1 00:00:00,090 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation, or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:21,580 --> 00:00:25,540 PROFESSOR: And today it's me, back again. 9 00:00:25,540 --> 00:00:28,600 And we'll study continuous types of stochastic processes. 10 00:00:28,600 --> 00:00:31,420 So far we were discussing discrete time processes. 11 00:00:39,270 --> 00:00:41,990 We studied the basics like variance, expectation, 12 00:00:41,990 --> 00:00:45,270 all this stuff-- moments, moment generating function, 13 00:00:45,270 --> 00:00:52,160 and some important concepts for Markov chains, and martingales. 14 00:01:01,260 --> 00:01:04,180 So I'm sure a lot of you would have forgot 15 00:01:04,180 --> 00:01:06,660 about what martingale and Markov chains were, 16 00:01:06,660 --> 00:01:09,680 but try to review this before the next few lectures. 17 00:01:09,680 --> 00:01:11,880 Because starting next week when we 18 00:01:11,880 --> 00:01:15,970 start discussing continuous types of stochastic processes-- 19 00:01:15,970 --> 00:01:16,490 not from me. 20 00:01:16,490 --> 00:01:18,740 You're not going to hear martingale from me that much. 21 00:01:18,740 --> 00:01:21,230 But from people-- say, outside speakers-- 22 00:01:21,230 --> 00:01:23,840 they're going to use this martingale concept 23 00:01:23,840 --> 00:01:25,590 to do pricing. 24 00:01:25,590 --> 00:01:28,250 So I will give you some easy exercises. 25 00:01:28,250 --> 00:01:30,180 You will have some problems on martingales. 26 00:01:30,180 --> 00:01:34,610 Just refer back to the notes that I had like a month ago, 27 00:01:34,610 --> 00:01:35,460 and just review. 28 00:01:35,460 --> 00:01:37,120 It won't be difficult problems, but try 29 00:01:37,120 --> 00:01:40,901 to make the concept comfortable. 30 00:01:40,901 --> 00:01:41,400 OK. 31 00:01:41,400 --> 00:01:46,320 And then Peter taught some time series analysis. 32 00:01:46,320 --> 00:01:49,625 Time series is just the same as discrete time process. 33 00:01:53,640 --> 00:02:04,600 And regression analysis, this was all done on discrete time. 34 00:02:04,600 --> 00:02:09,162 That means the underlying space was x_1, x_2, x_3, dot dot 35 00:02:09,162 --> 00:02:10,726 dot, x_t. 36 00:02:14,369 --> 00:02:16,410 But now we're going to talk about continuous time 37 00:02:16,410 --> 00:02:17,800 processes. 38 00:02:17,800 --> 00:02:18,512 What are they? 39 00:02:18,512 --> 00:02:20,720 They're just a collection of random variables indexed 40 00:02:20,720 --> 00:02:22,610 by time. 41 00:02:22,610 --> 00:02:24,520 But now the time is a real variable. 42 00:02:24,520 --> 00:02:27,430 Here, time was just in integer values. 43 00:02:27,430 --> 00:02:29,570 Here, we have real variable. 44 00:02:29,570 --> 00:02:35,750 So a stochastic process develops over time, 45 00:02:35,750 --> 00:02:39,606 and the time variable is continuous now. 46 00:02:39,606 --> 00:02:43,060 It doesn't necessarily mean that the process itself 47 00:02:43,060 --> 00:02:46,815 is continuous-- it may as well look like these jumps. 48 00:02:46,815 --> 00:02:48,690 It may as well have a lot of jumps like this. 49 00:02:51,280 --> 00:02:52,950 It just means that the underlying time 50 00:02:52,950 --> 00:02:54,850 variable is continuous. 51 00:02:54,850 --> 00:02:57,070 Whereas when it was discrete time, 52 00:02:57,070 --> 00:03:00,160 you were only looking at specific observations 53 00:03:00,160 --> 00:03:02,780 at some times. 54 00:03:02,780 --> 00:03:05,500 I'll draw it here. 55 00:03:05,500 --> 00:03:12,290 Discrete time looks more like that. 56 00:03:15,170 --> 00:03:16,482 OK. 57 00:03:16,482 --> 00:03:18,470 So the first difficulty when you try 58 00:03:18,470 --> 00:03:20,830 to understand continuous time stochastic processes when 59 00:03:20,830 --> 00:03:23,215 you look at it is, how do you describe the probability 60 00:03:23,215 --> 00:03:25,635 distribution? 61 00:03:25,635 --> 00:03:35,690 How to describe the probability distribution? 62 00:03:40,430 --> 00:03:43,935 So let's go back to discrete time processes. 63 00:03:43,935 --> 00:03:46,833 So the universal example was a simple random walk. 64 00:03:52,630 --> 00:03:56,470 And if you remember, how we described it was x_t minus 65 00:03:56,470 --> 00:04:04,650 x_(t-1), was either 1 or minus 1, probability half each. 66 00:04:04,650 --> 00:04:07,450 This was how we described it. 67 00:04:07,450 --> 00:04:10,290 And if you think about it, this is a slightly indirect way 68 00:04:10,290 --> 00:04:11,740 of describing the process. 69 00:04:11,740 --> 00:04:13,790 You're not describing the probability 70 00:04:13,790 --> 00:04:18,850 of this process following this path, it's like a path. 71 00:04:18,850 --> 00:04:20,390 Instead what you're doing is, you're 72 00:04:20,390 --> 00:04:24,850 describing the probability of this event happening. 73 00:04:24,850 --> 00:04:28,335 From time t to t plus 1, what is the probability 74 00:04:28,335 --> 00:04:30,220 that it will go down? 75 00:04:30,220 --> 00:04:33,760 And at each step you describe the probability altogether, 76 00:04:33,760 --> 00:04:36,280 when you combine them, you get the probability distribution 77 00:04:36,280 --> 00:04:39,170 over the process. 78 00:04:39,170 --> 00:04:41,930 But you can't do it for continuous time, right? 79 00:04:41,930 --> 00:04:44,280 The time variable is continuous so you can't just 80 00:04:44,280 --> 00:04:48,160 take intervals t and interval t prime 81 00:04:48,160 --> 00:04:49,954 and describe the difference. 82 00:04:49,954 --> 00:04:52,620 If you want to do that, you have to do it infinitely many times. 83 00:04:52,620 --> 00:04:54,939 You have to do it for all possible values. 84 00:04:54,939 --> 00:04:56,105 That's the first difficulty. 85 00:05:00,400 --> 00:05:02,955 Actually, that's the main difficulty. 86 00:05:08,540 --> 00:05:12,780 And how can we handle this? 87 00:05:18,060 --> 00:05:20,430 It's not an easy question. 88 00:05:20,430 --> 00:05:22,460 And you'll see a very indirect way to handle it. 89 00:05:22,460 --> 00:05:26,040 It's somewhat in the spirit of this thing. 90 00:05:26,040 --> 00:05:29,910 But it's not like you draw some path to describe a probability 91 00:05:29,910 --> 00:05:32,890 density of this path. 92 00:05:32,890 --> 00:05:35,150 That's the omega. 93 00:05:35,150 --> 00:05:37,365 What is the probability density at omega? 94 00:05:40,390 --> 00:05:42,480 Of course, it's not a discrete variable 95 00:05:42,480 --> 00:05:46,020 so you have a probability density function, not 96 00:05:46,020 --> 00:05:47,249 a probability mass function. 97 00:05:52,628 --> 00:05:56,362 In fact, can we even write it down? 98 00:05:56,362 --> 00:05:57,820 You'll later see that we won't even 99 00:05:57,820 --> 00:06:01,150 be able to write this down. 100 00:06:01,150 --> 00:06:04,340 So just have this in mind and you'll 101 00:06:04,340 --> 00:06:06,090 see what I was trying to say. 102 00:06:09,150 --> 00:06:14,000 So finally, I get to talk about Brownian processes, 103 00:06:14,000 --> 00:06:14,890 Brownian motion. 104 00:06:20,830 --> 00:06:24,360 Some outside speakers already started talking about it. 105 00:06:24,360 --> 00:06:26,480 I wish I already was able to cover it 106 00:06:26,480 --> 00:06:29,790 before they talked about it, but you'll see a lot more from now. 107 00:06:29,790 --> 00:06:31,370 And let's see what it actually is. 108 00:06:38,840 --> 00:06:40,852 So it's described as the following, 109 00:06:40,852 --> 00:06:42,310 it actually follows from a theorem. 110 00:06:44,920 --> 00:06:55,000 There exists a probability distribution 111 00:06:55,000 --> 00:07:13,240 over the set of continuous functions from positive reals 112 00:07:13,240 --> 00:07:21,973 to the reals such that first, B(0) is always 0. 113 00:07:21,973 --> 00:07:26,360 So probability of B(0) is equal to 0 is 1. 114 00:07:26,360 --> 00:07:29,770 Number two-- we call this stationary. 115 00:07:32,790 --> 00:07:43,540 For all s and t, B(t) minus B(s) has 116 00:07:43,540 --> 00:07:50,850 normal distribution with mean 0 and variance t minus s. 117 00:07:50,850 --> 00:07:53,080 And the third-- independent increment. 118 00:08:04,700 --> 00:08:19,410 That means if intervals [s i, t i] are not overlapping, 119 00:08:19,410 --> 00:08:25,845 then B(t_i) minus B(s_i) are independent. 120 00:08:31,160 --> 00:08:33,710 So it's actually a theorem saying 121 00:08:33,710 --> 00:08:35,960 that there is some strange probability 122 00:08:35,960 --> 00:08:40,530 distribution over the continuous functions 123 00:08:40,530 --> 00:08:42,490 from positive reals-- non-negative reals-- 124 00:08:42,490 --> 00:08:44,551 to the reals. 125 00:08:44,551 --> 00:08:48,457 So if you look at some continuous function, 126 00:08:48,457 --> 00:08:50,540 this theorem gives you a probability distribution. 127 00:08:50,540 --> 00:08:55,150 It describes the probability of this path happening. 128 00:08:55,150 --> 00:08:56,400 It doesn't really describe it. 129 00:08:56,400 --> 00:08:58,608 It just says that there exists some distribution such 130 00:08:58,608 --> 00:09:04,790 that it always starts at 0 and it's continuous. 