The main purpose of this chapter is to prove the rst and second fundamental theorem of asset pricing in a so called nite market model.
|
|
- Berenice Fox
- 5 years ago
- Views:
Transcription
1 1 2. Option pricing in a nite market model (February 14, 2012) 1 Introduction The main purpose of this chapter is to prove the rst and second fundamental theorem of asset pricing in a so called nite market model. 2.1 A nite market Throughout this chapter (; F; P ) is a probability space with a non-empty nite set. The time is discrete and consists of the points 0; 1; :::; T, where T is a positive integer. Moreover, there is given a ltration F 0 = f;, g F 1 F 2 ::: F T = F and n + 1 traded asset price processes S 1 = (S 1 (t)) T t=0; :::; S n+1 = (S n+1 (t)) T t=0: The corresponding vector price process is denoted by S; that is S = (S 1 (t); :::; S n+1 (t)) T t=0 and it is assumed that the random vector S(t) = (S 1 (t); :::; S n+1 (t)) is F t - measurable at each point of time t: It is appropriate to think of F t as the known information at time t: Throughout, it will be assumed that S n+1 (t) > 0; t = 0; :::; T: In particular, S n+1 can be used as a numéraire. Since is nite the -algebra F is generated by a partition of, and without loss of generality assume each member of this partition is of positive probability. Hence, we may assume = f! 1 ; :::;! d g ;
2 2 and F = 2 P [f! k g] > 0; k = 1; :::; d: In particular, if X is an F-measurable random variable, X = 0 a.s. if and only if X(!) = 0 for every! 2 : Finally, we assume there are no transaction costs or other frictions on the market. The above assumoptions de ne our nite market model. 2.2 Self- nancing strategies An R m -valued stochastic process (X(t)) T t=1 is predictable if X(t) is F t 1 - measurable for t = 1; :::; T: A (portfolio) strategy h = (h(t)) T t=0 is an R n+1 -valued stochastic process with (h(t)) T t=1 predictable and h(0) = h(1): The corresponding value process is given by Xn+1 V h (t) = h(t) S(t) = h i (t)s i (t); t = 0; :::; T: i=1 The strategy h is said to be self- nancing when V h (t) = h(t + 1) S(t); t = 1; :::; T 1: Since h(0) = h(1) this relation also holds for t = 0. If (a(t)) T t=0 is a sequence in a vector space, let 4a(t) = a(t) a(t 1); t = 1; :::; T: Theorem If h is a strategy h is self- nancing if and only if 4V h (t) = h(t) 4S(t); t = 1; :::; T:
3 3 PROOF. The defnitions yield 4V h (t) = h(t) S(t) h(t 1) S(t 1) and and, hence if and only if h(t) 4S(t) = h(t) S(t) h(t) S(t 1) 4V h (t) = h(t) 4S(t); t = 1; :::; T which proves the theorem. h(t 1) S(t 1) = h(t) S(t 1); t = 1; :::; T The so called gain process G h = (G h (t)) T t=0 corresponding to a strategy h is given by G h (0) = 0 and G h (t) = h(1) 4S(1) + ::: + h(t) 4S(t); t = 1; :::; T: Theorem A strategy h is self- nancing if and only if V h (t) = V h (0) + G h (t); t = 0; 1; :::; T: PROOF. Suppose t 2 f1; :::; T g and note that V h (t) V h (0) = tx 4V h (u): u=1 Now by Theorem 2.2.1, if h is self- nancing V h (t) V h (0) = tx h(u) 4S(u) = G h (t): u=1
4 4 Conversely, if the last identity holds 4V h (t) = 4(V h (t) V h (0)) = h(t) 4S(t) and Theorem shows that h is self- nancing. This completes the proof of the theorem. 2.3 The fundamental theorems of asset pricing An arbitrage (opportunity or strategy) is a self- nancing strategy h such that or, equivalently, V h (0) = 0; V h (T ) 0; and E [V h (T )] > 0 V h (0) = 0; V h (T ) 0; and max 1kd V h(t;! k ) > 0; Next we want to give conditions on the nite market model model so that it is free of arbitrage. To this end choose S n+1 as a numéraire and introduce the price processes S(t) = ( S 1 (t); :::; S n+1 (t)) = S(t) S n+1 (t) : Clearly, each S(t) is F t -measurable. The corresponding nite market model is called the normalized market. If h is a strategy, the value process and gain process in the normalized market are denoted by V h and G h, respectively. The equation is equivalent to the equation h(t) S(t) = h(t + 1) S(t) h(t) S(t) = h(t + 1) S(t)
5 and we conclude that h is self- nancing in the original market if and only if h is self- nancing in the normalized market. Furthermore, note that From S n+1 (t) = 0; t = 1; :::; T: V h (t) = V h(t) S n+1 (t) follows that our original nite market model is free of arbitrage if and only if the normalized market is free of arbitrage. A probability measure P ~ de ned on the -algebra F is said to be an equivalent martingale measure if the following two conditions hold: (i) P ~ and P are equivalent, that is P ~ and P have the same null sets, (ii) under P ~ ; ( S(t); F t ) T t=0 is martingale, that is ( S i (t); F t ) T t=0 is a martingale for every i = 1; :::; n + 1: If Q is a probability measure on F and X a random variable, we write Z XdQ = E Q [X]. Moreover, if ~ P is an equivalent martingale measure E ~P [X] is written ~ E [X] : 5 Theorem (First fundamental theorem of asset pricing) The market model is free of arbitrage if and only if there exists an equivalent martingale measure. To prove the rst fundamental theorem of asset pricing we need the following lemma with a = 0 (the general case is needed later to prove the second fundamental theorem of asset pricing). Lemma Suppose a is a real number and g = (g(t)) T t=1 a predictable process in R n : Then there exists a unique self- nancing strategy h such that V h (0) = a and (h 1 (t); :::; h n (t)) = g(t); t = 1; :::; T:
