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the (634), and (201), for (145), this (127), that (102), with (90), are (85), probability (79), which (71), can (70), not (61), barrier (51), space (51), #random (44), will (42), such (40), time (40), new (39), times (37), then (36), discrete (35), stochastic (33), all (33), measurable (32), where (32), sigma (32), has (31), process (31), have (30), opens (30), window (30), distribution (30), set (30), these (29), zero (29), concentration (27), brownian (26), value (26), over (25), there (25), motion (25), function (24), rademacher (23), from (23), values (23), more (22), bounds (22), email (21), independent (21), only (21), post (21), variables (21), measure (21), may (20), reddit (20), print (20), continuous (20), also (20), one (20), volatility (20), some (20), first (20), finite (20), about (19), processes (19), being (19), than (19), but (19), consider (19), each (19), same (19), optimal (19), theory (18), any (18), algebra (18), sets (17), general (17), positive (17), density 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Text of the page (random words):
andomly chosen finite sequences of coefficients very quickly converges on the optimal values as soon as we randomly select values close to those given by lemma 2 we obtain the exact bounds and any further simulations only serve to verify that these hold running additional simulations with coefficients chosen randomly in the neighbourhood of points where the bound is attained and near certain critical values where the bound looks close to being broken further strengthens the belief that they are indeed optimal at least this is how i originally found them before asking the mathoverflow question although this is still far from a proof concentration bounds figure 2 optimal concentration bounds continue reading rademacher concentration inequalities print opens in new window print tweet share on reddit opens in new window reddit email a link to a friend opens in new window email like loading george lowther probability theory leave a comment 8 october 23 12 october 23 non measurable sets probability and measure theory relies on the concept of measurable sets on the real numbers ℝ in particular there are several different sigma algebras which are commonly used and a set is said to be measurable if it lies in the one under consideration probabilities and measures are only defined for events lying in a specific sigma algebra so it is essential to know if sets are measurable fortunately most simply constructed events will indeed be measurable but this is not always the case in fact once we start working with more complex setups such as continuous time stochastic processes observed at random times non measurable sets occur more commonly than might be expected to avoid such issues it is usual to enlarge the underlying sigma algebra defining a probability space as much as possible the borel sets form the smallest sigma algebra containing the open sets or equivalently containing all intervals this is denoted as which i will also shorten to an explicit construction of a non borel set was given by lusin in 1927 every irrational real number can be expressed uniquely as a continued fraction where a 0 is an integer and a i are positive integers for i 1 lusin considered the set of irrationals whose continued fraction coefficients contain a subsequence a k 1 a k 2 such that each term is a divisor of the subsequent term other examples can be given along similar lines to lusin s every real number has a binary expansion where a 0 is an integer and a i is in 0 1 for each i 1 consider the set of reals having a binary expansion for which there is an infinite sequence of positive integers k 1 k 2 each term strictly dividing the next such that a k i 1 i will give proofs that these examples are non borel in this post there is a general method of enlarging sigma algebras known as the completion consider a measure μ defined on a measurable space consisting of sigma algebra on set x the completion consists of all subsets s x which can be sandwiched between sets in whose difference has zero measure that is a s b for with μ b a 0 it can be shown that is a sigma algebra containing and μ uniquely extends to a measure on this by taking μ s μ a μ b for s a b as above the lebesgue measure λ is uniquely defined on the borel sets by specifiying its value on intervals as λ a b b a the completion is the lebesgue sigma algebra which i will denote by usually when saying that a subset of the reals is measurable without further qualification it is understood to mean that it is in the non borel set constructed by lusin can be shown to be measurable in fact its complement has zero lebesgue measure while the lebesgue measure extends uniquely to this is not true for for all measures defined on the borel sigma algebra in particular it will not be true for singular measures which assign a positive value to some lebesgue null sets an example is the uniform probability measure or haar measure on the cantor middle thirds set c this has zero lebesgue measure so every subset of c is in but the uniform measure on c cannot be extended uniquely to all subsets for this reason universal completions are often used for a measurable space the universal completion consists of the subsets of x which lie in the completion of with respect