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Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: katastrofaTraden4Alpha:@"Yet it is still a blunt tool in the sense of having a very limited ability to resolve the features of the system"@"MC only gives statistical results and only for those specific parameters that were tried."When you model a stochastic system, like those I described, a distributional result is usually all you need and can get. When the model parameters are correlated, multistage and time-dependent, you won't see any straightforward trends or scaling laws in the results. Besides, from my experience MC has generally better convergence than deterministic methods (for high dimensional systems), and there are techniques (which you should probably know better than I, e.g. path-wise differentiation) to obtain dependencies on input parameters without rerunning the simulation. In specific cases you can sometimes come up with a better deterministic model, but it doesn't make MC blunt...The popular approximations usually don't work in complex systems in chemistry and biology. Not to mention the models of a population or a market microstructures.Part of MC's bluntness is that it does not actually give a distribution as an output, only a discrete set of simulated empirical events. The probability density for any outcomes that were not observed during the runs (e.g., those outside the min-to-max range or those between any adjacent pairs of samples) is unknown. If the true PDF has spikes or holes, these might not be apparent. If the true PDF does "bad things" in the tails -- a major concern for QF -- the sparsity of events in the tail can pose a problem. Of course, tricks like importance sampling can help in the tails and examining the analytic structure within the sample generator might let one surmise that those unobserved values can occur with a probability that can be estimated from the frequency of observed events in that neighborhood.MC is like a chainsaw: powerful and gets jobs done that other tools can't (e.g., complex systems in chemistry and biology) but it is blunt (and noisy).
Last edited by Traden4Alpha on April 15th, 2015, 10:00 pm, edited 1 time in total.

Cuchulainn
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### What does ?convergence? in Monte Carlo simulation mean?

From a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.
Last edited by Cuchulainn on April 14th, 2015, 10:00 pm, edited 1 time in total.

emac
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.What do you mean by "an appropriate norm" in 2?What do you mean by 2_A?

Cuchulainn
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.What do you mean by "an appropriate norm" in 2?What do you mean by 2_A?A norm is an estimate of the size of a mathematical entity, e.g. matrix. You can have L2, L infinity (max) norm.example finite element2_A: as T4A mentioned

emac
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.What do you mean by "an appropriate norm" in 2?What do you mean by 2_A?A norm is an estimate of the size of a mathematical entity, e.g. matrix. You can have L2, L infinity (max) norm.example finite element2_A: as T4A mentionedI know what a norm is. What do you mean that there are not convergence results in an appropriate norm? Which norm do you think is appropriate?T4A says that MC does not output a density. Is this what you mean by "It is missing the concept of a function as in real analysis."?I don't see this as a problem. There are also results on the convergence of the empirical MC distribution or density to the target distribution/density.

Cuchulainn
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.What do you mean by "an appropriate norm" in 2?What do you mean by 2_A?QuotePart of MC's bluntness is that it does not actually give a distribution as an output, only a discrete set of simulated empirical events. The probability density for any outcomes that were not observed during the runs (e.g., those outside the min-to-max range or those between any adjacent pair of samples) is unknown. If the true PDF and spikes or holes, these might not be apparent. If the true PDF does "bad things" in the tails -- a major concern for QF -- the sparsity of events in the tail can pose a problem

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.What do you mean by "an appropriate norm" in 2?What do you mean by 2_A?A norm is an estimate of the size of a mathematical entity, e.g. matrix. You can have L2, L infinity (max) norm.example finite element2_A: as T4A mentionedI know what a norm is. What do you mean that there are not convergence results in an appropriate norm? Which norm do you think is appropriate?T4A says that MC does not output a density. Is this what you mean by "It is missing the concept of a function as in real analysis."?I don't see this as a problem. There are also results on the convergence of the empirical MC distribution or density to the target distribution/density.A blunt knife will cut if you put in enough time and hard work. A blunt stochastic numerical tool will converge if you put in enough time and hard work. Of course, just because the finally converged results were OK for one set of parameter values does not mean they will be OK for parameter values an epsilon away.

emac
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.What do you mean by "an appropriate norm" in 2?What do you mean by 2_A?A norm is an estimate of the size of a mathematical entity, e.g. matrix. You can have L2, L infinity (max) norm.example finite element2_A: as T4A mentionedI know what a norm is. What do you mean that there are not convergence results in an appropriate norm? Which norm do you think is appropriate?T4A says that MC does not output a density. Is this what you mean by "It is missing the concept of a function as in real analysis."?I don't see this as a problem. There are also results on the convergence of the empirical MC distribution or density to the target distribution/density.A blunt knife will cut if you put in enough time and hard work. A blunt stochastic numerical tool will converge if you put in enough time and hard work. Of course, just because the finally converged results were OK for one set of parameter values does not mean they will be OK for parameter values an epsilon away.But what does that have to do with MC? That's related to how sensitive the system you are simulating is to the parameters.

savr
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.I'm eager to go further in this discussion.1. Let's dismiss as a "my car won't run unless I start it"-style tautology: last I checked seeds were easily communicatable and reusable. Those who require seeds written in Braille on the back of a horse are hereafter excluded from analysis.2. I'm especially interested in which error estimates you cannot find in the textbooks? If your worries are about floating-point calculation, I think Bouleau & Lepingle clarify what problems can be tackled and which, like the indicator set of rationals, are better not tackled numerically.2_A. Is there not a topology on $\Omega$ ?3. Outside of specific convex problems, I'm not aware of many deterministic methods that have monotonic convergence?