131 00:09:04,790 --> 00:09:12,250 Second, the distribution for all fixed s and t, the distribution 132 00:09:12,250 --> 00:09:17,150 of this difference is normally distributed 133 00:09:17,150 --> 00:09:20,440 with mean 0 and variance t minus s, which 134 00:09:20,440 --> 00:09:24,140 scales according to the time. 135 00:09:24,140 --> 00:09:28,320 And then third, independent increment means 136 00:09:28,320 --> 00:09:31,850 what happened between this interval, [s1, t1], 137 00:09:31,850 --> 00:09:37,450 and [s2, t2], this part and this part, 138 00:09:37,450 --> 00:09:39,850 is independent as long as intervals do not overlap. 139 00:09:45,050 --> 00:09:47,780 It sounds very similar to the simple random walk. 140 00:09:47,780 --> 00:09:50,800 But the reason we have to do this very complicated process 141 00:09:50,800 --> 00:09:53,500 is because the time is continuous. 142 00:09:53,500 --> 00:09:57,260 You can't really describe at each time what's happening. 143 00:09:57,260 --> 00:10:01,470 Instead, what you're describing is over all possible intervals 144 00:10:01,470 --> 00:10:03,270 what's happening. 145 00:10:03,270 --> 00:10:06,530 When you have a fixed interval, it describes the probability 146 00:10:06,530 --> 00:10:07,352 distribution. 147 00:10:07,352 --> 00:10:09,060 And then when you have several intervals, 148 00:10:09,060 --> 00:10:12,421 as long as they don't overlap, they're independent. 149 00:10:12,421 --> 00:10:12,920 OK? 150 00:10:12,920 --> 00:10:15,780 And then by this theorem, we call this probability 151 00:10:15,780 --> 00:10:18,980 distribution a Brownian motion. 152 00:10:18,980 --> 00:10:27,820 So probability distribution, the definition, distribution 153 00:10:27,820 --> 00:10:36,290 given by this theorem is called the Brownian motion. 154 00:10:44,731 --> 00:10:46,230 That's why I'm saying it's indirect. 155 00:10:46,230 --> 00:10:48,500 I'm not saying Brownian motion is this probability 156 00:10:48,500 --> 00:10:49,200 distribution. 157 00:10:49,200 --> 00:10:53,140 It satisfies these conditions, but we are reversing it. 158 00:10:53,140 --> 00:10:55,060 Actually, we have these properties in mind. 159 00:10:55,060 --> 00:10:57,309 We're not sure if such a probability distribution even 160 00:10:57,309 --> 00:10:58,690 exists or not. 161 00:10:58,690 --> 00:11:01,080 And actually this theorem is very, very difficult. 162 00:11:01,080 --> 00:11:03,580 I don't know how to prove it right now. 163 00:11:03,580 --> 00:11:06,170 I have to go through a book. 164 00:11:06,170 --> 00:11:09,432 And even graduate probability courses 165 00:11:09,432 --> 00:11:11,640 usually don't cover it because it's really technical. 166 00:11:14,610 --> 00:11:18,690 That means this just shows how continuous time stochastic 167 00:11:18,690 --> 00:11:24,170 processes can be so much more complicated than discrete time. 168 00:11:24,170 --> 00:11:29,170 Then why are you-- why are we studying continuous time 169 00:11:29,170 --> 00:11:31,380 processes when it's so complicated? 170 00:11:31,380 --> 00:11:33,700 Well, you'll see in the next few lectures. 171 00:11:37,487 --> 00:11:38,070 Any questions? 172 00:11:41,290 --> 00:11:42,289 OK. 173 00:11:42,289 --> 00:11:44,080 So let's go through this a little bit more. 174 00:11:44,080 --> 00:11:44,990 AUDIENCE: Excuse me. 175 00:11:44,990 --> 00:11:45,746 PROFESSOR: Yes. 176 00:11:45,746 --> 00:11:50,030 AUDIENCE: So when you talk about the probability distribution, 177 00:11:50,030 --> 00:11:51,458 what's the underlying space? 178 00:11:51,458 --> 00:11:52,890 Is it the space of-- 179 00:11:52,890 --> 00:11:54,810 PROFESSOR: Yes, that's a very good question. 180 00:11:54,810 --> 00:11:59,250 The space is the space of all functions. 181 00:11:59,250 --> 00:12:01,770 That means it's a space of all possible paths, 182 00:12:01,770 --> 00:12:03,630 if you want to think about it this way. 183 00:12:03,630 --> 00:12:06,090 Just think about all possible ways 184 00:12:06,090 --> 00:12:09,880 your variable can evolve over time. 185 00:12:09,880 --> 00:12:12,900 And for some fixed drawing for this path, 186 00:12:12,900 --> 00:12:17,050 there's some probability that this path will happen. 187 00:12:17,050 --> 00:12:21,540 It's not the probability spaces that you have been looking at. 188 00:12:21,540 --> 00:12:26,750 It's not one point-- well, a point is now a path. 189 00:12:26,750 --> 00:12:28,240 And your probability distribution 190 00:12:28,240 --> 00:12:33,310 is given over paths, not for a fixed point. 191 00:12:33,310 --> 00:12:35,940 And that's also a reason why it makes it so complicated. 192 00:12:40,494 --> 00:12:41,160 Other questions? 193 00:12:44,880 --> 00:12:47,735 So the main thing you have to remember-- well, intuitively 194 00:12:47,735 --> 00:12:49,170 you will just know it. 195 00:12:49,170 --> 00:12:53,930 But one thing you want to try to remember is this property. 196 00:12:53,930 --> 00:12:58,610 As your time scales, what happens between that interval 197 00:12:58,610 --> 00:13:01,910 is it's like a normal variable. 198 00:13:01,910 --> 00:13:05,360 So this is a collection of a bunch of normal variables. 199 00:13:05,360 --> 00:13:08,870 And the mean is always 0, but the variance 200 00:13:08,870 --> 00:13:12,800 is determined by the length of your interval. 201 00:13:12,800 --> 00:13:16,800 Exactly that will be the variance. 202 00:13:16,800 --> 00:13:18,700 So try to remember this property. 203 00:13:23,550 --> 00:13:28,375 A few more things, it has a lot of different names. 204 00:13:28,375 --> 00:13:31,220 It's also called Wiener process. 205 00:13:36,710 --> 00:13:38,990 And let's see, there was one more. 206 00:13:43,962 --> 00:13:45,170 Is there another name for it? 207 00:13:48,070 --> 00:13:52,100 I thought I had one more name in mind, but maybe not. 208 00:13:52,100 --> 00:13:54,492 AUDIENCE: Norbert Wiener was an MIT professor. 209 00:13:54,492 --> 00:13:55,325 PROFESSOR: Oh, yeah. 210 00:13:55,325 --> 00:13:56,440 That's important. 211 00:13:56,440 --> 00:13:58,200 AUDIENCE: Of course. 212 00:13:58,200 --> 00:14:02,420 PROFESSOR: Yeah, a professor at MIT. 213 00:14:02,420 --> 00:14:04,650 But apparently he wasn't the first person 214 00:14:04,650 --> 00:14:06,570 who discovered this process. 215 00:14:06,570 --> 00:14:10,450 I was some other person in 1900. 216 00:14:10,450 --> 00:14:13,260 And actually, in the first paper that appeared, 217 00:14:13,260 --> 00:14:15,700 of course, they didn't know about each other's result. 218 00:14:15,700 --> 00:14:17,200 In that paper the reason he studied 219 00:14:17,200 --> 00:14:20,156 this was to evaluate stock prices and auction prices. 220 00:14:36,620 --> 00:14:40,080 And here's another slightly different description, 221 00:14:40,080 --> 00:14:41,740 maybe a more intuitive description 222 00:14:41,740 --> 00:14:44,580 of the Brownian motion. 223 00:14:44,580 --> 00:14:46,720 So here is this philosophy. 224 00:14:46,720 --> 00:14:56,260 Philosophy is that Brownian motion is the limit 225 00:14:56,260 --> 00:14:57,550 of simple random walks. 226 00:15:04,460 --> 00:15:08,429 The limit-- it's a very vague concept. 227 00:15:08,429 --> 00:15:09,720 You'll see what I mean by this. 228 00:15:12,570 --> 00:15:18,210 So fix a time interval of 0 up to 1 229 00:15:18,210 --> 00:15:21,470 and slice it into very small pieces. 230 00:15:21,470 --> 00:15:23,480 So I'll say, into n pieces. 231 00:15:23,480 --> 00:15:27,290 1 over n, 2 over n, 3 over n, dot dot dot, to n minus 1 232 00:15:27,290 --> 00:15:29,520 over n. 233 00:15:29,520 --> 00:15:31,400 And consider a simple random walk, 234 00:15:31,400 --> 00:15:33,560 n-step simple random walk. 235 00:15:33,560 --> 00:15:35,860 So from time 0 you go up or down, up or down. 236 00:15:39,290 --> 00:15:41,190 Then you get something like that. 237 00:15:41,190 --> 00:15:43,940 OK? 238 00:15:43,940 --> 00:15:47,491 So let me be a little bit more precise. 239 00:15:47,491 --> 00:16:01,380 Let Y_0, Y_1, to Y_n, be a simple random walk, 240 00:16:01,380 --> 00:16:06,690 and let Z be the function such that at time t over n, 241 00:16:06,690 --> 00:16:10,440 we let it to be Y of t. 242 00:16:10,440 --> 00:16:13,630 That's exactly just written down in formula what it means. 243 00:16:13,630 --> 00:16:17,445 So this process is Z. I take a simple random walk 244 00:16:17,445 --> 00:16:20,220 and scale it so that it goes from time 0 to time 1. 245 00:16:24,180 --> 00:16:27,150 And then in the intermediate values-- 246 00:16:27,150 --> 00:16:29,040 for values that are not this, just 247 00:16:29,040 --> 00:16:35,480 linearly extended-- linearly extend in intermediate values. 248 00:16:41,570 --> 00:16:44,120 It's a complicated way of saying just connect the dots. 249 00:16:49,390 --> 00:16:50,555 And take n to infinity. 250 00:16:55,870 --> 00:17:00,810 Then the resulting distribution is a Brownian motion. 251 00:17:19,420 --> 00:17:21,660 So mathematically, that's just saying 252 00:17:21,660 --> 00:17:24,369 the limit of simple random walks is a Brownian motion. 