6 6 PROOF. De ne (h 1 (t); :::; h n (t)) = g(t); t = 1; :::; T; h i (0) = h i (1); i = 1; :::; n; and, moreover, h n+1 (0) such that nx h i (0) S i (0) + h n+1 (0) = a: i=1 Observe that S n+1 (t) = 1 for every t and (h 1 (0); :::; h n+1 (0)) is a constant vector. Put h n+1 (1) = h n+1 (0) so that h(0) = h(1): Next we use induction to de ne h n+1 (t + 1); t = 1; :::; T 1; such that nx h i (t) S nx i (t) + h n+1 (t) = h i (t + 1) S i (t) + h n+1 (t + 1); t = 1; :::; T 1: i=1 i=1 It follows that h = (h(t)) T t=1 is predictable and self- nancing. The uniqueness is obvious from the construction of h n+1 ; which completes the proof of the theorem. PROOF THEOREM (: Suppose P ~ is an equivalent martingale measure and let h be a self- nancing strategy satisfying V h (0) = 0 and V h (T ) 0: We must show that V h (T ) = 0 or, stated otherwise V h (T ) = V h (T )=S n+1 (T ) = 0. Now given t 2 f1; ::; T g ; ~E h(t) 4S(t) = E ~ h ~E h(t) 4S(t) i j Ft 1 = E ~ h h(t) E ~ 4S(t) i j F t 1 = E ~ [h(t) 0] = 0: Since it follows that V h (T ) = V h (0) + G h (T ) = 0 + ~E Vh (T ) = 0: TX h(t) 4S(t) But V h (T ) 0 and we conclude that V h (T ) = 0 as ~ P [f! k g] > 0; k = 1; :::; d: t=1 ): Let S n (t) = ( S 1 (t); :::; S n (t)); t = 0; :::; T: If g = (g(t)) T t=1 is an R n -valued predictable process we de ne Hg = g(1) 4 S n (1) + ::: + g(t ) 4 S n (T )
7 and use Lemma to get a self- nancing strategy such that V h (0) = 0 and = 0 + V h (T ) = V h (0) + G h (T ) TX h(t) 4S(t) = Hg: t=1 Here recall that 4 S n+1 (t) = 0; t = 1; :::; T: Since the market model is free of arbitrage Hg =2 C, where C = Y 2 L 2 (; F; P ); Y 0 och Y (! k ) > 0 for some k = 1; :::; d : Next introduce the vector space L = Hg; g = (g(t)) T t=1 predictable process in R n and the compact convex set Then K = fy 2 C; E [Y ] = 1g L 2 (; F; P ): K \ L = ;: and from the Theorem we obtain Z 2 L 2 (; F; P ) such that and Here E [(Hg)Z] = 0 if g = (g(t)) T t=1 predictable process in R n E [Y Z] > 0 if Y 2 K: 1f!k g Z(! k ) = E P [f! k g] Z > 0; k = 1; :::; d and we de ne a probability measure ~ P on F by setting 7 d ~ P = Clearly, P and ~ P are equivalent and Z E [Z] dp: ~E [Hg] = 0 if g = (g(t)) T t=1 predictable process in R n :
8 8 Finally, to prove that( S(t); F t ) T t=0 is a ~ P -martingale, choose u 2 f1; :::; T g and pick a predictable process g = (g(t)) T t=1 in R n such that g(t) = 0 if t 6= u: Then 0 = ~ E [Hg] = ~ E g(u) 4 S n (u) and it follows that ( S n (t); F t ) T t=0 is a ~ P -martingal. Clearly, then the process ( S(t); F t ) T t=0 is a ~ P -martingal. This completes the proof of the rst fundamental theorem of asset pricing. A contingent claim is an F T -measurable random variable Y, representing a contract that pays out the amount Y (!) at time T if! occurs: In the following Y and the contract are identi ed. We will say the contract is replicable if there exists a self- nancing strategy h such that V h (T ) = Y: The model is called complete if every contingent claim is replicable. In a nite market the class of all contingent claims is equal to L 2 (; F T ; Q) for every probability measure Q on F T : Theorem Suppose the market model is free of arbitrage. (a) If P ~ is an equivalent martingale measure and h a self- nancing strategy, then under P ~ T ( V h (t)) T Vh (t) t=0 = S n+1 (t) t=0 is a martingale. (b) If h is a self- nancing strategy which replicates the contengent claim Y; then V h (t) = S n+1 (t) E ~ Y S n+1 (T ) j F t ; t = 0; 1; ::; T for every equivalent martingale measure ~ P : PROOF. (a) By Theorem applied to the normalized market, we have V h (t) = V h (0) + G h (t)
9 9 and it follows that V h (T ) = V h (t) + TX u=t+1 h(u) 4 S(u): Now suppose ~ P is an equivalent martingale measure and let t+1 u T: Since S = ( S(t); F t ) T t=0 is a ~ P -martingale ~E h(u) 4 S(u) j F t = ~ E h ~E h(u) 4 S(u) j Fu 1 j Ft i = E ~ hh(u) E ~ 4S(u) i j F u 1 j Ft = E ~ [h(u) 0 j F t ] = 0 and thus ~E Vh (T ) j F t = ~ E Vh (t) j F t = Vh (t) that is ~E Vh (T ) S n+1 (T ) j F t = V h(t) S n+1 (t) : This proves Part (a). Part (b) is an immediate consequence of Part (a), which completes the proof of Theorem Theorem (Second fundamental theorem of asset pricing) Suppose the market model is free of arbitrage. Then the model is complete if and only if there is a unique martingale measure. PROOF. ): Suppose X is an F T -measurable random variable and P ~ 0 and ~P 1 are two equivalent martingale measures. Set Y = S n+1 (T )X and let h be a self- nancing strategy which replicates Y: By applying Theorem 2.3.2, we get P V h (0) = S n+1 (0)E ~ i Y ; i = 0; 1 S n+1 (T ) and, hence P E ~ 0 P [X] = E ~ 0 Y P = E ~ 1 Y P = E ~ 1 [X] : S n+1 (T ) S n+1 (T )