to every possible sigma finite measure the intersection is taken over all sigma finite measures μ on it is enough to take the intersection over finite or even probability measures since every sigma finite measure is equivalent to such the universal completion is a bit tricky to understand in an explicit way but by construction all sigma finite measures on a sigma algebra extend uniquely to its universal completion it can be shown that lusin s example of a non borel set does lie in the universal completion finally the power set consisting of all subsets of a set x is a sigma algebra for uncountable sets such as the reals this is often too large to be of use and measures cannot be extended in any unique way however we have four common sigma algebras of the real numbers 1 in this post i show that each of these inclusions is strict that is there are subsets of the reals which are not lebesgue measurable there are lebesgue sets which are not in the universal completion and there are sets in which are not borel lusin s construction is an example of the latter the strictness of the other two inclusions does depend crucially on the axiom of choice so unlike lusin s set the examples demonstrating that these are strict are not explicit continue reading non measurable sets print opens in new window print tweet share on reddit opens in new window reddit email a link to a friend opens in new window email like loading george lowther probability theory 2 comments 16 july 23 16 july 23 model independent discrete barrier adjustments i continue the investigation of discrete barrier approximations started in an earlier post the idea is to find good approximations to a continuous barrier condition while only sampling the process at a discrete set of times the difference now is that i will look at model independent methods which do not explicitly depend on properties of the underlying process such as the volatility this will enable much more generic adjustments which can be applied more easily and more widely i point out now the techniques that i will describe here are original research and cannot currently be found in the literature outside of this blog to the best of my knowledge recall that the problem is to compute the expected value of a function of a stochastic process x 1 which depends on whether or not the process crosses a continuous barrier level k in many applications such as with monte carlo simulation we typically only sample x at a discrete set of times 0 t 1 t 2 t n t in that case the continuous barrier is necessarily approximated by a discrete one 2 as we saw this converges slowly as the number n of sampling times increases with the error between this and the limiting continuous barrier 1 only going to zero at rate 1 n a barrier adjustment as described in the earlier post is able to improve this convergence rate if x is a brownian motion with constant drift μ and positive volatility σ then the discrete barrier level k is shifted down by an amount βσ δt where β 0 5826 is a constant and δt t n is the sampling width we are assuming for now that the sampling times are equally spaced as was seen using the shifted barrier level in 2 improves the rate of convergence although we did not theoretically derive the new convergence rate numerical experiment suggests that it is close to 1 n another way to express this is to shift the values of x up 3 then 2 is replaced to use these shifted values which are a proxy for the maximum value of x across each of the intervals t i 1 t i 4 as it is equivalent to shifting the level k down we still obtain the improved rate of convergence this idea is especially useful because of its generality for non equally spaced sampling times the adjustment 3 can still be applied now we just set δt t i t i 1 to be the spacing for the specific time so depends on index i it can also be used for much more general expressions than 1 any function of x which depends on whether or not it crosses a continuous barrier can potentially make use of the adjustment described even if x is an ito process with time dependent drift and volatility 5 the method can be applied now the volatility in 3 is replaced by an average value across the interval t i 1 t i the methods above are very useful but there is a further improvement that can be made ideally we would not have to specify an explicit value of the volatility σ that is it should be model independent there are many reasons why this is desirable suppose that we are running a monte carlo simulation and generate samples of x at the times t i if the simulation only outputs values of x then this is not sufficient to compute 3 so it will be necessary to update the program running the simulation to also output the volatility in some situations this might not be easy for example x could be a complicated function of various other processes and although we could use ito s lemma to compute the volatility of x from the other processes it could be messy in some situations we might not even have access to the volatility or any method of computing it for example the values of x could be computed from historical data we could be looking at the probability of stock prices crossing a level by looking at historical close fixings without access to the complete intra day data in any case a model independent discrete barrier adjustment would make applying