253 00:17:24,369 --> 00:17:26,480 But it's more than that. 254 00:17:26,480 --> 00:17:29,430 That means if you have some suspicion 255 00:17:29,430 --> 00:17:33,600 that some physical quantity follows a Brownian motion, 256 00:17:33,600 --> 00:17:36,890 and then you observe the variable 257 00:17:36,890 --> 00:17:41,640 at discrete times at very, very fine scales-- 258 00:17:41,640 --> 00:17:44,500 so you observe it really, really often, like a million times 259 00:17:44,500 --> 00:17:46,640 in one second. 260 00:17:46,640 --> 00:17:51,877 Then once you see-- if you see that and take it to the limit, 261 00:17:51,877 --> 00:17:53,210 it looks like a Brownian motion. 262 00:17:53,210 --> 00:17:56,140 Then now you can conclude that it's a Brownian motion. 263 00:17:56,140 --> 00:18:01,300 What I'm trying to say is this continuous time process, 264 00:18:01,300 --> 00:18:06,560 whatever the strange thing is, it follows from something 265 00:18:06,560 --> 00:18:07,790 from a discrete world. 266 00:18:07,790 --> 00:18:10,200 It's not something new. 267 00:18:10,200 --> 00:18:13,350 It's the limit of these objects that you already now. 268 00:18:17,440 --> 00:18:21,320 So this tells you that it might be a reasonable model for stock 269 00:18:21,320 --> 00:18:23,380 prices because for stock prices, no matter 270 00:18:23,380 --> 00:18:27,120 how-- there's only a finite amount of time scale 271 00:18:27,120 --> 00:18:29,360 that you can observe the prices. 272 00:18:29,360 --> 00:18:32,220 But still, if you observe it infinitely as 273 00:18:32,220 --> 00:18:34,750 much as you can, and the distribution looks 274 00:18:34,750 --> 00:18:37,560 like a Brownian motion, then you can use 275 00:18:37,560 --> 00:18:40,150 a Brownian motion to model it. 276 00:18:40,150 --> 00:18:43,860 So it's not only the theoretical observation. 277 00:18:43,860 --> 00:18:46,930 It also has implication when you want 278 00:18:46,930 --> 00:18:49,610 to use Brownian motion as a physical model 279 00:18:49,610 --> 00:18:53,000 for some quantity. 280 00:18:53,000 --> 00:18:56,097 It also tells you why Brownian motion might 281 00:18:56,097 --> 00:18:57,180 appear in some situations. 282 00:19:00,960 --> 00:19:02,155 So here's an example. 283 00:19:05,480 --> 00:19:07,420 Here's a completely different context 284 00:19:07,420 --> 00:19:11,180 where Brownian motion was discovered, 285 00:19:11,180 --> 00:19:14,220 and why it has the name Brownian motion. 286 00:19:14,220 --> 00:19:18,960 So a botanist-- I don't know if I'm pronouncing it correctly-- 287 00:19:18,960 --> 00:19:26,210 named Brown in the 1800s, what he did was he 288 00:19:26,210 --> 00:19:39,630 observed a pollen particle in water. 289 00:19:43,895 --> 00:19:46,020 So you have a cup of water and there's some pollen. 290 00:19:48,660 --> 00:19:53,960 Of course you have gravity that pulls the pollen down. 291 00:19:53,960 --> 00:19:56,690 And pollen is heavier than water so eventually it 292 00:19:56,690 --> 00:19:59,460 will go down, eventually. 293 00:19:59,460 --> 00:20:01,360 But that only explains the vertical action, 294 00:20:01,360 --> 00:20:02,976 it will only go down. 295 00:20:02,976 --> 00:20:04,850 But in fact, if you observe what's happening, 296 00:20:04,850 --> 00:20:07,500 it just bounces back and forth crazily 297 00:20:07,500 --> 00:20:12,100 until it finally reaches down the bottom of your cup. 298 00:20:12,100 --> 00:20:15,460 And this motion, if you just look 299 00:20:15,460 --> 00:20:17,760 at a two-dimension picture, it's a Brownian motion 300 00:20:17,760 --> 00:20:18,810 to the left and right. 301 00:20:23,300 --> 00:20:29,522 So it moves as according to Brownian motion. 302 00:20:35,370 --> 00:20:38,820 Well, first of all, I should say a little bit more. 303 00:20:38,820 --> 00:20:40,550 What Brown did was he observed it. 304 00:20:40,550 --> 00:20:44,595 He wasn't able to explain the horizontal actions because he 305 00:20:44,595 --> 00:20:47,920 only understood gravity, but then people 306 00:20:47,920 --> 00:20:49,120 tried to explain it. 307 00:20:49,120 --> 00:20:53,960 They suspected that it was the water molecules that 308 00:20:53,960 --> 00:20:58,210 caused this action, but weren't able to really explain it. 309 00:20:58,210 --> 00:21:01,100 But the first person to actually rigorously explain it 310 00:21:01,100 --> 00:21:08,155 was, surprisingly, Einstein, that relativity 311 00:21:08,155 --> 00:21:11,430 guy, that famous guy. 312 00:21:11,430 --> 00:21:13,680 So I was really surprised. 313 00:21:13,680 --> 00:21:16,595 He's really smart, apparently. 314 00:21:19,720 --> 00:21:21,870 And why? 315 00:21:21,870 --> 00:21:23,930 So why will this follow a Brownian motion? 316 00:21:23,930 --> 00:21:25,760 Why is it a reasonable model? 317 00:21:25,760 --> 00:21:30,790 And this gives you a fairly good reason for that. 318 00:21:30,790 --> 00:21:35,020 This description, where it's the limit of simple random walks. 319 00:21:35,020 --> 00:21:37,030 Because if you think about it, what's happening 320 00:21:37,030 --> 00:21:38,488 is there is a big molecule that you 321 00:21:38,488 --> 00:21:42,050 can observe, this big particle. 322 00:21:42,050 --> 00:21:45,030 But inside there's tiny water molecules, 323 00:21:45,030 --> 00:21:49,970 tiny ones that don't really see, but it's filling the space. 324 00:21:49,970 --> 00:21:51,750 And they're just moving crazily. 325 00:21:51,750 --> 00:21:54,660 Even though the water looks still, what's really happening 326 00:21:54,660 --> 00:21:56,990 is these water molecules are just 327 00:21:56,990 --> 00:22:00,300 crazily moving inside the cup. 328 00:22:00,300 --> 00:22:07,170 And each water molecule, when they collide with the pollen, 329 00:22:07,170 --> 00:22:10,510 it will change the action of the pollen a little bit, 330 00:22:10,510 --> 00:22:13,170 by a tiny amount. 331 00:22:13,170 --> 00:22:18,770 So if you think about each collision as one step, 332 00:22:18,770 --> 00:22:23,070 then each step will either push this pollen to the left 333 00:22:23,070 --> 00:22:26,850 or to the right by some tiny amount. 334 00:22:26,850 --> 00:22:28,810 And it just accumulates over time. 335 00:22:28,810 --> 00:22:31,244 So you're looking at a very, very fine time scale. 336 00:22:31,244 --> 00:22:33,160 Of course, the times will differ a little bit, 337 00:22:33,160 --> 00:22:35,770 but let's just forget about it, assume that it's uniform. 338 00:22:35,770 --> 00:22:38,360 And at each time it just pushes to the left or right 339 00:22:38,360 --> 00:22:40,030 by a tiny amount. 340 00:22:40,030 --> 00:22:43,360 And you look at what accumulates, as we saw, 341 00:22:43,360 --> 00:22:47,050 the limit of a simple random walk is a Brownian motion. 342 00:22:47,050 --> 00:22:49,130 And that tells you why we should get something 343 00:22:49,130 --> 00:22:50,550 like a Brownian motion here. 344 00:22:54,500 --> 00:23:03,515 So the action of pollen particle is determined 345 00:23:03,515 --> 00:23:07,536 by infinitesimal-- I don't know if that's the right word-- 346 00:23:07,536 --> 00:23:18,250 but just, quote, "infinitesimal" interactions 347 00:23:18,250 --> 00:23:19,310 with water molecules. 348 00:23:25,680 --> 00:23:29,670 That explains, at least intuitively, 349 00:23:29,670 --> 00:23:31,430 why it follows Brownian motion. 350 00:23:35,610 --> 00:23:45,932 And the second example is-- any questions here-- 351 00:23:45,932 --> 00:23:47,045 is stock prices. 352 00:23:51,806 --> 00:23:55,670 At least to give you some reasonable reason, some reason 353 00:23:55,670 --> 00:24:03,040 that Brownian motion is not so bad a model for stock prices. 354 00:24:03,040 --> 00:24:11,980 Because if you look at a stock price, S, 355 00:24:11,980 --> 00:24:14,700 the price is determined by buying actions or selling 356 00:24:14,700 --> 00:24:16,100 actions. 357 00:24:16,100 --> 00:24:18,220 Each action kind of pulls down the price 358 00:24:18,220 --> 00:24:20,180 or pulls up the price, pushes down the price 359 00:24:20,180 --> 00:24:21,660 or pulls up the price. 360 00:24:21,660 --> 00:24:26,110 And if you look at very, very tiny scales, what's happening 361 00:24:26,110 --> 00:24:29,690 is at a very tiny amount they will go up or down. 362 00:24:29,690 --> 00:24:32,230 Of course, it doesn't go up and down by a uniform amount, 363 00:24:32,230 --> 00:24:34,640 but just forget about that technicality. 364 00:24:34,640 --> 00:24:37,640 It just bounces back and forth infinitely often, 365 00:24:37,640 --> 00:24:40,045 and then you're taking these tiny scales 366 00:24:40,045 --> 00:24:42,590 to be tinier, so very, very small. 367 00:24:42,590 --> 00:24:45,185 So again, you see this limiting picture. 368 00:24:45,185 --> 00:24:47,060 Where you have a discrete-- something looking 369 00:24:47,060 --> 00:24:51,540 like a random walk, and you take t as infinity. 370 00:24:51,540 --> 00:24:55,830 So if that's the only action causing the price, 371 00:24:55,830 --> 00:25:00,020 then Brownian motion will be the right model to use. 372 00:25:00,020 --> 00:25:02,930 Of course, there are many other things involved 373 00:25:02,930 --> 00:25:06,900 which makes this deviate from Brownian motion, 374 00:25:06,900 --> 00:25:09,850 but at least, theoretically, it's a good starting point. 