An Introduction to Stochastic Calculus

Transcription

1 An Introduction to Stochastic Calculus Haijun Li Department of Mathematics Washington State University Week 5 Haijun Li An Introduction to Stochastic Calculus Week 5 1 / 20

2 Outline 1 Martingales Filtration Martingale Martingale Transform Haijun Li An Introduction to Stochastic Calculus Week 5 2 / 20

3 Filtration = Increasing Stream of Information Let (F t, t 0) be a collection of σ-fields on the same probability space (Ω, F, P) with F t F, for all t 0. Definition The collection (F t, t 0) of σ-fields on Ω is called a filtration if F s F t, 0 s t. A filtration represents an increasing stream of information. The index t can be discrete, for example, the filtration (F n, n = 0, 1,...) is a sequence of σ-fields on Ω with F n F n+1 for all n 0. Haijun Li An Introduction to Stochastic Calculus Week 5 3 / 20

4 Adapted Processes A filtration is usually linked up with a stochastic process. Definition The stochastic process Y = (Y t, t 0) is said to be adapted to the filtration (F t, t 0) if σ(y t ) F t, t 0. The stochastic process Y is always adapted to the natural filtration generated by Y : F t = σ(y s, s t). For a discrete-time process Y = (Y n, n = 0, 1,... ), the adaptedness means σ(y n ) F n for all n 0. Haijun Li An Introduction to Stochastic Calculus Week 5 4 / 20

5 Example Let (B t, t 0) denote Brownian motion and (F t, t 0) denote the corresponding natural filtration. Stochastic processes of the form X t = f (t, B t ), t 0, where f is a function of two variables, are adapted to (F t, t 0). Examples: X (1) t = B t and X (2) t = Bt 2 t. More Examples: X (3) t = max 0 s t B s and X (4) t = max 0 s t Bs 2. Examples that are not adapted to the Brownian motion filtration: X (5) t = B t+1 and X (6) t = B t + B T for some fixed number T > 0. Definition If the stochastic process Y = (Y t, t 0) is adapted to the natural Brownian filtration (F t, t 0) (that is, Y t is a function of (B s, s t) for all t 0), we will say that Y is adapted to Brownian motion. Haijun Li An Introduction to Stochastic Calculus Week 5 5 / 20

6 Adapted to Different Filtrations Consider Brownian motion (B t, t 0) and the corresponding natural filtration F t = σ(b s, s t). The stochastic process X t := B 2 t, t 0, generates its own natural filtration F t = σ(b 2 s, s t), t 0. The process (X t, t 0) is adapted to both F t and F t. Observe that F t F t. For example, we can only reconstruct the whole information about B t from B 2 t ;, but not about B t: we can say nothing about the sign of B t. Haijun Li An Introduction to Stochastic Calculus Week 5 6 / 20

7 Market Information or Information Histories Share prices, exchange rates, interest rates, etc., can be modelled by solutions of stochastic differential equations which are driven by Brownian motion. These solutions are then functions of Brownian motion. The fluctuations of these processes actually represent the information about the market. This relevant knowledge is contained in the natural filtration. In finance there are always people who know more than the others. For example, they might know that an essential political decision will be taken in the very near future which will completely change the financial landscape. This enables the informed persons to act with more competence than the others. Thus they have their own filtrations which can be bigger than the natural filtration. Haijun Li An Introduction to Stochastic Calculus Week 5 7 / 20

8 Martingale If information F s and X are dependent, we can expect that knowing F s reduces the uncertainty about the values of X t at t > s. That is, X t can be better predicted via E(X t F s ) with the information F s than without it. Definition The stochastic process X = (X t, t 0) adapted to the filtration (F t, t 0) is called a continuous-time martingale with respect to (F t, t 0), if 1 E X t < for all t 0. 2 X s is the best prediction of X t given F s : E(X t F s ) = X s for all 0 s t. The discrete-time martingale can be similarly defined by replacing the second condition by E(X n+1 F n ) = X n, n = 0, 1,.... A martingale has the remarkable property that its expectation function is constant: EX s = E[E(X t F s )] = EX t for all s, t. Haijun Li An Introduction to Stochastic Calculus Week 5 8 / 20

9 Example: Gambler s Ruin A class of betting strategies that was popular in 18th-century France. The gambler wins his stake if a fair coin comes up heads and loses it if the coin comes up tails Winning (or Financial Ruin) Strategy: the gambler double his bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler s available time goes to infinity, his probability of eventually flipping heads approaches 1. Problem: the gambler s wealth must be infinity; otherwise, the exponential growth of the bets eventually bankrupts the gambler. Haijun Li An Introduction to Stochastic Calculus Week 5 9 / 20

10 Gambler s Ruin: Partial Sums Let (Z n ) be a sequence of independent random variables with finite expectations and Z 0 = 0. Consider the partial sums R n = n Z i, n 0. i=0 and the corresponding natural filtration F n = σ(r 0,..., R n ) = σ(z 0,..., Z n ), n 0. Observe that E(R n+1 F n ) = E(R n F n ) + E(Z n+1 F n ) = R n + EZ n+1. and hence, if EZ n = 0 for all n 0, then (R n, n 0) is a martingale with respect to the filtration (F n, n 0). Haijun Li An Introduction to Stochastic Calculus Week 5 10 / 20