it much easier removing volatility dependence how can the volatility term be removed from adjustment 3 one idea is to replace it by an estimator computed from the samples of x such as while this would work at least for a constant volatility process it does not meet the requirements for a general ito process 5 with stochastic volatility using an estimator computed over the whole time interval 0 t may not be a good approximation for the volatility at the time that the barrier is hit a possible way around this is for the adjustment 3 applied at time t i to only depend on a volatility estimator computed from samples near the time this would be possible although it is not clear what is the best way to select these times besides an important point to note is that we do not need a good estimate of the volatility since that is not the goal here as explained in the previous post adjustment 3 works because it corrects for the expected overshoot when the barrier is hit specifically at the first time for which m i k the overshoot is r x t i k if there was no adjustment then the overshoot is positive and the leading order term in the discrete barrier approximation error is proportional to 𝔼 r the positive shift added to x t i is chosen to compensate for this giving zero expected overshoot to leading order and reducing the barrier approximation error the same applies to any similar adjustment as long as there is sufficient freedom in choosing m i then it should be possible to do it in a way that has zero expected overshoot taking this to the extreme it should be possible to compute the adjustment at time t i using only the sampled values x t i 1 and x t i figure 1 barrier overshoot consider adjustments of the form for θ ℝ 2 ℝ by model independence if this adjustment applies to a process x then it should equally apply to the shifted and scaled processes x a and bx for constants a and b 0 equivalently θ satisfies the scaling and translation invariance 6 this restricts the possible forms that θ can take lemma 1 a function θ ℝ 2 ℝ satisfies 6 if and only if for constants p c proof write θ 0 u as the sum of its antisymmetric and symmetric parts by scaling invariance the first term on the right is proportional to u and the second is proportional to u hence for constants p and c using translation invariance as required i will therefore only consider adjustments where the maximum of the process across the interval t i 1 t i is replaced by 7 according to 3 the barrier condition sup t t x t k is replaced by the discrete approximation max i m i k there are various ways in which 7 can be parameterized but this form is quite intuitive the term px t i 1 p x t i 1 is an interpolation of the path of x and c x t i x t i 1 represents a shift proportional to the sample deviation across the interval replacing the σ δt term of the simple shift 3 the purpose of this post is to find values for p and c giving a good adjustment improving convergence of the discrete approximation figure 2 adjusted barrier overshoot the discrete barrier condition m i k given by 7 can be satisfied while the process is below the barrier level giving a negative barrier overshoot r x t i k as in figure 2 as we will see this is vital to obtaining an accurate approximation for the hitting probability continue reading model independent discrete barrier adjustments print opens in new window print tweet share on reddit opens in new window reddit email a link to a friend opens in new window email like loading george lowther stochastic calculus leave a comment 9 july 23 12 july 23 discrete barrier approximations it is quite common to consider functions of real time stochastic process which depend on whether or not it crosses a specified barrier level k this can involve computing expectations involving a real valued process x of the form 1 for a positive time t and function f ℝ ℝ i am using the notation 𝔼 a s to denote the expectation of random variable a restricted to event s or 𝔼 a 1 s one example is computing prices of financial derivatives such as barrier options where t represents the expiration time and f is the payoff at expiry conditional on hitting upper barrier level k a knock in call option would have the final payoff f x x a for a contractual strike of a knock out options are similar except that the payoff is conditioned on not hitting the barrier level as the sum of knock in and knock out options is just an option with no barrier both cases involve similar calculations alternatively the barrier can be discrete meaning that it only involves sampling the process at a finite set of times 0 t 1 t n t then equation 1 is replaced by 2 naturally sampling at a finite set of times will reduce the probability of the barrier being reached and so if f is nonnegative then 2 will have a lower value than 1 it should still converge though as n goes to infinity and the sampling times become dense in the interval if the underlying process x is brownian motion or geometric brownian motion possibly with a constant drift then there are exact expressions for computing 1 in terms of integrating f against a normal density see the post on the reflection principle for more information however it is difficult to find exact expressions for the discrete barrier 2 other than integrating over high dimensional joint normal distributions so it can be useful to ...