10 10 Thus ~ P 0 = ~ P 1. (: Let ~ P denote the equivalent martingale measure: As above for each R n - valued predictable process g = (g(t)) T t=1 set and introduce the vector space Hg = g(1) 4 S n (1) + ::: + g(t ) 4 S n (T ) L = a + Hg; a 2 R and g = (g(t)) T t=0 a predictable process in R n : We now claim that L = L 2 (; F; ~ P ): In fact, if this is wrong by Theorem there is a Z 2 L 2 (; F; ~ P ) such that Z 6= 0 and ~E [Y Z] = 0 if Y 2 L: The choice Y = 1 yields E ~ [Z] = 0: Moreover, choosing a positive real number M > Z we get a probability measure P ~ 0 on F by setting d ~ P 0 = (1 + Z M )d ~ P : Clearly, P ~ 0 and P ~ are equivalent and if Y 2 L; P E ~ 0 [Y ] = E Y ~ + 1M Y Z In particular, = ~ E [Y ] + 1 M ~ E [Y Z] = ~ E [Y ] : E ~ P 0 [Hg] = 0; if g = (g(t)) T t=1 is a predictable process in R n. From this, as in the proof of the rst fundamental theorem of asset pricing, ( S(t); F t ) T t=0 is a ~ P 0 -martingale and we have got an equivalent martingale measure ~ P 0 di erent from ~ P. From this contradiction we conclude that L = L 2 (; F; ~ P ): Next suppose Y 2 L 2 (; F; ~ P ). To show that Y is replicable we introduce X = Y S n+1 (T ) :
11 Since L = L 2 (; F; ~ P ) there is a real number a and a predictable process g = (g(t)) T t=1 in R n such that X = a + Hg: Now by Lemma there exists a self- nancing strategy h satisfying V h (0) = a and (h 1 (t); :::; h n (t)) = g(t); t = 1; :::; T: But then X = V h (0) + G h (T ) = V h (T ) = h(t ) S(T ) and it follows that Y = h(t ) S(T ) = V h (T ): 11 Exercises 1. Suppose the nite market is free of arbitrage. Show that the restricted market with with time set f0; :::; T 1g is free of arbitrage. 2. Suppose the nite market is free of arbitrage and complete. Show that the restricted market with time set f0; :::; T 1g is complete. 2.4 The price of a contingent claim To de ne a price of a contingent claim Y in our nite market model we will require that there are no arbitrages or, stated otherwise, there is at least one equivalent martingale measure. If, in addition, Y is replicable by Theorem the expression S n+1 (t) E ~ Y S n+1 (T ) j F t is independent of the underlying equivalent martingale measure ~ P and equals V h (t) for every strategy h which replicates Y: This leads us to the following De nition Suppose the market is free of arbitrage. A replicable contingent claim Y has the price Y (t) = S n+1 (t) E ~ Y S n+1 (T ) j F t
12 12 at time t; where the expectation is with respect to an arbitrary equivalent martingale measure. The next example shows that the quantity S n+1 (t) E ~ Y S n+1 (T ) j F t may depend on the martingale measure ~ P even if the market is free from arbitrage. Example Consider a nite market model with one stock and one bond in a single period from t = 0 to t = 1: The stock (bond) price at time t equals S(t) (B(t)): Furthermore, suppose x 1 ; x 2 ; x 3 2R; x 1 < x 2 < x 3 ; and S(1) = S(0)e X where S(0) is a positive constant and X:! fx 1 ; x 2 ; x 3 g a random variable such that p i = P [X = x i ] > 0; i = 1; 2; 3: The bond price is deterministic and B(1) = B(0)e r where B(0) and r are positive constants. Finally, suppose x 1 < r < x 3 : Next assume the strategy h is an arbitrage and h(1) = (h S ; h B ); that is and h S S(0) + h B B(0) = 0 h S S(0)e x i + h B B(0)e r 0; i = 1; 2; 3 where strict inequality occurs for some i. Setting a = h S S(0), a(e x i e r ) 0; i = 1; 2; 3 where strict inequality occurs for some i. Hence a 6= 0 and we have got a contradiction. Thus by the rst fundamental theorem of asset pricing there is at least one equivalent martingale measure. In the following, for simplicity, choose X as the identity map in = fx 1 ; x 2 ; x 3 g so that P is a linear combination of Dirac measures P = p 1 x1 + p 2 x2 + p 3 x3 :
13 13 If P and ~ P are equivalent probability measures where ~p i > 0; i = 1; 2; 3, Moreover, if and only if Here ~P = ~p 1 x1 + ~p 2 x2 + ~p 3 x3 ~p 1 + ~p 2 + ~p 3 = 1: S(0) = e r ~ E [S(1)] e x 1 r ~p 1 + e x 2 r ~p 2 + e x 3 r ~p 3 = 1: e x 1 r < 1 < e x 3 r and it is evident that there exists in nitely many martingale measures. In the following discussion assume r < x 2. Drawing a gure it is simple to see that the map ~ P! ~p 3 is injective. De ning Y such that Y (x 1 ) = Y (x 0 ) = 0; and Y (x 3 ) = 1 it follows that the quantity e r E ~P [Y ] = e r ~p 3 is di erent for di erent equivalent martingale measures. Exercises 1. Suppose the market is free of arbitrage and let Y be a replicable contingent claim. Add the price process ( Y (t)) T t=0 to the other price processes and show that the augmented market model does not admit arbitrage. 2.5 American options An American derivative may be exercised at each time point. If the exercise takes place at time t the holder of the contract gets the amount X(t) and
