Theorem, provided by Shreve
in Section 5.4, is that the process B∗ deﬁned by B∗
t = Bt +
t η
0
s ds, is a standard Brownian motion
under Q.

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173
pricing, I am convinced that even more advanced scholars would proﬁt from pure
enjoyment of the exposition.
References
[1] Bachelier, L. (1900). Th´
eorie de la Speculation. Annales Scientiﬁques de L’ ´
Ecole Normale
Sup´
erieure 3d ser., 17, 21–88. Translation in The Random Character of Stock Market Prices,
ed. Paul Cootner, pp. 17–79. Cambridge, MA: MIT Press, 1964.
[2] Black, F. and M. Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal
of Political Economy 81, 637–654.
[3] Cox, J., J. Ingersoll, and S. Ross (1985). A Theory of the Term Structure of Interest Rates.
Econometrica 53, 385–408. MR785475
[4] Cox, J. and S. Ross (1976). The Valuation of Options for Alternative Stochastic Processes.
Journal of Financial Economics 3, 145–166.
[5] Cox, J., S. Ross, and M. Rubinstein (1979). Option Pricing: A Simpliﬁed Approach. Journal
of Financial Economics 7, 229–263.
[6] Delbaen, F. and W. Schachermayer (1999). A General Version of the Fundamental Theorem
of Asset Pricing. Mathematische Annalen 300, 463–520. MR1304434 (95m:90022b)
[7] Doeblin, V. (1940). Sur l’Equation de Kolmogorov. Comptes Rendus de L’Acad´
emie de
Sciences, Ser. I 331, 1059–1102.
[8] Einstein, A. (1905). Uber die von der Molekularkinetischen Theorie der W¨
arme gefordete
Bewegung von in ruhenden Fl¨
ussigkeiten suspendieren Teilchen. Annalen der Physik und
Chemie 17, 549–560, Reprinted as “On the Movement of Small Particles Suspended in Static
Liquids as Claimed in the Molecular Kinetic Theory of Heat” in Investigations of the Theory
of Brownian Movement, ed. R. F¨
urth, New York: Dover, 1956.
[9] Ethier, S. and T. Kurtz (1986). Markov Processes: Characterization and Convergence. New
York: Wiley. MR838085 (88a:60130)
[10] Harrison, M. and D. Kreps (1979). Martingales and Arbitrage in Multiperiod Securities
Markets. Journal of Economic Theory 20, 381–408. MR540823 (80h:90025)
[11] Harrison, M. and S. Pliska (1981). Martingales and Stochastic Integrals in the Theory of
Continuous Trading. Stochastic Processes and Their Applications 11, 215–260. MR622165
(83a:90022)
[12] Itˆ
o, K. (1944). Stochastic Integral. Proceedings of the Imperial Academy of Tokyo 20, 519–
524. MR0014633 (7:313c)
[13] Itˆ
o, K. (1951). On a Formula Concerning Stochastic Diﬀerentials. Nagoya Mathematics
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Darrell Duffie
Graduate School of Business, Stanford University, Stanford, California 94305-5015

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