375 00:25:15,270 --> 00:25:16,155 Any questions? 376 00:25:19,420 --> 00:25:20,020 OK. 377 00:25:20,020 --> 00:25:21,145 So you saw Brownian motion. 378 00:25:21,145 --> 00:25:23,750 You already know that it's used in the financial market a lot. 379 00:25:23,750 --> 00:25:27,130 It's also being used in science and other fields like that. 380 00:25:27,130 --> 00:25:31,562 And really big names, like Einstein, is involved. 381 00:25:31,562 --> 00:25:33,770 So it's a really, really important theoretical thing. 382 00:25:37,750 --> 00:25:41,270 Now that you've learned it, it's time to get used to it. 383 00:25:43,890 --> 00:25:45,600 So I'll tell you some properties, 384 00:25:45,600 --> 00:25:48,620 and actually prove a little bit-- just some propositions 385 00:25:48,620 --> 00:25:50,650 to show you some properties. 386 00:25:50,650 --> 00:25:53,970 Some of them are quite surprising if you never 387 00:25:53,970 --> 00:25:56,160 saw it before. 388 00:25:56,160 --> 00:25:56,660 OK. 389 00:25:56,660 --> 00:25:57,826 So here are some properties. 390 00:26:05,990 --> 00:26:17,146 Crosses the x-axis infinitely often, 391 00:26:17,146 --> 00:26:18,270 or I should say the t-axis. 392 00:26:21,390 --> 00:26:25,375 Because you start from 0, it will never go to infinity, 393 00:26:25,375 --> 00:26:27,230 or get to negative infinity. 394 00:26:27,230 --> 00:26:29,330 It will always go balanced positive and negative 395 00:26:29,330 --> 00:26:31,496 infinitely often. 396 00:26:31,496 --> 00:26:42,636 And the second, it does not deviate too much 397 00:26:42,636 --> 00:26:48,748 from t equals y squared. 398 00:26:48,748 --> 00:26:51,380 We'll call this y. 399 00:26:51,380 --> 00:26:52,880 Now, this is a very vague statement. 400 00:26:52,880 --> 00:26:57,554 What I'm trying to say is draw this curve as this. 401 00:27:03,530 --> 00:27:08,430 If you start at time 0, at some time t_0, 402 00:27:08,430 --> 00:27:10,120 the probability distribution here 403 00:27:10,120 --> 00:27:11,860 is given as a normal random variable 404 00:27:11,860 --> 00:27:16,400 with mean 0 and variance t_0. 405 00:27:16,400 --> 00:27:21,309 And because of that, the standard deviation 406 00:27:21,309 --> 00:27:22,100 is square root t_0. 407 00:27:26,740 --> 00:27:30,960 So the typical value will be around the standard deviation. 408 00:27:30,960 --> 00:27:32,030 And it won't deviate. 409 00:27:32,030 --> 00:27:33,720 It can be 100 times this. 410 00:27:33,720 --> 00:27:37,370 It won't really be a million times that or something. 411 00:27:37,370 --> 00:27:42,475 So most likely it will look something like that. 412 00:27:45,325 --> 00:27:48,170 So it plays around this curve a lot, 413 00:27:48,170 --> 00:27:50,195 but it crosses the axis infinitely often. 414 00:27:50,195 --> 00:27:52,962 It goes back and forth. 415 00:27:52,962 --> 00:27:53,830 What else? 416 00:27:53,830 --> 00:27:56,710 The third one is quite really interesting. 417 00:27:56,710 --> 00:27:59,010 It's more theoretical interest, but it also 418 00:27:59,010 --> 00:28:01,510 has real-life implications. 419 00:28:01,510 --> 00:28:12,140 It's not differentiable anywhere. 420 00:28:12,140 --> 00:28:15,500 It's nowhere differentiable. 421 00:28:15,500 --> 00:28:18,140 So this curve, whatever that curve is, 422 00:28:18,140 --> 00:28:21,691 it's a continuous path, but it's nowhere differentiable, really 423 00:28:21,691 --> 00:28:22,190 surprising. 424 00:28:22,190 --> 00:28:24,970 It's hard to imagine even one such path. 425 00:28:24,970 --> 00:28:27,640 What it's saying is if you take one path according 426 00:28:27,640 --> 00:28:30,270 to this probability distribution, 427 00:28:30,270 --> 00:28:32,580 then more than likely you'll obtain a path which 428 00:28:32,580 --> 00:28:33,663 is nowhere differentiable. 429 00:28:36,560 --> 00:28:40,972 That just sounds nice, but why it does it matter? 430 00:28:40,972 --> 00:28:44,860 It matters because we can't use calculus anymore. 431 00:28:53,680 --> 00:28:55,180 Because all the theory of calculus 432 00:28:55,180 --> 00:28:58,360 is based on differentiation. 433 00:28:58,360 --> 00:29:02,210 However, our paths have some nice things, it's universal, 434 00:29:02,210 --> 00:29:05,190 and it appears in very different contexts. 435 00:29:05,190 --> 00:29:07,600 But if you want to do analysis on it, 436 00:29:07,600 --> 00:29:09,500 it's just not differentiable. 437 00:29:09,500 --> 00:29:12,020 So the standard tools of calculus 438 00:29:12,020 --> 00:29:15,270 can't be used here, which is quite unfortunate 439 00:29:15,270 --> 00:29:16,340 if you think about it. 440 00:29:16,340 --> 00:29:19,860 You have this nice model, which can describe many things, 441 00:29:19,860 --> 00:29:21,710 you can't really do analysis on it. 442 00:29:25,052 --> 00:29:26,510 We'll later see that actually there 443 00:29:26,510 --> 00:29:34,780 is a variant, a different calculus that works. 444 00:29:37,640 --> 00:29:40,920 And I'm sure many of you would have heard about it. 445 00:29:40,920 --> 00:29:42,100 It's called Ito's calculus. 446 00:29:48,810 --> 00:29:50,210 So we have this nice object. 447 00:29:50,210 --> 00:29:52,040 Unfortunately, it's not differentiable, 448 00:29:52,040 --> 00:29:54,710 so the standard calculus does not work here. 449 00:29:54,710 --> 00:29:57,640 However, there is a modified version 450 00:29:57,640 --> 00:30:02,110 of calculus called Ito's calculus, which 451 00:30:02,110 --> 00:30:04,310 extends the classical calculus to this setting. 452 00:30:04,310 --> 00:30:06,930 And it's really powerful and it's really cool. 453 00:30:06,930 --> 00:30:10,310 But unfortunately, we don't have that much time to cover it. 454 00:30:10,310 --> 00:30:13,920 I will only be able to tell you really basic properties 455 00:30:13,920 --> 00:30:17,270 and basic computations of it. 456 00:30:17,270 --> 00:30:22,150 And you'll see how this calculus is 457 00:30:22,150 --> 00:30:24,730 being used in the financial world 458 00:30:24,730 --> 00:30:26,330 in the coming-up lectures. 459 00:30:33,080 --> 00:30:34,950 But before going into Ito's calculus, 460 00:30:34,950 --> 00:30:38,220 let's talk about the property of Brownian motion a little bit 461 00:30:38,220 --> 00:30:40,316 because we have to get used to it. 462 00:30:47,760 --> 00:30:52,960 Suppose I'm using it as a model of a stock price. 463 00:30:52,960 --> 00:31:00,190 So I'm using-- use Brownian motion 464 00:31:00,190 --> 00:31:12,390 as a model for stock price-- say, daily stock price. 465 00:31:16,730 --> 00:31:21,570 The market opens at 9:30 AM. 466 00:31:21,570 --> 00:31:24,890 It closes at 4:00 PM. 467 00:31:24,890 --> 00:31:31,228 It starts at some price, and then moves 468 00:31:31,228 --> 00:31:32,602 according to the Brownian motion. 469 00:31:37,690 --> 00:31:43,370 And then you want to obtain the distribution of the min value 470 00:31:43,370 --> 00:31:45,340 and the max value for the stock. 471 00:31:49,510 --> 00:31:52,720 So these are very useful statistics. 472 00:31:52,720 --> 00:31:56,080 So a daily stock price, what will 473 00:31:56,080 --> 00:31:58,610 the minimum and the maximum-- what will 474 00:31:58,610 --> 00:32:01,490 the distribution of those be? 475 00:32:01,490 --> 00:32:02,380 So let's compute it. 476 00:32:02,380 --> 00:32:03,570 We can actually compute it. 477 00:32:10,920 --> 00:32:14,670 What we want to do is-- I'll just compute the maximum. 478 00:32:14,670 --> 00:32:20,166 I want to compute this thing over s smaller 479 00:32:20,166 --> 00:32:23,415 than t of the Brownian motion. 480 00:32:28,250 --> 00:32:33,650 So I define this new process from the Brownian motion, 481 00:32:33,650 --> 00:32:35,630 and I want to compute the distribution 482 00:32:35,630 --> 00:32:39,830 of this new stochastic process. 483 00:32:39,830 --> 00:32:40,930 And here's the theorem. 484 00:32:44,300 --> 00:32:51,160 So for all t, the probability that you 485 00:32:51,160 --> 00:33:04,410 have M(t) greater than a and positive a is equal to 2 times 486 00:33:04,410 --> 00:33:11,620 the probability that you have the Brownian motion greater 487 00:33:11,620 --> 00:33:12,120 than a. 488 00:33:17,274 --> 00:33:18,190 It's quite surprising. 489 00:33:20,710 --> 00:33:22,650 If you just look at this, there's 490 00:33:22,650 --> 00:33:26,290 no reason to expect that such a nice formula should 491 00:33:26,290 --> 00:33:27,050 exist at all. 492 00:33:31,340 --> 00:33:34,640 And notice that maximum is always at least 0, 493 00:33:34,640 --> 00:33:37,390 so we don't have to worry about negative values. 494 00:33:37,390 --> 00:33:38,585 It starts at 0. 495 00:33:41,100 --> 00:33:42,295 How do we prove it? 496 00:33:48,328 --> 00:33:48,828 Proof. 497 00:33:52,650 --> 00:33:53,820 Take this tau. 498 00:33:53,820 --> 00:33:57,350 It's a stopping time, if you remember what it is. 499 00:33:57,350 --> 00:34:09,389 It's a minimum value of t such that the Brownian motion 500 00:34:09,389 --> 00:34:10,630 at time t is equal to a. 501 00:34:13,594 --> 00:34:15,260 That's a complicated way of saying, just 502 00:34:15,260 --> 00:34:17,587 record the first time you hit the line a. 503 00:34:21,403 --> 00:34:24,760 Line a, with some Brownian motion, 504 00:34:24,760 --> 00:34:26,780 and you record this time. 505 00:34:26,780 --> 00:34:28,380 That will be your tau of a. 506 00:34:56,389 --> 00:35:00,610 So now here's some strange thing. 