14 14 after that point of time the contract expires. Here X(t) is an F t -measurable random variable for each t = 0; 1; :::; T. Next we want to de ne the price V (t) of the derivative at an arbitrary time t given that the contract is alive at this time. Clearly, V (T ) = X(T ): To de ne the price at a time point before maturity T from now on in this section we assume that the market is complete with the equivalent martingale measure P ~. Under this assumption de ne V (t) = max(x(t); S n+1 (t) E ~ V (t + 1) S n+1 (t + 1) j F t ); t = T 1; T 2; :::; 1; 0: It is optimal to exercise the derivative at time t T X(t) > S n+1 (t) E ~ V (t + 1) S n+1 (t + 1) j F t since it costs the amount S n+1 (t) ~ E V (t + 1) S n+1 (t + 1) j F t 1; if to buy a portfolio at time t which replicates V (t + 1) at time t + 1: Note that V (t) S n+1 (t) E ~ V (t + 1) S n+1 (t + 1) j F t and we conclude that the sequence T V (t) S n+1 (t) ; F t t=0 is a ~ P -supermartingale. If the same sequence is a ~ P -martingale there is no extra gain to exercise the American derivative before maturity. Furthermore note that if (U(t); F t ) T t=0 is a ~ P -supermartingale such that then U(t) X(t) ; t = 0; :::; T S n+1 (t) U(t) max( X(t) S n+1 (t) ; ~ E [U(t + 1) j F t ]); t = 0; :::; T 1:
15 15 Hence the price process V (t) T S n+1 (t) t=0 is equal to the smallest P ~ -supermartingale, which dominates the process T X(t) S n+1 (t) : t=0 The aim in this section is to describe the price of the derivative in terms of so called stopping times. Let t 0 2 f0; 1; :::; T g : A map :! ft 0 ; :::; T g is called a stopping time with respect to the ltration (F t ) T t=0 if the event [ t] 2 F t for every t 2 ft 0 ; :::; T g : Note that a constant map from into ft 0 ; ; :::; T gis a stopping time. Moreover, if 0 ; 1 :! ft 0 ; :::; T g are stopping times and 0 _ 1 = max( 0 ; 1 ) 0 ^ 1 = min( 0 ; 1 ) are stopping times. The following result is a special case of a martingale theorem by J. L. Snell (see e.g. J. Neveu, Discrete-Parameter Martingales, North-Holland Publishing Company, 1972). Theorem If :! ft 0 ; :::; T g is a stopping time with respect to the ltration (F t ) T t=0; then V (t 0 ) S n+1 (t 0 ) E ~ X() S n+1 () j F t 0 where equality occurs if = min t 2 ft 0 ; :::; T 1g ; X(t) > S n+1 (t) ~ E V (t + 1) S n+1 (t + 1) j F t with the convention that the minimum over the empty set equals T:
16 16 PROOF. To simplify notation let S n+1 (t) = 1 for every t: We rst prove that V (t 0 ) ~ E [X() j F t0 ] where :! ft 0 ; :::; T g is a stopping time with respect to the ltration (F t ) N t=0: The case t 0 = T yields = T and as V (T ) = X(T ) this case is obvious. Next assume t 0 < T and V (t 0 + 1) ~ E [X() j F t0 +1] for every stopping time :! ft 0 + 1; :::; T g with respect to the ltration (F t ) N t=0: Now if :! ft 0 ; :::; T g is a stopping time with respect to the ltration (F t ) N t=0we have ~E [X() j F t0 ] = ~ E X()1 [=t0 ] j F t0 + ~ E X()1[>t0 ] j F t0 = ~ E X(t 0 )1 [=t0 ] j F t0 + ~ E X_(t0 +1)1 [>t0 ] j F t0 = X(t 0 )1 [=t0 ] + 1 ~ [>t0 ] E X _(t0 +1) j F t0 = X(t 0 )1 [=t0 ] + 1 ~ h i [>t0 ] E ~E X_(t0 +1) j F t0 +1 j Ft0 Now by induction we get V (t 0 )1 [=t0 ] + 1 [>t0 ] ~ E [V (t 0 + 1) j F t0 ] V (t 0 )1 [=t0 ] + 1 [>t0 ]V (t 0 ) = V (t 0 ): V (t 0 ) E [X() j F t0 ] if :! ft 0 ; :::; T g is a stopping time and t 0 2 f0; :::; T g : Finally, if n = min t 2 ft 0 ; :::; T 1g ; X(t) > E ~ o [V (t + 1) j F t ] it is readily seen that there is equality in each step in the induction proof above. This concludes the proof of the theorem. Exercises
17 1. Let (V (t)) T t=0 denote the price process of an American styled derivative. Show that V (t) ~E E S n+1 (t) ~ V (t + 1) ; t = 0; :::; T 1: S n+1 (t) 17
Discrete-Time Market Models
Discrete-Time Market Models 80-646-08 Stochastic Calculus I Geneviève Gauthier HEC Montréal I We will study in depth Section 2: The Finite Theory in the article Martingales, Stochastic Integrals and Continuous
More informationSnell envelope and the pricing of American-style contingent claims
Snell envelope and the pricing of -style contingent claims 80-646-08 Stochastic calculus I Geneviève Gauthier HEC Montréal Notation The The riskless security Market model The following text draws heavily
More informationThe Asymptotic Theory of Transaction Costs
The Asymptotic Theory of Transaction Costs Lecture Notes by Walter Schachermayer Nachdiplom-Vorlesung, ETH Zürich, WS 15/16 1 Models on Finite Probability Spaces In this section we consider a stock price
More information1 The Well Ordering Principle, Induction, and Equivalence Relations
1 The Well Ordering Principle, Induction, and Equivalence Relations The set of natural numbers is the set N = f1; 2; 3; : : :g. (Some authors also include the number 0 in the natural numbers, but number
More informationStochastic Processes Calcul stochastique. Geneviève Gauthier. HEC Montréal. Stochastic processes. Stochastic processes.
