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Notes on WI4430 Martingales and Brownian Motion Robbert Fokkink

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Notes on WI4430 Martingales and Brownian Motion Robbert Fokkink Notes on WI4430 Martingales and Brownian Motion Robbert Fokkink TU Delft E-mail address: [email protected] Abstract. These notes accompany the course WI4430 on Martingales and Brownian Motion that I teach in the fall of 2016 at Delft University. Normally, Frank Redig teaches this course, but he has a sabbatical and I step in for one time. The course is mainly based on chapter 10 of Gut's book Probability, a graduate course which is available through the WI4430 homepage. Apart from Gut, Brze´zniak and Zastawniak's text on Basic Stochastic Processes is also recommended. It has many exercises and solutions. I wrote these notes to help you digest Gut. I have included exercises, but they are not meant to be very hard. Frank Redig has also prepared notes { you can find them under `scribenotes' { and sets of more difficult exercises. This is your `homework', which I include in between my notes. There is a homework session once every week in which you can seek assistance in solving these exercises. The difficult exercises are marked as challenges, to warn you. Sometimes a challenge offers a financial reward as an incentive. You can claim the reward by handing in a written solution and a transfer of copyright. These challenges are usually open research problems. Contents Chapter 1. A recap of measure theory 1 1.1. The Riemann integral ... and beyond 1 1.2. Putting the sigma into the algebra 3 1.3. A call for better integration 5 1.4. Swap until you drop 7 1.5. A final word 10 Chapter 2. E-learning 15 2.1. E-definitions 15 2.2. E-properties 17 2.3. E-laboration 20 Chapter 3. Meet the Martingales 25 3.1. Coming to terms with terminology 26 3.2. Thirteen examples 28 3.3. Linear Algebra Again 29 Chapter 4. Welcome to the California Casino 35 4.1. You can check out any time you like 35 4.2. But you can never lose 38 Chapter 5. A discourse on inequalities 47 5.1. Doob's optional stopping (or sampling) theorem 47 5.2. Doob's maximal inequality 48 5.3. Doob's p inequality 50 5.4. Doob's upcrossingL inequality 51 5.5. And beyond 53 Chapter 6. Don't Stop Me Now 55 6.1. 2 convergence 55 6.2. AlmostL sure convergence 56 6.3. p convergence 57 6.4. L1 convergence 57 6.5. WrappingL it up 59 Chapter 7. Sum ergo cogito 65 7.1. All, or nothing at all 65 7.2. The domino effect 66 7.3. Show me the money, Jerry 67 7.4. Take it to the limit, one more time 69 v vi CONTENTS Chapter 8. Get used to it 75 8.1. Are we there yet? 75 8.2. A deviation 78 Chapter 9. Walk the Line 83 9.1. Putting PDE's in your PC 84 9.2. Time's Arrow 86 9.3. Measuring motion on the atomic scale 89 Chapter 10. Get Real 93 10.1. A fishy frog 94 10.2. Get on up, get on the scene 97 Chapter 11. Meet the real Martingales 105 11.1. Continuous martingales 107 11.2. Crooked paths 108 Chapter 12. A tale of two thinkers 115 12.1. Cracking up crooked paths 116 12.2. The stochastic integral 119 Chapter 13. Stochastic Calculus 125 13.1. It^o'srule 127 13.2. Properties of the Stochastic Integral 130 13.3. The It^oformula 130 13.4. EXERCISES CHAPTER 13 135 Chapter 14. The End is Here 137 14.1. A review of lecture 9 137 14.2. Building Bridges 138 14.3. A final word 141 CHAPTER 1 A recap of measure theory This material is covered by Chapter 1 of Brze´zniakand Zastawniak's text on Basic Stochastic Processes Measure Theory was developed around the turn of the 20th century by the French mathematicians Emile´ Borel and Henri Lebesgue.1 Most of you have already learned this theory, for instance in the courses TW2090 Real Analysis, TW3560 Advanced Probability, TW3570 Fourier Analysis, or if you have followed the Minor Finance. If you dit not learn the theory yet, you need to catch up, because this here is only a partial recap to refresh your memory. Fortunately, there is plenty of material available on the internet. You could for instance consult the first part of the notes by Terry Tao, the world's most famous mathematician. You could also try Probability with Martingales, by David Williams, which is insightful but demanding. Finally, there are the very elegant notes on Probability by Varadhan. 