507 00:35:00,610 --> 00:35:54,530 The probability that B(t), B(tau_a), given this-- OK. 508 00:35:54,530 --> 00:36:01,850 So what this is saying is, if you're interested at time t, 509 00:36:01,850 --> 00:36:05,080 if your tau_a happened before time t, 510 00:36:05,080 --> 00:36:10,100 so if your Brownian motion hit the line a before time t, 511 00:36:10,100 --> 00:36:14,280 then afterwards you have the same probability of ending up 512 00:36:14,280 --> 00:36:18,350 above a and ending up below a. 513 00:36:18,350 --> 00:36:21,150 The reason is because you can just reflect the path. 514 00:36:21,150 --> 00:36:24,220 Whatever path that ends over a, you 515 00:36:24,220 --> 00:36:29,450 can reflect it to obtain a path that ends below a. 516 00:36:29,450 --> 00:36:31,475 And by symmetry, you just have this property. 517 00:36:34,520 --> 00:36:36,888 Well, it's not obvious how you'll use this right now. 518 00:36:49,110 --> 00:36:51,540 And then we're almost done. 519 00:36:51,540 --> 00:36:56,840 The probability that maximum at time t is greater than a 520 00:36:56,840 --> 00:37:00,710 that's equal to the probability that you're stopping time 521 00:37:00,710 --> 00:37:02,985 is less than t, just by definition. 522 00:37:06,510 --> 00:37:12,425 And that's equal to the probability that B(t) minus 523 00:37:12,425 --> 00:37:18,940 B(tau_a) is positive given tau a is less than t-- 524 00:37:35,840 --> 00:37:39,350 Because if you know that tau is less than t, 525 00:37:39,350 --> 00:37:41,270 there's only two possible ways. 526 00:37:41,270 --> 00:37:44,780 You can either go up afterwards, or you can go down afterwards. 527 00:37:44,780 --> 00:37:47,410 But these two are the same probability. 528 00:37:47,410 --> 00:37:59,070 What you obtain is 2 times the probability that-- and that's 529 00:37:59,070 --> 00:38:00,920 just equal to 2 times the probability 530 00:38:00,920 --> 00:38:03,620 that B(t) is greater than a. 531 00:38:14,790 --> 00:38:15,500 What happened? 532 00:38:15,500 --> 00:38:16,630 Some magic happened. 533 00:38:16,630 --> 00:38:18,463 First of all, these two are the same because 534 00:38:18,463 --> 00:38:20,230 of this property by symmetry. 535 00:38:20,230 --> 00:38:25,560 Then from here to here, B(tau_a) is always equal to a, as long 536 00:38:25,560 --> 00:38:27,230 as tau_a is less than t. 537 00:38:27,230 --> 00:38:32,110 This is just-- I rewrote this as a, and I got this thing. 538 00:38:32,110 --> 00:38:35,980 And then I can just remove this because if I already 539 00:38:35,980 --> 00:38:42,480 know that tau_a is less than t-- order is reversed. 540 00:38:42,480 --> 00:38:45,850 If I already know that B at time t is greater than a, 541 00:38:45,850 --> 00:38:47,460 then I know that tau is less than t. 542 00:38:47,460 --> 00:38:51,780 Because if you want to reach a because of continuity, 543 00:38:51,780 --> 00:38:55,340 if you want to go over a, you have to reach a at some point. 544 00:38:55,340 --> 00:38:58,790 That means you hit a before time t. 545 00:38:58,790 --> 00:39:03,320 So that event is already inside that event. 546 00:39:03,320 --> 00:39:05,350 And you just get rid of it. 547 00:39:23,542 --> 00:39:27,740 Sorry, all this should be-- something looks weird. 548 00:39:31,853 --> 00:39:32,660 Not conditioned. 549 00:39:39,980 --> 00:39:40,480 OK. 550 00:39:40,480 --> 00:39:41,476 That makes more sense. 551 00:39:46,589 --> 00:39:48,255 Just the intersection of two properties. 552 00:39:52,850 --> 00:39:54,140 Any questions here? 553 00:40:01,190 --> 00:40:04,890 So again, you just want to compute the probability 554 00:40:04,890 --> 00:40:08,760 that the maximum is greater than a at time t. 555 00:40:11,460 --> 00:40:14,050 In other words, just by definition of tau_a, 556 00:40:14,050 --> 00:40:18,310 that's equal to the problem that tau_a is less than t. 557 00:40:18,310 --> 00:40:20,940 And if tau_a is less than t, afterwards, 558 00:40:20,940 --> 00:40:23,470 depending on afterwards what happens, 559 00:40:23,470 --> 00:40:24,960 it increases or decreases. 560 00:40:24,960 --> 00:40:26,450 So there's only two possibilities. 561 00:40:26,450 --> 00:40:29,490 It increases or it decreases. 562 00:40:29,490 --> 00:40:31,470 But these two events have the same probability 563 00:40:31,470 --> 00:40:32,511 because of this property. 564 00:40:35,490 --> 00:40:38,180 Here's a bar and that's an intersection. 565 00:40:38,180 --> 00:40:43,430 But it doesn't matter, because if you have the B of X_1 bar y 566 00:40:43,430 --> 00:40:48,000 equals B of x_2 bar y then probability 567 00:40:48,000 --> 00:40:51,580 of X_1 intersection Y over probability of Y 568 00:40:51,580 --> 00:41:00,110 is equal to-- these two cancel. 569 00:41:00,110 --> 00:41:05,400 So this bar can just be replaced by intersection. 570 00:41:05,400 --> 00:41:07,920 That means these two events have the same probability. 571 00:41:07,920 --> 00:41:09,140 So you can just take one. 572 00:41:09,140 --> 00:41:12,040 What I'm going to take is one that goes above 0. 573 00:41:12,040 --> 00:41:16,007 So after tau_a, it accumulates more value. 574 00:41:16,007 --> 00:41:18,090 And if you rewrite it, what that means is just B_t 575 00:41:18,090 --> 00:41:21,640 is greater than a given that tau_a is less than t. 576 00:41:21,640 --> 00:41:24,920 But now that just became redundant. 577 00:41:24,920 --> 00:41:27,440 Because if you already know that B(t) is greater than a, 578 00:41:27,440 --> 00:41:30,380 tau_a has to be less than t. 579 00:41:30,380 --> 00:41:31,880 And that's just the conclusion. 580 00:41:35,696 --> 00:41:39,910 And it's just some nice result about the maximum 581 00:41:39,910 --> 00:41:41,065 over some time interval. 582 00:41:44,660 --> 00:41:51,425 And actually, I think Peter uses distribution in your lecture, 583 00:41:51,425 --> 00:41:51,925 right? 584 00:41:51,925 --> 00:41:53,656 AUDIENCE: Yes. 585 00:41:53,656 --> 00:42:00,052 [INAUDIBLE] is that the distribution of the max 586 00:42:00,052 --> 00:42:03,988 minus the movement of the Brownian motion. 587 00:42:03,988 --> 00:42:07,520 And use that range of the process as a scaling 588 00:42:07,520 --> 00:42:13,942 for [INAUDIBLE] and get more precise measures of volatility 589 00:42:13,942 --> 00:42:17,888 than just using, say, the close-to-close price 590 00:42:17,888 --> 00:42:18,388 [INAUDIBLE]. 591 00:42:23,064 --> 00:42:23,730 PROFESSOR: Yeah. 592 00:42:32,250 --> 00:42:34,301 That was one property. 593 00:42:34,301 --> 00:43:15,290 And another property is-- and that's what I already told you, 594 00:43:15,290 --> 00:43:16,570 but I'm going to prove this. 595 00:43:16,570 --> 00:43:18,904 So at each time the Brownian motion 596 00:43:18,904 --> 00:43:20,320 is not differentiable at that time 597 00:43:20,320 --> 00:43:23,910 with probability equal to 1. 598 00:43:23,910 --> 00:43:26,520 Well, not very strictly, but I will 599 00:43:26,520 --> 00:43:28,830 use this theorem to prove it. 600 00:43:28,830 --> 00:43:29,330 OK? 601 00:43:41,940 --> 00:43:49,260 Suppose the Brownian motion has a differentiation at time t 602 00:43:49,260 --> 00:43:50,660 and it's equal to a. 603 00:43:59,114 --> 00:44:01,530 Then what you just see is that the Brownian motion at time 604 00:44:01,530 --> 00:44:10,480 t plus epsilon, minus Brownian motion at time t, 605 00:44:10,480 --> 00:44:15,260 has to be less than or equal to epsilon times a. 606 00:44:15,260 --> 00:44:19,435 Not precisely, so I'll say just almost. 607 00:44:23,011 --> 00:44:24,510 Can make it mathematically rigorous. 608 00:44:24,510 --> 00:44:26,410 But what I'm trying to say here is 609 00:44:26,410 --> 00:44:29,250 by-- is it mean value theorem? 610 00:44:29,250 --> 00:44:36,430 So from t to t plus epsilon, you expect to gain a times epsilon. 611 00:44:36,430 --> 00:44:40,540 That's-- OK? 612 00:44:40,540 --> 00:44:43,160 You should have this-- then. 613 00:44:45,770 --> 00:44:46,960 In fact, for all epsilon. 614 00:44:53,690 --> 00:44:57,141 Greater than epsilon prime'. 615 00:44:57,141 --> 00:44:59,250 Let's write it like that. 616 00:44:59,250 --> 00:45:04,960 So in other words, the maximum in this interval, 617 00:45:04,960 --> 00:45:07,470 B(t+epsilon) minus t, this distribution is the same 618 00:45:07,470 --> 00:45:09,880 as the maximum at epsilon prime. 619 00:45:09,880 --> 00:45:14,640 That has to be less than epsilon times A. So 620 00:45:14,640 --> 00:45:18,670 what I'm trying to say is if this differentiable, depending 621 00:45:18,670 --> 00:45:23,430 on the slope, your Brownian motion should have always been 622 00:45:23,430 --> 00:45:29,160 inside this cone from t up to time t plus epsilon. 623 00:45:29,160 --> 00:45:34,150 If you draw this slope, it must have been inside this cone. 624 00:45:34,150 --> 00:45:38,180 I'm trying to say that this cannot happen. 625 00:45:38,180 --> 00:45:40,340 From here to here, it should have passed this line 626 00:45:40,340 --> 00:45:41,640 at some point. 