Processes 80-646-08 Calcul stochastique Geneviève Gauthier HEC Montréal Let (Ω, F) be a measurable space. A stochastic process X = fx t : t 2 T g is a family of random variables, all built on the same
More informationOptimal Stopping Problems and American Options
Optimal Stopping Problems and American Options Nadia Uys A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master
More informationDynamic Risk Measures and Nonlinear Expectations with Markov Chain noise
Dynamic Risk Measures and Nonlinear Expectations with Markov Chain noise Robert J. Elliott 1 Samuel N. Cohen 2 1 Department of Commerce, University of South Australia 2 Mathematical Insitute, University
More informationValuation and Pricing of Electricity Delivery Contracts The Producer s View
SWM ORCOS Valuation and Pricing of Electricity Delivery Contracts The Producer s View Raimund M. Kovacevic Research Report 2015-04 February, 2015 Operations Research and Control Systems Institute of Statistics
More informationTrajectorial Martingales, Null Sets, Convergence and Integration
Trajectorial Martingales, Null Sets, Convergence and Integration Sebastian Ferrando, Department of Mathematics, Ryerson University, Toronto, Canada Alfredo Gonzalez and Sebastian Ferrando Trajectorial
More informationElimination of Arbitrage States in Asymmetric Information Models
Elimination of Arbitrage States in Asymmetric Information Models Bernard CORNET, Lionel DE BOISDEFFRE Abstract In a financial economy with asymmetric information and incomplete markets, we study how agents,
More informationSample of Ph.D. Advisory Exam For MathFinance
Sample of Ph.D. Advisory Exam For MathFinance Students who wish to enter the Ph.D. program of Mathematics of Finance are required to take the advisory exam. This exam consists of three major parts. The
More informationStochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier.
Ito 8-646-8 Calculus I Geneviève Gauthier HEC Montréal Riemann Ito The Ito The theories of stochastic and stochastic di erential equations have initially been developed by Kiyosi Ito around 194 (one of
More informationLecture 1- The constrained optimization problem
Lecture 1- The constrained optimization problem The role of optimization in economic theory is important because we assume that individuals are rational. Why constrained optimization? the problem of scarcity.
More informationCLASSICAL PROBABILITY MODES OF CONVERGENCE AND INEQUALITIES
CLASSICAL PROBABILITY 2008 2. MODES OF CONVERGENCE AND INEQUALITIES JOHN MORIARTY In many interesting and important situations, the object of interest is influenced by many random factors. If we can construct
More information1 Uncertainty and Insurance
Uncertainty and Insurance Reading: Some fundamental basics are in Varians intermediate micro textbook (Chapter 2). A good (advanced, but still rather accessible) treatment is in Kreps A Course in Microeconomic
More information2 Lecture Span, Basis and Dimensions
2 Lecture 2 2.1 Span, Basis and Dimensions Related to the concept of a linear combination is that of the span. The span of a collection of objects is the set of all linear combinations of those objects
More informationGeneralized Hypothesis Testing and Maximizing the Success Probability in Financial Markets
Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung 1, Qingshuo Song 2, and Jie Yang 3 1 Columbia University, New York, USA; leung@ieor.columbia.edu 2 City
More informationMarch 16, Abstract. We study the problem of portfolio optimization under the \drawdown constraint" that the
ON PORTFOLIO OPTIMIZATION UNDER \DRAWDOWN" CONSTRAINTS JAKSA CVITANIC IOANNIS KARATZAS y March 6, 994 Abstract We study the problem of portfolio optimization under the \drawdown constraint" that the wealth
More informationCompetitive Equilibrium and the Welfare Theorems
Competitive Equilibrium and the Welfare Theorems Craig Burnside Duke University September 2010 Craig Burnside (Duke University) Competitive Equilibrium September 2010 1 / 32 Competitive Equilibrium and
More information1 Selected Homework Solutions
Selected Homework Solutions Mathematics 4600 A. Bathi Kasturiarachi September 2006. Selected Solutions to HW # HW #: (.) 5, 7, 8, 0; (.2):, 2 ; (.4): ; (.5): 3 (.): #0 For each of the following subsets
More informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 3. Calculaus in Deterministic and Stochastic Environments Steve Yang Stevens Institute of Technology 01/31/2012 Outline 1 Modeling Random Behavior
More information2.2 Some Consequences of the Completeness Axiom
60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that
More informationNonlinear Programming (NLP)
Natalia Lazzati Mathematics for Economics (Part I) Note 6: Nonlinear Programming - Unconstrained Optimization Note 6 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 3 and 5) and Simon and Blume
More informationRisk-Minimality and Orthogonality of Martingales
Risk-Minimality and Orthogonality of Martingales Martin Schweizer Universität Bonn Institut für Angewandte Mathematik Wegelerstraße 6 D 53 Bonn 1 (Stochastics and Stochastics Reports 3 (199, 123 131 2
More informationMacroeconomics IV Problem Set I
14.454 - Macroeconomics IV Problem Set I 04/02/2011 Due: Monday 4/11/2011 1 Question 1 - Kocherlakota (2000) Take an economy with a representative, in nitely-lived consumer. The consumer owns a technology
More informationStochastic processes and stopping time Exercises
Stochastic processes and stopping time Exercises Exercise 2.1. Today is Monday and you have one dollar in your piggy bank. Starting tomorrow, every morning until Friday (inclusively), you toss a coin.