1.1. The Riemann integral ... and beyond You have received excessive training on calculating the integral b (1.1) a f(x)dx But what does the integral mean? You need to remember your definitions. In R 1Delft pride: the Dutch mathematician Thomas Stieltjes, who studied in Delft but failed to get his BSA, got some of these ideas first. Figure 1. Upper and lower Riemann sum 1 2 1. A RECAP OF MEASURE THEORY particular, you need to remember your Riemann sums. Divide the interval [a; b] into a finite union of subintervals [a; b] = [x0; x1) [x1; x2) ::: [xn−1; xn] [ [ [ with x0 = a and xn = b and approximate the integral by b f(x)dx f(ξ0)(x1 x0) + f(ξ1)(x2 x1) + + f(ξn−1)(xn xn−1) ≈ − − ··· − Za As the mesh of the subintervals gets smaller, the approximation gets better and better, and (hopefully) converges to the integral. The summands f(ξi)(xi+1 xi) have two factors: a function value f(ξi) and a − length xi+1 xi. Suppose that we change the integral as follows: − b 2 (1.2) a f(x)d(x ) Then the function values remain the same, but we have changed the lengths. The Riemann sum with d(x2) insteadR of dx is equal to b 2 2 2 2 2 2 2 f(x)d(x ) f(ξ0)(x x ) + f(ξ1)(x x ) + + f(ξn−1)(x x ) ≈ 1 − 0 2 − 1 ··· n − n−1 Za So here is what you need to observe: the definition of the integral involves the definition of the lengths of the intervals. There are many possible notions of length. We can define the length of [x; y] to be y x or y2 x2 or ey ex and in general b − − − we may consider a f(x)d(g(x)) where the length of [x; y] is equal to g(y) g(x) for some monotonic function g. A measure is a more general notion of− length. R I did not tell you yet what the ξi are. They are arbitrary points in [xi; xi+1). If f is continuous and if the interval is very small, then f(ξi) hardly varies and the b choice of ξi is not really important. This is why a f(x)dx is well defined if f is continous. But if f is discontinuous, then the Riemann sums may not converge. You probably remember this example: R b (1.3) a 1Q(x)dx is not defined where the indicator function 1Q(x) is equal to 1 if x is rational and it is equal to 0 if x is irrational. However,R we can fix this if we can decide what the length of Q is. The rational numbers are countable, so we can enumerate them Q = r1; r2; r3;::: f g To decide what the length of Q is, we cover it by intervals (r1 0:01; r1 + 0:01) (r2 0:001; r2 + 0:001) (r3 0:0001; r3 + 0:0001);::: − [ − [ − −n−1 In other words, we cover each rn by an interval of length 2:10 . The total length of these intervals adds up to 0:0222 . We can do even better and cover Q by ··· intervals with lengths that add up to nearly nothing. So the length of Q must be b equal to zero and that is why a 1Q(x)dx = 0. Or to put this in technical terms: b the Lebesgue integral a 1QR(x)dx is equal to zero. R 1.2. PUTTING THE SIGMA INTO THE ALGEBRA 3 The advantages of the Lebesgue integral over the Riemann integral are: Lebesgue is an upgrade: it is well defined for many more functions and • allows you to integrate over spaces that are much more general than R Lebesgue is downward compatible: it produces the same value as Rie- • mann, if Riemann is well defined. Lebesgue extends accross platforms: it treats and Σ in the same • manner, and it connects Analysis to Probability. R Lebesgue has no ambiguity: there is no need for points ξi • Lebesgue has improved functionality: the swap between limn!1 fn • and limn!1 fn is handled in a much more transparent manner. R In short, the LebesgueR environment is more user friendly than the Riemann environ- ment. We are talking about a genuine upgrade here, we are not going Microsoft. The Lebesgue integral hinges on the idea of a measure, which is some kind of length function for sets rather than intervals. We need to describe these sets first. 1.2. Putting the sigma into the algebra Let Ω be any arbitrary set, but let it be [0; 1] in particular, or let it be R if you like. We would like to define the measure of all subsets of Ω. Unfortunately, this is impossible without running into fundamental difficulties, because there are so many subsets, more than you can ever describe.
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