627 00:45:41,640 --> 00:45:42,345 OK? 628 00:45:42,345 --> 00:45:44,220 So to do that I'm looking at the distribution 629 00:45:44,220 --> 00:45:47,560 of the maximum value over this time interval. 630 00:45:47,560 --> 00:45:50,066 And I want to say that it's even greater than that. 631 00:45:50,066 --> 00:45:52,460 So if your maximum is greater than that, 632 00:45:52,460 --> 00:45:55,520 you definitely can't have this control. 633 00:45:55,520 --> 00:46:04,650 So if differentiable, then maximum of epsilon 634 00:46:04,650 --> 00:46:17,595 prime-- the maximum of epsilon, actually, and just compute it. 635 00:46:17,595 --> 00:46:23,520 So the probability that M epsilon is less than epsilon*A 636 00:46:23,520 --> 00:46:27,390 is equal to 2 times the probability of that, 637 00:46:27,390 --> 00:46:33,297 the Brownian motion at epsilon is less than or equal to a. 638 00:46:33,297 --> 00:46:34,505 This has normal distribution. 639 00:46:39,750 --> 00:46:44,440 And if you normalize it to N(0, 1), 640 00:46:44,440 --> 00:46:47,691 divide by the standard deviation so you get the square root 641 00:46:47,691 --> 00:46:51,390 of epsilon A. 642 00:46:51,390 --> 00:46:55,227 As epsilon goes to 0, this goes to 0. 643 00:46:55,227 --> 00:46:56,435 That means this goes to half. 644 00:46:59,037 --> 00:47:00,120 The whole thing goes to 1. 645 00:47:05,760 --> 00:47:06,624 What am I missing? 646 00:47:06,624 --> 00:47:07,540 I did something wrong. 647 00:47:07,540 --> 00:47:10,744 I flipped it. 648 00:47:10,744 --> 00:47:11,410 This is greater. 649 00:47:18,590 --> 00:47:20,880 Now, if you combine it, if it was differentiable, 650 00:47:20,880 --> 00:47:24,120 your maximum should have been less than epsilon*A. 651 00:47:24,120 --> 00:47:26,860 But what we saw here is your maximum is always greater than 652 00:47:26,860 --> 00:47:29,770 that epsilon times A. With probability 1, 653 00:47:29,770 --> 00:47:31,200 you take epsilon goes to 0. 654 00:47:40,980 --> 00:47:41,958 Any questions? 655 00:47:46,380 --> 00:47:48,530 OK. 656 00:47:48,530 --> 00:47:50,750 So those are some interesting things, 657 00:47:50,750 --> 00:47:53,280 properties of Brownian motion that I want to talk about. 658 00:47:53,280 --> 00:47:56,510 I have one final thing, and this one it's 659 00:47:56,510 --> 00:48:00,420 really important theoretically. 660 00:48:00,420 --> 00:48:07,170 And also, it will be the main lemma for Ito's calculus. 661 00:48:07,170 --> 00:48:11,090 So the theorem is called quadratic variation. 662 00:48:19,507 --> 00:48:21,590 And it's something that doesn't happen that often. 663 00:48:29,860 --> 00:49:14,050 So let 0-- let me write it down even more clear. 664 00:49:39,894 --> 00:49:42,390 Now that's something strange. 665 00:49:42,390 --> 00:49:47,220 Let me just first parse it before proving it. 666 00:49:47,220 --> 00:49:50,084 Think about it as just a function, function f. 667 00:49:54,730 --> 00:49:55,720 What is this quantity? 668 00:49:55,720 --> 00:49:59,141 This quantity means that from 0 up to time T, 669 00:49:59,141 --> 00:50:03,240 you chop it up into n pieces. 670 00:50:03,240 --> 00:50:07,810 You get T over n, 2T over n, 3T over n, 671 00:50:07,810 --> 00:50:10,570 and you look at the function. 672 00:50:10,570 --> 00:50:15,350 The difference between each consecutive points, 673 00:50:15,350 --> 00:50:19,340 record these differences, and then square it. 674 00:50:19,340 --> 00:50:21,770 And you sum it as n goes to infinity. 675 00:50:21,770 --> 00:50:26,740 So you take smaller and smaller scales take it to infinity. 676 00:50:26,740 --> 00:50:28,840 What the theorem says is for Brownian motion 677 00:50:28,840 --> 00:50:31,560 this goes to T, the limit. 678 00:50:31,560 --> 00:50:32,875 Why is this something strange? 679 00:50:42,640 --> 00:50:44,500 Assume f is a lot better function. 680 00:50:44,500 --> 00:50:49,220 Assume f is continuously differentiable. 681 00:50:49,220 --> 00:50:52,880 That means it's differentiable, and its differentiation 682 00:50:52,880 --> 00:50:54,880 is continuous. 683 00:50:54,880 --> 00:50:56,109 Derivative is continuous. 684 00:51:01,600 --> 00:51:04,420 Then let's compute the exact same property, 685 00:51:04,420 --> 00:51:05,260 exact same thing. 686 00:51:05,260 --> 00:51:08,770 I'll just call this-- maybe i will be better. 687 00:51:11,510 --> 00:51:20,490 This time t_i and time t_(i-1), then the sum over i of f 688 00:51:20,490 --> 00:51:24,446 at t_(i+1) minus f at t_i. 689 00:51:24,446 --> 00:51:31,330 If you square it, this is at most sum from i equal 1 to n, 690 00:51:31,330 --> 00:51:42,190 f of t_(i+1) minus f of t_i, times-- by mean value theorem-- 691 00:51:42,190 --> 00:51:43,620 f prime of s_i. 692 00:52:02,810 --> 00:52:06,340 So by mean value theorem, there exists a point s_i such that 693 00:52:06,340 --> 00:52:09,740 f(t_(i+1)) minus f(t_i) is equal to f prime s_i, times that. 694 00:52:09,740 --> 00:52:11,300 s_i belongs to that interval. 695 00:52:21,020 --> 00:52:22,500 Yes. 696 00:52:22,500 --> 00:52:24,660 And then you take this term out. 697 00:52:24,660 --> 00:52:32,850 You take the maximum, from 0 up to t, f prime of s squared, 698 00:52:32,850 --> 00:52:39,930 times i equal 1 to n, t_(i+1) minus t_i squared. 699 00:52:39,930 --> 00:52:43,710 This thing is T over n because we chopped it up 700 00:52:43,710 --> 00:52:44,950 into n intervals. 701 00:52:44,950 --> 00:52:47,220 Each consecutive difference is T over n. 702 00:52:47,220 --> 00:52:51,480 If you square it, that's equal to T squared over n squared. 703 00:52:51,480 --> 00:52:55,310 If you had n of them, you get T squared over n. 704 00:52:55,310 --> 00:53:01,320 So you get whatever that maximum is times T squared over n. 705 00:53:01,320 --> 00:53:03,430 If you take n to infinity, that goes to 0. 706 00:53:06,170 --> 00:53:08,160 So if you have a reasonable function, which 707 00:53:08,160 --> 00:53:11,290 is differentiable, this variation-- 708 00:53:11,290 --> 00:53:14,930 this is called the quadratic variation-- quadratic variation 709 00:53:14,930 --> 00:53:16,320 is 0. 710 00:53:16,320 --> 00:53:19,800 So all these classical functions that you've been studying 711 00:53:19,800 --> 00:53:22,610 will not even have this quadratic variation. 712 00:53:22,610 --> 00:53:24,800 But for Brownian motion, what's happening 713 00:53:24,800 --> 00:53:28,010 is it just bounced back and forth too much. 714 00:53:28,010 --> 00:53:30,270 Even if you scale it smaller and smaller, 715 00:53:30,270 --> 00:53:32,900 the variation is big enough to accumulate. 716 00:53:32,900 --> 00:53:36,460 They won't disappear like if it was a differentiable function. 717 00:53:39,110 --> 00:53:42,550 And that pretty much-- it's a slightly stronger version 718 00:53:42,550 --> 00:53:44,175 than this that it's not differentiable. 719 00:53:47,290 --> 00:53:49,354 We saw that it's not differentiable. 720 00:53:49,354 --> 00:53:50,770 And this a different way of saying 721 00:53:50,770 --> 00:53:51,978 that it's not differentiable. 722 00:53:54,596 --> 00:53:56,310 It has very important implications. 723 00:54:00,530 --> 00:54:06,246 And another way to write it is-- so here's a difference of B, 724 00:54:06,246 --> 00:54:10,060 it's dB squared is equal to dt. 725 00:54:13,340 --> 00:54:15,090 So if you take the differential-- whatever 726 00:54:15,090 --> 00:54:17,650 that means-- if you take the infinitesimal difference 727 00:54:17,650 --> 00:54:21,690 of each side, this part is just dB squared, 728 00:54:21,690 --> 00:54:27,130 the Brownian motion difference squared; this part is d of t. 729 00:54:27,130 --> 00:54:31,662 And that we'll see again. 730 00:54:31,662 --> 00:54:33,620 But before that, let's just prove this theorem. 731 00:54:57,730 --> 00:55:04,360 So we're looking at the sum of B of t_(i+1), minus B of t_i, 732 00:55:04,360 --> 00:55:06,100 squared. 733 00:55:06,100 --> 00:55:10,140 Where t of i is i over n times the time. 734 00:55:14,496 --> 00:55:17,958 From 1 to n-- 0 to n minus 1. 735 00:55:22,248 --> 00:55:22,748 OK. 736 00:55:26,647 --> 00:55:27,980 What's the distribution of this? 737 00:55:30,926 --> 00:55:32,400 AUDIENCE: Normal. 738 00:55:32,400 --> 00:55:40,310 PROFESSOR: Normal, meaning 0, variance t_(i+1) minus t_i. 739 00:55:40,310 --> 00:55:41,455 But that was just T over n. 740 00:55:44,020 --> 00:55:46,720 Is the distribution. 741 00:55:46,720 --> 00:55:48,290 So I'll write it like this. 742 00:55:48,290 --> 00:55:51,666 You sum from i equal 1 to n minus 1, 743 00:55:51,666 --> 00:55:55,770 X_i squared for X_i is normal variable. 744 00:56:07,306 --> 00:56:07,806 OK? 745 00:56:13,190 --> 00:56:15,550 And what's the expectation of X_i squared? 746 00:56:19,674 --> 00:56:21,500 It's T squared over n squared. 747 00:56:27,590 --> 00:56:28,460 OK. 748 00:56:28,460 --> 00:56:31,034 So maybe it's better to write it like this. 749 00:56:31,034 --> 00:56:34,940 So I'll just write it again-- the sum from i equals 0 to n 750 00:56:34,940 --> 00:56:39,720 minus 1 of random variables Y_i, such that expectation of Y_i-- 751 00:56:42,748 --> 00:56:43,664 AUDIENCE: [INAUDIBLE]. 