More informationTime is discrete and indexed by t =0; 1;:::;T,whereT<1. An individual is interested in maximizing an objective function given by. tu(x t ;a t ); (0.
Chapter 0 Discrete Time Dynamic Programming 0.1 The Finite Horizon Case Time is discrete and indexed by t =0; 1;:::;T,whereT
More informationFundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales
Fundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales Prakash Balachandran Department of Mathematics Duke University April 2, 2008 1 Review of Discrete-Time
More informationLecture 21 Representations of Martingales
Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let
More informationConditional Expectation and Independence Exercises
Conditional Expectation and Independence Exercises Exercise 3.1. Show the four following properties : for any random variables and Y and for any real numbers a and b, (E1) E P [a + by ] = ae P [] + be
More informationRandom Times and Their Properties
Chapter 6 Random Times and Their Properties Section 6.1 recalls the definition of a filtration (a growing collection of σ-fields) and of stopping times (basically, measurable random times). Section 6.2
More informationEconomics Bulletin, 2012, Vol. 32 No. 1 pp Introduction. 2. The preliminaries
1. Introduction In this paper we reconsider the problem of axiomatizing scoring rules. Early results on this problem are due to Smith (1973) and Young (1975). They characterized social welfare and social
More informationSOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker CHAPTER 2. UNIVARIATE DISTRIBUTIONS
SOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker CHAPTER. UNIVARIATE DISTRIBUTIONS. Random Variables and Distribution Functions. This chapter deals with the notion of random variable, the distribution
More informationThe Inclusion Exclusion Principle and Its More General Version
The Inclusion Exclusion Principle and Its More General Version Stewart Weiss June 28, 2009 1 Introduction The Inclusion-Exclusion Principle is typically seen in the context of combinatorics or probability
More informationFlow Network. The following figure shows an example of a flow network:
Maximum Flow 1 Flow Network The following figure shows an example of a flow network: 16 V 1 12 V 3 20 s 10 4 9 7 t 13 4 V 2 V 4 14 A flow network G = (V,E) is a directed graph. Each edge (u, v) E has a
More informationWe suppose that for each "small market" there exists a probability measure Q n on F n that is equivalent to the original measure P n, suchthats n is a
Asymptotic Arbitrage in Non-Complete Large Financial Markets Irene Klein Walter Schachermayer Institut fur Statistik, Universitat Wien Abstract. Kabanov and Kramkov introduced the notion of "large nancial
More informationExercise Exercise Homework #6 Solutions Thursday 6 April 2006
Unless otherwise stated, for the remainder of the solutions, define F m = σy 0,..., Y m We will show EY m = EY 0 using induction. m = 0 is obviously true. For base case m = : EY = EEY Y 0 = EY 0. Now assume
More informationADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS
ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements
More informationA Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1
Chapter 3 A Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1 Abstract We establish a change of variable
More informationEconomics 241B Review of Limit Theorems for Sequences of Random Variables
Economics 241B Review of Limit Theorems for Sequences of Random Variables Convergence in Distribution The previous de nitions of convergence focus on the outcome sequences of a random variable. Convergence
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationTopics in Mathematical Economics. Atsushi Kajii Kyoto University
Topics in Mathematical Economics Atsushi Kajii Kyoto University 25 November 2018 2 Contents 1 Preliminary Mathematics 5 1.1 Topology.................................. 5 1.2 Linear Algebra..............................
More informationElements of Stochastic Analysis with application to Finance (52579) Pavel Chigansky
Elements of Stochastic Analysis with application to Finance 52579 Pavel Chigansky Department of Statistics, The Hebrew University, Mount Scopus, Jerusalem 9195, Israel E-mail address: pchiga@mscc.huji.ac.il
More informationTopics in Mathematical Economics. Atsushi Kajii Kyoto University
Topics in Mathematical Economics Atsushi Kajii Kyoto University 26 June 2018 2 Contents 1 Preliminary Mathematics 5 1.1 Topology.................................. 5 1.2 Linear Algebra..............................
More informationStochastic Calculus for Finance II - some Solutions to Chapter VII
Stochastic Calculus for Finance II - some Solutions to Chapter VII Matthias hul Last Update: June 9, 25 Exercise 7 Black-Scholes-Merton Equation for the up-and-out Call) i) We have ii) We first compute
More informationMarkov processes Course note 2. Martingale problems, recurrence properties of discrete time chains.
Institute for Applied Mathematics WS17/18 Massimiliano Gubinelli Markov processes Course note 2. Martingale problems, recurrence properties of discrete time chains. [version 1, 2017.11.1] We introduce
More information3 hours UNIVERSITY OF MANCHESTER. 22nd May and. Electronic calculators may be used, provided that they cannot store text.