11 Collecting Information About a Random Variable Let Z be a random variable on Ω with E Z < and (F t, t 0) be a filtration on Ω. Define X t = E(Z F t ), t 0. Since F t increases when time goes by, X t gives us more and more information about the random variable Z. In particular, if σ(z ) F t for some t, then X t = Z. An appeal to Jensen s inequality yields σ(x t ) F t. E X t = E[ E(Z F t ) ] E[E( Z F t )] = E Z <. E(X t F s ) = E[E(Z F t ) F s ] = E(Z F s ) = X s. X is a martingale with respect to (F t, t 0). Haijun Li An Introduction to Stochastic Calculus Week 5 11 / 20

12 Brownian Motion is a Martingale Let B = (B t, t 0) be Brownian motion with the natural filtration F t = σ(b s, s t). B and (B 2 t t, t 0) are martingales with respect to the natural filtration. (B 3 t 3tB t, t 0) is a martingale. The stopped Brownian motion (B min{t,τ}, t 0) is a martingale (Gambler s betting game with the possibility of bankruptcy). Haijun Li An Introduction to Stochastic Calculus Week 5 12 / 20

13 Stopped Brownian Motion is a Martingale Haijun Li An Introduction to Stochastic Calculus Week 5 13 / 20

14 A Simulated Martingale Haijun Li An Introduction to Stochastic Calculus Week 5 14 / 20

15 Compensated Poisson Process is a Martingale Let (N t, t 0) be a Poisson process of intensity λ with the natural filtration F t = σ(n s, s t). The compensated Poisson process (N t λt, t 0) is a martingales with respect to the natural filtration. Figure : Left = Poisson Process Right = Compensated Poisson Process Haijun Li An Introduction to Stochastic Calculus Week 5 15 / 20

16 Martingale Transform Let X = (X n, n = 0, 1,... ) be a discrete-time martingale with respect to the filtration (F n, n = 0, 1,... ). Let Y n := X n X n 1, n 1, and Y 0 := X 0. The sequence Y = (Y n, n = 0, 1,... ) is called a martingale difference sequence with respect to the filtration (F n, n = 0, 1,... ). Consider a stochastic process C = (C n, n = 1, 2,... ), satisfying that σ(c n ) F n 1, n 1. Given F n 1, we completely know C n at time n 1. Such a sequence is called predictable with respect to (F n, n = 0, 1,... ). Define n n Z 0 = 0, Z n = C i Y i = C i (X i X i 1 ), n 1. i=1 The process C Y := (Z n, n 0) is called the martingale transform of Y by C. Note that if C n = 1 for all n 1, then C Y = X is the original martingale. Haijun Li An Introduction to Stochastic Calculus Week 5 16 / 20 i=1

17 Martingale Transform Leads to a Martingale Assume that the second moments of C n and Y n are finite. It follows from the Cauchy-Schwarz inequality that E Z n n E C i Y i i=1 n i=1 [EC 2 i EY 2 i ] 1/2 <. Since Y 1,..., Y n do not carry more information than F n, and σ(c 1,..., C n ) F n 1 (predictability), we have σ(z n ) F n. Due to the predictability of C, E(Z n Z n 1 F n 1 ) = E(C n Y n F n 1 ) = C n E(Y n F n 1 ) = 0. (Z n Z n 1, n 0) is a martingale difference sequence, and (Z n, n 0) is a martingale with respect to (F n, n = 0, 1,... ). Haijun Li An Introduction to Stochastic Calculus Week 5 17 / 20

18 A Brownian Martingale Transform Consider Brownian motion B = (B s, s t) and a partition 0 = t 0 < t 1 < < t n 1 < t n = t. The σ-fields at these time instants are described by the filtration: F 0 = {, Ω}, F i = σ(b tj, 1 j i), i = 1,..., n. The sequence B := ( i B, 1 i n) defined by 0 B = 0, i B = B ti B ti 1, i = 1,..., n, forms a martingale difference sequence with respect to the filtration (F i, 1 i n). B := (B ti 1, 1 i n) is predictable with respect to (F i, 1 i n). The martingale transform B B is then a martingale: ( B B) k = k i=1 B t i 1 (B ti B ti 1 ), k = 1,..., n. This is precisely a discrete-time analogue of the Itô stochastic integral t 0 B sdb s. Haijun Li An Introduction to Stochastic Calculus Week 5 18 / 20

19 Martingale as a Fair Game Let X = (X n, n = 0, 1,... ) be a discrete-time martingale with respect to the filtration (F n, n = 0, 1,... ). Let Y n = X n X n 1, n 0, denote the martingale difference, and C n, n 1, be predictable with respect to (F n, n = 0, 1,... ). Think of Y, as your net winnings per unit stake at the n-th game which are adapted to a filtration (F n, n = 0, 1,... ). At the n-th game, your stake C n, does not contain more information than F n 1 does. At time n 1, this is the best information we have about the game. C n Y n is the net winnings for stake C n at the n-th game. (C Y ) n = n i=1 C iy i is the net winnings up to time n. The game is fair because the best prediction of the net winnings C n Y n of the n-th game, just before the n-th game starts, is zero: E(C n Y n F n 1 ) = 0. Haijun Li An Introduction to Stochastic Calculus Week 5 19 / 20

20 History Notes: Martingale The concept of martingale in probability theory was introduced by Paul Pierre Lévy, Émile Borel (French Probability School). The development of the theory was done by Joseph Leo Doob. The martingale is the first general class of integrable stochastic processes, satisfying a particular conditional expectation property. Widely used in finance: A financial market is viable (i.e., no arbitrage opportunities) if and only if there exists a probability measure under which the realized prices are martingales. Haijun Li An Introduction to Stochastic Calculus Week 5 20 / 20