752 00:56:48,935 --> 00:56:52,040 PROFESSOR: Did I make a mistake somewhere? 753 00:56:52,040 --> 00:56:55,549 AUDIENCE: The expected value of X_i squared is the variance. 754 00:56:55,549 --> 00:56:56,590 PROFESSOR: It's T over n. 755 00:56:56,590 --> 00:56:57,920 Oh, yeah, you're right. 756 00:57:00,920 --> 00:57:01,530 Thank you. 757 00:57:10,590 --> 00:57:12,230 OK. 758 00:57:12,230 --> 00:57:17,666 So divide by n and multiply by n. 759 00:57:21,240 --> 00:57:21,820 What is this? 760 00:57:21,820 --> 00:57:23,217 What will this go to? 761 00:57:33,654 --> 00:57:36,139 AUDIENCE: [INAUDIBLE]. 762 00:57:36,139 --> 00:57:38,140 PROFESSOR: No. 763 00:57:38,140 --> 00:57:39,790 Remember strong law of large numbers. 764 00:57:42,539 --> 00:57:44,080 You have a bunch of random variables, 765 00:57:44,080 --> 00:57:45,496 which are independent, identically 766 00:57:45,496 --> 00:57:48,896 distributed, and mean T over n. 767 00:57:48,896 --> 00:57:52,960 You sum n of them and divide by n. 768 00:57:52,960 --> 00:57:55,860 You know that it just converges to T over 769 00:57:55,860 --> 00:57:58,240 n, just this one number. 770 00:57:58,240 --> 00:58:02,240 It doesn't-- it's a distribution, 771 00:58:02,240 --> 00:58:05,630 but most of the time it's just T over n. 772 00:58:05,630 --> 00:58:06,590 OK? 773 00:58:06,590 --> 00:58:12,540 If you take-- that's equal to T, because these 774 00:58:12,540 --> 00:58:14,180 are random variables accumulating 775 00:58:14,180 --> 00:58:16,210 these squared terms. 776 00:58:16,210 --> 00:58:17,200 That's what's happened. 777 00:58:17,200 --> 00:58:21,880 Just a nice application of strong law of large numbers, 778 00:58:21,880 --> 00:58:24,080 or just law of large numbers. 779 00:58:24,080 --> 00:58:25,574 To be precise, you'll have to use 780 00:58:25,574 --> 00:58:26,740 strong law of large numbers. 781 00:58:42,806 --> 00:58:43,306 OK. 782 00:58:46,290 --> 00:58:48,650 So I think that's enough for Brownian motion. 783 00:58:54,669 --> 00:58:55,460 And final question? 784 00:58:58,418 --> 00:58:58,918 OK. 785 00:59:02,880 --> 00:59:03,920 Now, let's move on-- 786 00:59:03,920 --> 00:59:05,086 AUDIENCE: I have a question. 787 00:59:05,086 --> 00:59:05,808 PROFESSOR: Yes. 788 00:59:05,808 --> 00:59:10,056 AUDIENCE: So this [INAUDIBLE], is it 789 00:59:10,056 --> 00:59:12,667 for all Brownian motions B? 790 00:59:12,667 --> 00:59:13,500 PROFESSOR: Oh, yeah. 791 00:59:13,500 --> 00:59:15,460 That's a good question. 792 00:59:15,460 --> 00:59:17,252 This is what happens with probability one. 793 00:59:17,252 --> 00:59:20,282 So always-- I'll just say always. 794 00:59:20,282 --> 00:59:21,490 It's not a very strict sense. 795 00:59:21,490 --> 00:59:24,180 But if you take one path according to the Brownian 796 00:59:24,180 --> 00:59:28,950 motion, in that path you'll have this. 797 00:59:28,950 --> 00:59:31,730 No matter what path you get, it always happens. 798 00:59:34,496 --> 00:59:35,817 AUDIENCE: With probability one. 799 00:59:35,817 --> 00:59:37,150 PROFESSOR: With probability one. 800 00:59:37,150 --> 00:59:41,516 So there's a hiding statement-- with probability. 801 00:59:45,970 --> 00:59:49,500 And you'll see why you need this with probability one 802 00:59:49,500 --> 00:59:52,776 is because we're using this probability statement here. 803 00:59:56,060 --> 01:00:00,465 But for all practical means, like with probability one 804 01:00:00,465 --> 01:00:01,260 just means always. 805 01:00:11,000 --> 01:00:13,520 Now, I want to motivate Ito's calculus. 806 01:00:19,760 --> 01:00:21,330 First of all, this. 807 01:00:24,620 --> 01:00:28,080 So now, I was saying that Brownian motion, at least, 808 01:00:28,080 --> 01:00:32,660 is not so bad a model for stock prices. 809 01:00:32,660 --> 01:00:35,080 But if you remember what I said before, 810 01:00:35,080 --> 01:00:37,190 and what people are actually doing, 811 01:00:37,190 --> 01:00:39,380 a better way to describe it is instead 812 01:00:39,380 --> 01:00:42,970 of the differences being a normal distribution, what 813 01:00:42,970 --> 01:00:45,580 we want is the percentile difference. 814 01:00:45,580 --> 01:01:02,190 So for stock prices we want the percentile difference 815 01:01:02,190 --> 01:01:03,535 to be normally distributed. 816 01:01:11,930 --> 01:01:15,150 In other words, you want to find the distribution of S_t 817 01:01:15,150 --> 01:01:24,230 such that the difference of S_t divided by S_t 818 01:01:24,230 --> 01:01:25,500 is a normal distribution. 819 01:01:25,500 --> 01:01:27,250 So it's like a Brownian motion. 820 01:01:29,770 --> 01:01:31,607 That's the differential equation for it. 821 01:01:41,560 --> 01:01:45,520 So the percentile difference follows Brownian motion. 822 01:01:45,520 --> 01:01:46,572 That's what it's saying. 823 01:01:49,740 --> 01:01:54,259 Question, is S_t equal to e sub B_t? 824 01:01:59,580 --> 01:02:05,020 Because in classical calculus this is not a very absurd thing 825 01:02:05,020 --> 01:02:05,840 to say. 826 01:02:05,840 --> 01:02:08,800 If you differentiate each side, what you get is dS_t 827 01:02:08,800 --> 01:02:13,110 equals e to the B_t, times dB_t. 828 01:02:13,110 --> 01:02:15,140 That's S_t times dB_t. 829 01:02:18,020 --> 01:02:20,080 It doesn't look that wrong. 830 01:02:20,080 --> 01:02:24,710 Actually, it looks right, but it's wrong. 831 01:02:24,710 --> 01:02:28,710 For reasons that you don't know yet, OK? 832 01:02:28,710 --> 01:02:33,740 So this is wrong and you'll see why. 833 01:02:33,740 --> 01:02:36,650 First of all, Brownian motion is not differentiable. 834 01:02:36,650 --> 01:02:38,445 So what does it even mean to say that? 835 01:02:42,630 --> 01:02:46,740 And then that means if you want to solve this equation, 836 01:02:46,740 --> 01:02:49,950 or in other words, if you want to model this thing, 837 01:02:49,950 --> 01:02:51,265 you need something else. 838 01:03:01,840 --> 01:03:04,878 And that's where Ito's calculus comes in. 839 01:03:22,970 --> 01:03:24,972 OK. 840 01:03:24,972 --> 01:03:26,651 I'll try not to rush too much. 841 01:03:29,420 --> 01:03:48,700 So suppose-- now we're talking about Ito's calculus-- 842 01:03:48,700 --> 01:03:49,810 you want to compute. 843 01:04:08,990 --> 01:04:10,380 So here is a motivation. 844 01:04:10,380 --> 01:04:12,240 You have a function f. 845 01:04:12,240 --> 01:04:15,340 I will call it a very smooth function f. 846 01:04:15,340 --> 01:04:16,800 Just think about the best function 847 01:04:16,800 --> 01:04:19,420 you can imagine, like an exponential function. 848 01:04:19,420 --> 01:04:23,230 Then you have a Brownian motion, and then you 849 01:04:23,230 --> 01:04:24,220 apply this function. 850 01:04:24,220 --> 01:04:25,886 As an input, you put the Brownian motion 851 01:04:25,886 --> 01:04:27,240 inside the input. 852 01:04:27,240 --> 01:04:29,070 And you want to estimate the outcome. 853 01:04:32,990 --> 01:04:34,795 More precisely, you want to estimate 854 01:04:34,795 --> 01:04:35,878 infinitesimal differences. 855 01:04:43,360 --> 01:04:45,260 Why will we want to do that? 856 01:04:45,260 --> 01:04:49,521 For example, f can be the price of an option. 857 01:04:49,521 --> 01:04:53,090 More precisely, let f be this thing. 858 01:04:57,770 --> 01:04:58,270 OK. 859 01:04:58,270 --> 01:04:59,020 You have some s_0. 860 01:05:02,190 --> 01:05:04,910 Up to s_0, the value of f is equal to 0. 861 01:05:04,910 --> 01:05:10,520 After s_0, it's just a line with slope 1. 862 01:05:10,520 --> 01:05:14,590 Then f of Brownian motion is just 863 01:05:14,590 --> 01:05:19,510 the price exercise-- what is it-- value of the option 864 01:05:19,510 --> 01:05:21,095 at the expiration. 865 01:05:23,600 --> 01:05:25,660 T is the expiration time. 866 01:05:28,750 --> 01:05:29,680 It's a call option. 867 01:05:33,592 --> 01:05:35,059 That's the call option. 868 01:05:42,400 --> 01:05:46,830 So if your stock at time T goes over s_0, you make that much. 869 01:05:46,830 --> 01:05:50,210 If it's below s_0, you'll lose that much. 870 01:05:50,210 --> 01:05:52,635 More precisely, you have to put it below like that. 871 01:05:52,635 --> 01:05:56,864 Let's just do it like that. 872 01:05:56,864 --> 01:05:58,340 And it looks like that. 873 01:06:02,770 --> 01:06:04,860 So that's like a financial derivative. 874 01:06:04,860 --> 01:06:06,700 You have an underlying stock and then 875 01:06:06,700 --> 01:06:08,780 some function applies to it. 876 01:06:08,780 --> 01:06:12,130 And then what you have, the financial asset you have, 877 01:06:12,130 --> 01:06:14,135 actually can be described as this function. 878 01:06:14,135 --> 01:06:17,190 A function of an underlying stock, that's 879 01:06:17,190 --> 01:06:21,030 called financial derivatives. 880 01:06:21,030 --> 01:06:22,550 And then in the mathematical world, 881 01:06:22,550 --> 01:06:27,574 it's just a function applied to the underlying financial asset. 882 01:06:27,574 --> 01:06:29,240 And then, of course, what you want to do 883 01:06:29,240 --> 01:06:31,280 is understand the difference of the value, 884 01:06:31,280 --> 01:06:36,500 in terms of the difference of the underlying asset. 