3 hours MATH40512 UNIVERSITY OF MANCHESTER DYNAMICAL SYSTEMS AND ERGODIC THEORY 22nd May 2007 9.45 12.45 Answer ALL four questions in SECTION A (40 marks in total) and THREE of the four questions in SECTION
More informationNo-Arbitrage Criteria for Financial Markets with Transaction Costs and Incomplete Information
Noname manuscript No. (will be inserted by the editor) No-Arbitrage Criteria for Financial Markets with Transaction Costs and Incomplete Information Dimitri De Vallière 1, Yuri Kabanov 1,2, Christophe
More informationOn the dual problem of utility maximization
On the dual problem of utility maximization Yiqing LIN Joint work with L. GU and J. YANG University of Vienna Sept. 2nd 2015 Workshop Advanced methods in financial mathematics Angers 1 Introduction Basic
More informationNear convexity, metric convexity, and convexity
Near convexity, metric convexity, and convexity Fred Richman Florida Atlantic University Boca Raton, FL 33431 28 February 2005 Abstract It is shown that a subset of a uniformly convex normed space is nearly
More informationSummer Jump-Start Program for Analysis, 2012 Song-Ying Li
Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture 6: Uniformly continuity and sequence of functions 1.1 Uniform Continuity Definition 1.1 Let (X, d 1 ) and (Y, d ) are metric spaces and
More informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
More informationLecture Notes on Game Theory
Lecture Notes on Game Theory Levent Koçkesen Strategic Form Games In this part we will analyze games in which the players choose their actions simultaneously (or without the knowledge of other players
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationReal Analysis: Homework # 12 Fall Professor: Sinan Gunturk Fall Term 2008
Eduardo Corona eal Analysis: Homework # 2 Fall 2008 Professor: Sinan Gunturk Fall Term 2008 #3 (p.298) Let X be the set of rational numbers and A the algebra of nite unions of intervals of the form (a;
More informationAdvanced Microeconomics Fall Lecture Note 1 Choice-Based Approach: Price e ects, Wealth e ects and the WARP
Prof. Olivier Bochet Room A.34 Phone 3 63 476 E-mail olivier.bochet@vwi.unibe.ch Webpage http//sta.vwi.unibe.ch/bochet Advanced Microeconomics Fall 2 Lecture Note Choice-Based Approach Price e ects, Wealth
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationMA 8101 Stokastiske metoder i systemteori
MA 811 Stokastiske metoder i systemteori AUTUMN TRM 3 Suggested solution with some extra comments The exam had a list of useful formulae attached. This list has been added here as well. 1 Problem In this
More informationChapter 2: Preliminaries and elements of convex analysis
Chapter 2: Preliminaries and elements of convex analysis Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Website: http://home.deib.polimi.it/amaldi/opt-14-15.shtml Academic year 2014-15
More informationBasic Definitions: Indexed Collections and Random Functions
Chapter 1 Basic Definitions: Indexed Collections and Random Functions Section 1.1 introduces stochastic processes as indexed collections of random variables. Section 1.2 builds the necessary machinery
More informationSTRICT LOCAL MARTINGALE DEFLATORS AND PRICING AMERICAN CALL-TYPE OPTIONS. 0. Introduction
STRICT LOCAL MARTINGALE DEFLATORS AND PRICING AMERICAN CALL-TYPE OPTIONS ERHAN BAYRAKTAR, CONSTANTINOS KARDARAS, AND HAO XING Abstract. We solve the problem of pricing and optimal exercise of American
More informationAddendum to: International Trade, Technology, and the Skill Premium
Addendum to: International Trade, Technology, and the Skill remium Ariel Burstein UCLA and NBER Jonathan Vogel Columbia and NBER April 22 Abstract In this Addendum we set up a perfectly competitive version
More informationSTOCHASTIC DIFFERENTIAL EQUATIONS WITH EXTRA PROPERTIES H. JEROME KEISLER. Department of Mathematics. University of Wisconsin.
STOCHASTIC DIFFERENTIAL EQUATIONS WITH EXTRA PROPERTIES H. JEROME KEISLER Department of Mathematics University of Wisconsin Madison WI 5376 keisler@math.wisc.edu 1. Introduction The Loeb measure construction
More informationThe Kuhn-Tucker Problem
Natalia Lazzati Mathematics for Economics (Part I) Note 8: Nonlinear Programming - The Kuhn-Tucker Problem Note 8 is based on de la Fuente (2000, Ch. 7) and Simon and Blume (1994, Ch. 18 and 19). The Kuhn-Tucker
More informationA Simple Computational Approach to the Fundamental Theorem of Asset Pricing
Applied Mathematical Sciences, Vol. 6, 2012, no. 72, 3555-3562 A Simple Computational Approach to the Fundamental Theorem of Asset Pricing Cherng-tiao Perng Department of Mathematics Norfolk State University
More informationSolving Extensive Form Games
Chapter 8 Solving Extensive Form Games 8.1 The Extensive Form of a Game The extensive form of a game contains the following information: (1) the set of players (2) the order of moves (that is, who moves
More informationChapter 5 Linear Programming (LP)
Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize f(x) subject to x R n is called the constraint set or feasible set. any point x is called a feasible point We consider
More informationStochastic Processes
Introduction and Techniques Lecture 4 in Financial Mathematics UiO-STK4510 Autumn 2015 Teacher: S. Ortiz-Latorre Stochastic Processes 1 Stochastic Processes De nition 1 Let (E; E) be a measurable space
More informationIntroduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION
Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from
More informationSolution to Homework 8, Math 2568
Solution to Homework 8, Math 568 S 5.4: No. 0. Use property of heorem 5 to test for linear independence in P 3 for the following set of cubic polynomials S = { x 3 x, x x, x, x 3 }. Solution: If we use
More informationArbitrage and Duality in Nondominated Discrete-Time Models
Arbitrage and Duality in Nondominated Discrete-Time Models Bruno Bouchard Marcel Nutz February 9, 2014 Abstract We consider a nondominated model of a discrete-time nancial market where stocks are traded
More informationNested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model
Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model Xiaowei Chen International Business School, Nankai University, Tianjin 371, China School of Finance, Nankai