885 01:06:36,500 --> 01:06:40,700 If B_t was a very nice function as well. 886 01:06:40,700 --> 01:06:50,790 If B_t was differentiable, then the classical world calculus 887 01:06:50,790 --> 01:06:59,085 tells us that d of f is equal to d of B_t over d of t times dt. 888 01:07:04,530 --> 01:07:05,130 Yes. 889 01:07:05,130 --> 01:07:08,740 So if you can differentiate it over the time difference, 890 01:07:08,740 --> 01:07:09,860 over a small time scale. 891 01:07:09,860 --> 01:07:13,920 All we have to do is understand the differentiation. 892 01:07:13,920 --> 01:07:16,530 Unfortunately, we can't do that. 893 01:07:16,530 --> 01:07:21,530 We cannot do this. 894 01:07:21,530 --> 01:07:23,990 Because we don't know what-- we don't even 895 01:07:23,990 --> 01:07:25,134 have this differentiation. 896 01:07:30,051 --> 01:07:30,550 OK. 897 01:07:34,880 --> 01:07:39,590 Try one, take one failed, take two. 898 01:07:39,590 --> 01:07:42,220 Second try, OK? 899 01:07:42,220 --> 01:07:44,230 This is not differentiable, but still I 900 01:07:44,230 --> 01:07:46,830 understand the minuscule difference of dB_t. 901 01:07:46,830 --> 01:07:51,920 So what about this? 902 01:07:51,920 --> 01:07:55,210 df-- maybe I didn't write something, 903 01:07:55,210 --> 01:08:08,550 f prime-- is equal to just dB_t of f prime. 904 01:08:17,708 --> 01:08:19,180 OK? 905 01:08:19,180 --> 01:08:21,370 What is this? 906 01:08:21,370 --> 01:08:24,899 We can't differentiate Brownian motion, 907 01:08:24,899 --> 01:08:28,040 but still we understand the minuscule and infinitesimal 908 01:08:28,040 --> 01:08:30,630 difference of the Brownian motion. 909 01:08:30,630 --> 01:08:34,140 So I just gave up trying to compute the differentiation. 910 01:08:34,140 --> 01:08:38,060 But instead, I'm going to just compute how much the Brownian 911 01:08:38,060 --> 01:08:44,040 motion changed over this small time scale, this difference, 912 01:08:44,040 --> 01:08:47,029 and describe the change of our function 913 01:08:47,029 --> 01:08:49,479 in terms of the differentiation of our function f. 914 01:08:49,479 --> 01:08:53,119 f is a very good function, so it's differentiable. 915 01:08:53,119 --> 01:08:53,785 So we know this. 916 01:08:53,785 --> 01:08:55,990 This is computable. 917 01:08:55,990 --> 01:08:58,670 This is computable. 918 01:08:58,670 --> 01:09:02,359 It's the difference of Brownian motion over a very small time 919 01:09:02,359 --> 01:09:04,580 scale. 920 01:09:04,580 --> 01:09:07,029 So that at least now is reasonable. 921 01:09:07,029 --> 01:09:08,040 We can expect it. 922 01:09:08,040 --> 01:09:10,399 It might be true. 923 01:09:10,399 --> 01:09:12,104 Here, it didn't make sense at all. 924 01:09:12,104 --> 01:09:16,470 Here, it at least make sense, but it's wrong. 925 01:09:16,470 --> 01:09:19,284 And why is it wrong? 926 01:09:19,284 --> 01:09:21,529 It's precisely because of this. 927 01:09:24,590 --> 01:09:33,260 The reason it's wrong, the reason it is not valid 928 01:09:33,260 --> 01:09:46,190 is because of the fact dB squared equals dt. 929 01:09:46,190 --> 01:09:51,930 And let's see how this comes into play, this factor. 930 01:09:51,930 --> 01:09:55,390 I think that will be the last thing that we'll cover today. 931 01:10:03,190 --> 01:10:03,840 OK. 932 01:10:03,840 --> 01:10:06,500 So if you remember where you got this formula from, 933 01:10:06,500 --> 01:10:07,840 you probably won't remember. 934 01:10:07,840 --> 01:10:12,640 But from calculus, this follows from Taylor's expansion. 935 01:10:12,640 --> 01:10:21,310 f of t plus x, I'll say, is equal to f of t plus 936 01:10:21,310 --> 01:10:28,000 f prime of t times x, plus f double prime of t over 2, 937 01:10:28,000 --> 01:10:42,000 times x squared plus-- over 3 factorial x cubed plus-- df is 938 01:10:42,000 --> 01:10:43,411 just this difference. 939 01:10:48,710 --> 01:10:50,690 Over a very small time increase, we 940 01:10:50,690 --> 01:10:53,395 want to understand the difference of the function. 941 01:10:53,395 --> 01:10:55,698 That's equal to f prime t times x. 942 01:11:02,000 --> 01:11:03,530 OK. 943 01:11:03,530 --> 01:11:08,950 In classical calculus we were able to ignore all these terms. 944 01:11:12,630 --> 01:11:24,924 So in the classical world f(t+x) minus f(t) was about f prime t 945 01:11:24,924 --> 01:11:27,890 times x. 946 01:11:27,890 --> 01:11:31,730 And that's precisely this formula. 947 01:11:31,730 --> 01:11:33,765 But if you use Brownian motion here-- so what 948 01:11:33,765 --> 01:11:38,000 I'm trying to say is if B at some time t plus x, 949 01:11:38,000 --> 01:11:42,350 minus Brownian motion B at time t, 950 01:11:42,350 --> 01:11:44,360 then let's just write down the Taylor formula. 951 01:11:44,360 --> 01:11:48,840 We get f prime at B_t. 952 01:11:48,840 --> 01:11:55,010 x will be this difference, B at t plus x minus B at t. 953 01:11:55,010 --> 01:11:58,980 That's like the difference in B_t. 954 01:11:58,980 --> 01:12:02,200 So up to this much we see this formula. 955 01:12:02,200 --> 01:12:08,010 And the next term, we get the second derivative 956 01:12:08,010 --> 01:12:12,850 of this function over 2 and x squared, x 957 01:12:12,850 --> 01:12:15,070 plus this difference. 958 01:12:15,070 --> 01:12:19,870 So what we get is dB_t squared. 959 01:12:19,870 --> 01:12:22,570 OK? 960 01:12:22,570 --> 01:12:26,370 But as you saw, this is no longer ignorable. 961 01:12:26,370 --> 01:12:34,110 That is like a dt, as we deduced. 962 01:12:34,110 --> 01:12:37,210 And that comes into play. 963 01:12:37,210 --> 01:12:46,090 So the correct-- then by Taylor expansion, the right way 964 01:12:46,090 --> 01:12:54,020 to do it is df is equal to the first derivative term, dB_t, 965 01:12:54,020 --> 01:13:01,168 plus the second derivative term, double prime over 2 dt. 966 01:13:06,086 --> 01:13:07,210 This is called Ito's lemma. 967 01:13:15,812 --> 01:13:17,520 And now let's say if you want to remember 968 01:13:17,520 --> 01:13:20,251 one thing from the math part, try to make it this one. 969 01:13:25,130 --> 01:13:26,080 This had great impact. 970 01:13:31,330 --> 01:13:34,010 If you follow the logic it makes sense. 971 01:13:39,777 --> 01:13:44,850 It's really amazing how somebody came up with for the first time 972 01:13:44,850 --> 01:13:48,250 because it all makes sense. 973 01:13:48,250 --> 01:13:52,950 It all fits together if you think about it for a long time. 974 01:13:52,950 --> 01:13:58,440 But actually, I once saw that Ito's lemma 975 01:13:58,440 --> 01:14:01,790 is one of the most cited lemmas, like most cited paper. 976 01:14:01,790 --> 01:14:05,030 The paper that's containing this thing. 977 01:14:05,030 --> 01:14:06,575 Because people think it's nontrivial. 978 01:14:09,420 --> 01:14:11,110 Of course, there are facts that are 979 01:14:11,110 --> 01:14:13,160 being used more than this, classical facts, 980 01:14:13,160 --> 01:14:15,590 like trigonometric functions, exponential functions. 981 01:14:15,590 --> 01:14:17,732 They are being used a lot more than this, 982 01:14:17,732 --> 01:14:19,940 but people think that's trivial so they don't cite it 983 01:14:19,940 --> 01:14:22,430 in their research and paper. 984 01:14:22,430 --> 01:14:25,280 But this, people respect the result. 985 01:14:25,280 --> 01:14:27,390 It's a highly nontrivial result. 986 01:14:27,390 --> 01:14:31,330 And it's really amazing how just by adding this term, 987 01:14:31,330 --> 01:14:37,170 all this theory of calculus all now fit together. 988 01:14:37,170 --> 01:14:41,300 Without this-- maybe it's a too strong statement-- 989 01:14:41,300 --> 01:14:47,270 but really Brownian motion becomes much more rich 990 01:14:47,270 --> 01:14:48,400 because of this fact. 991 01:14:48,400 --> 01:14:49,796 Now we can do calculus with it. 992 01:14:53,540 --> 01:14:56,205 So there's two things to remember. 993 01:14:56,205 --> 01:14:58,705 Well, if you want to remember one thing, that's Ito's lemma. 994 01:14:58,705 --> 01:15:00,163 If you want to remember two things, 995 01:15:00,163 --> 01:15:04,803 it's just quadratic variation, dB_t squared is equal to dt. 996 01:15:08,590 --> 01:15:12,130 And I remember that's exactly because B_t 997 01:15:12,130 --> 01:15:16,150 is like a normal variable with 0, t. 998 01:15:16,150 --> 01:15:19,343 And time scale-- B_t is like a normal random variable 0, t. 999 01:15:19,343 --> 01:15:22,120 dB_t squared is like the variance of it. 1000 01:15:22,120 --> 01:15:25,760 So it's t, and if you differentiate it, you get dt. 1001 01:15:25,760 --> 01:15:27,605 That was exactly how we computed it. 1002 01:15:30,400 --> 01:15:33,240 So, yeah, I'll just quickly go over it again next time 1003 01:15:33,240 --> 01:15:36,300 just to try to make it stick in to your head. 1004 01:15:36,300 --> 01:15:38,430 But please, think about it. 1005 01:15:38,430 --> 01:15:40,980 This is really cool stuff. 1006 01:15:40,980 --> 01:15:43,080 Of course, because of that computation, 1007 01:15:43,080 --> 01:15:46,290 calculus using Brownian motion becomes a lot more complicated. 1008 01:15:50,310 --> 01:15:53,370 Anyway, so I'll see you on Thursday. 1009 01:15:53,370 --> 01:15:56,770 Any last minute questions? 1010 01:15:56,770 --> 01:15:57,947 Great.