More informationLecture 6: Contraction mapping, inverse and implicit function theorems
Lecture 6: Contraction mapping, inverse and implicit function theorems 1 The contraction mapping theorem De nition 11 Let X be a metric space, with metric d If f : X! X and if there is a number 2 (0; 1)
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More informationWidely applicable periodicity results for higher order di erence equations
Widely applicable periodicity results for higher order di erence equations István Gy½ori, László Horváth Department of Mathematics University of Pannonia 800 Veszprém, Egyetem u. 10., Hungary E-mail: gyori@almos.uni-pannon.hu
More informationCompleteness of bond market driven by Lévy process
Completeness of bond market driven by Lévy process arxiv:812.1796v1 [math.p] 9 Dec 28 Michał Baran Mathematics Department of Cardinal Stefan Wyszyński University in Warsaw, Poland Jerzy Zabczyk Institute
More informationIOANNIS KARATZAS Mathematics and Statistics Departments Columbia University
STOCHASTIC PORTFOLIO THEORY IOANNIS KARATZAS Mathematics and Statistics Departments Columbia University ik@math.columbia.edu Joint work with Dr. E. Robert FERNHOLZ, C.I.O. of INTECH Enhanced Investment
More informationThe integers. Chapter 3
Chapter 3 The integers Recall that an abelian group is a set A with a special element 0, and operation + such that x +0=x x + y = y + x x +y + z) =x + y)+z every element x has an inverse x + y =0 We also
More informationSets with two associative operations
CEJM 2 (2003) 169{183 Sets with two associative operations Teimuraz Pirashvili A.M. Razmadze Mathematical Inst. Aleksidze str. 1, Tbilisi, 380093, Republic of Georgia Received 7 January 2003; revised 3
More informationSARD S THEOREM ALEX WRIGHT
SARD S THEOREM ALEX WRIGHT Abstract. A proof of Sard s Theorem is presented, and applications to the Whitney Embedding and Immersion Theorems, the existence of Morse functions, and the General Position
More informationP (A G) dp G P (A G)
First homework assignment. Due at 12:15 on 22 September 2016. Homework 1. We roll two dices. X is the result of one of them and Z the sum of the results. Find E [X Z. Homework 2. Let X be a r.v.. Assume
More informationBrownian Motion and Stochastic Calculus
ETHZ, Spring 17 D-MATH Prof Dr Martin Larsson Coordinator A Sepúlveda Brownian Motion and Stochastic Calculus Exercise sheet 6 Please hand in your solutions during exercise class or in your assistant s
More informationGALOIS THEORY I (Supplement to Chapter 4)
GALOIS THEORY I (Supplement to Chapter 4) 1 Automorphisms of Fields Lemma 1 Let F be a eld. The set of automorphisms of F; Aut (F ) ; forms a group (under composition of functions). De nition 2 Let F be
More informationChapter 1. GMM: Basic Concepts
Chapter 1. GMM: Basic Concepts Contents 1 Motivating Examples 1 1.1 Instrumental variable estimator....................... 1 1.2 Estimating parameters in monetary policy rules.............. 2 1.3 Estimating
More information5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
More information2019 Spring MATH2060A Mathematical Analysis II 1
2019 Spring MATH2060A Mathematical Analysis II 1 Notes 1. CONVEX FUNCTIONS First we define what a convex function is. Let f be a function on an interval I. For x < y in I, the straight line connecting
More informationNotes on the Thomas and Worrall paper Econ 8801
Notes on the Thomas and Worrall paper Econ 880 Larry E. Jones Introduction The basic reference for these notes is: Thomas, J. and T. Worrall (990): Income Fluctuation and Asymmetric Information: An Example
More informationLecture 4. f X T, (x t, ) = f X,T (x, t ) f T (t )
LECURE NOES 21 Lecture 4 7. Sufficient statistics Consider the usual statistical setup: the data is X and the paramter is. o gain information about the parameter we study various functions of the data
More informationMicroeconomics, Block I Part 1
Microeconomics, Block I Part 1 Piero Gottardi EUI Sept. 26, 2016 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, 2016 1 / 53 Choice Theory Set of alternatives: X, with generic elements x,
More informationCOM S 330 Homework 05 Solutions. Type your answers to the following questions and submit a PDF file to Blackboard. One page per problem.
Type your answers to the following questions and submit a PDF file to Blackboard. One page per problem. Problem 1. [5pts] Consider our definitions of Z, Q, R, and C. Recall that A B means A is a subset
More informationR = µ + Bf Arbitrage Pricing Model, APM
4.2 Arbitrage Pricing Model, APM Empirical evidence indicates that the CAPM beta does not completely explain the cross section of expected asset returns. This suggests that additional factors may be required.
More informationNotes on uniform convergence
Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean
More informationLecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.
Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;
More informationLinear Algebra. Linear Algebra. Chih-Wei Yi. Dept. of Computer Science National Chiao Tung University. November 12, 2008
Linear Algebra Chih-Wei Yi Dept. of Computer Science National Chiao Tung University November, 008 Section De nition and Examples Section De nition and Examples Section De nition and Examples De nition
More informationAssignment 8. Section 3: 20, 24, 25, 30, Show that the sum and product of two simple functions are simple. Show that A\B = A B
Andrew van Herick Math 710 Dr. Alex Schuster Nov. 2, 2005 Assignment 8 Section 3: 20, 24, 25, 30, 31 20. Show that the sum and product of two simple functions are simple. Show that A\B = A B A[B = A +
More informationPROGRESSIVE ENLARGEMENTS OF FILTRATIONS AND SEMIMARTINGALE DECOMPOSITIONS
PROGRESSIVE ENLARGEMENTS OF FILTRATIONS AND SEMIMARTINGALE DECOMPOSITIONS Libo Li and Marek Rutkowski School of Mathematics and Statistics University of Sydney NSW 26, Australia July 1, 211 Abstract We
More informationNOTES ON VECTOR-VALUED INTEGRATION MATH 581, SPRING 2017
NOTES ON VECTOR-VALUED INTEGRATION MATH 58, SPRING 207 Throughout, X will denote a Banach space. Definition 0.. Let ϕ(s) : X be a continuous function from a compact Jordan region R n to a Banach space
More informationVerona Course April Lecture 1. Review of probability
Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is
More information