Cuchulainn
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: savrQuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.I'm eager to go further in this discussion.1. Let's dismiss as a "my car won't run unless I start it"-style tautology: last I checked seeds were easily communicatable and reusable. Those who require seeds written in Braille on the back of a horse are hereafter excluded from analysis.2. I'm especially interested in which error estimates you cannot find in the textbooks? If your worries are about floating-point calculation, I think Bouleau & Lepingle clarify what problems can be tackled and which, like the indicator set of rationals, are better not tackled numerically.2_A. Is there not a topology on $\Omega$ ?3. Outside of specific convex problems, I'm not aware of many deterministic methods that have monotonic convergence?Let's be concrete. I think we are on different wavelengths.So, price a one-factor option by applying 1-3 toA. SDE, Euler and Monte CarloB. PDE model with Crank Nicolson
Last edited by Cuchulainn on April 15th, 2015, 10:00 pm, edited 1 time in total.

savr
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Joined: January 21st, 2013, 3:28 pm

### What does ?convergence? in Monte Carlo simulation mean?

I have carried out the A,B exercise you set, and hope we can resume the discussion.You made a series of statements which I urged you to substantiate.Now, can you exhibit eithera) a Cauchy sequence of "sharp error results in an appropriate norm" that does not converge within this space, orb) two different equivalent martingale measures on the set of "sharp error results in an appropriate norm"? P.S. Seriously I still don't understand 2A_?
Last edited by savr on April 15th, 2015, 10:00 pm, edited 1 time in total.

Cuchulainn
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### What does ?convergence? in Monte Carlo simulation mean?

QuoteI have carried out the A,B exerciseThat's fast. What are your findings?

emac
Posts: 89
Joined: July 7th, 2009, 7:15 pm

### What does ?convergence? in Monte Carlo simulation mean?

QuoteOriginally posted by: CuchulainnQuoteI have carried out the A,B exerciseThat's fast. What are your findings?Maybe you think this is some form of Socratic dialogue, but it would be easier if you just explained what you meant (esp. about 2_A). I think the people replying have all used MC and PDE methods before.

savr
Posts: 48
Joined: January 21st, 2013, 3:28 pm

### What does ?convergence? in Monte Carlo simulation mean?

Please don't withhold your opinion, and elaborate onQuoteOriginally posted by: CuchulainnFrom a numerical analysis viewpoint, MC is strange:1. The results are not reproducible (unless the same seed is used).2. Sharp error results in an appropriate norm are incomplete. 2_A. It is missing the concept of a function as in real analysis.3. Convergence is not monotonic.I think there is some interesting discussion to be had.

katastrofa
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### What does ?convergence? in Monte Carlo simulation mean?

@ Traden4AlphaQuoteQuoteOriginally posted by: katastrofaTraden4Alpha:@"Yet it is still a blunt tool in the sense of having a very limited ability to resolve the features of the system"@"MC only gives statistical results and only for those specific parameters that were tried."When you model a stochastic system, like those I described, a distributional result is usually all you need and can get. When the model parameters are correlated, multistage and time-dependent, you won't see any straightforward trends or scaling laws in the results. Besides, from my experience MC has generally better convergence than deterministic methods (for high dimensional systems), and there are techniques (which you should probably know better than I, e.g. path-wise differentiation) to obtain dependencies on input parameters without rerunning the simulation. In specific cases you can sometimes come up with a better deterministic model, but it doesn't make MC blunt...The popular approximations usually don't work in complex systems in chemistry and biology. Not to mention the models of a population or a market microstructures.Part of MC's bluntness is that it does not actually give a distribution as an output, only a discrete set of simulated empirical events.  The probability density for any outcomes that were not observed during the runs (e.g., those outside the min-to-max range or those between any adjacent pairs of samples) is unknown.  If the true PDF has spikes or holes, these might not be apparent.  If the true PDF does "bad things" in the tails -- a major concern for QF -- the sparsity of events in the tail can pose a problem.  Of course, tricks like importance sampling can help in the tails and examining the analytic structure within the sample generator might let one surmise that those unobserved values can occur with a probability that can be estimated from the frequency of observed events in that neighborhood.I think we start walking in circles. I don't find any of these a valid argument against the MC technique, mostly because the same criticism applies to deterministic numerical methods (even when you're solving a differential equation, you do this on a lattice). As I said, one should ideally solve the problem analytically as far as possible, but when this cannot be done, MC is generally the most effective, efficient and flexible approach. When you poke your nose out of the tiny plot of quantitative finance, MC is not only used as a "tool" but often it is the most natural representation of the actual simulated phenomenon, like in the examples I gave you. But thanks anyway for elaborating on the author's behalf what the esoteric statement meant.QuoteMC is like a chainsaw: powerful and gets jobs done that other tools can't (e.g., complex systems in chemistry and biology) but it is blunt (and noisy).Not a good metaphor either: if chainsaws were blunt The Texas Chainsaw Massacre wouldn't be such a good movie.@CuchulainnQuote1. The results are not reproducible (unless the same seed is used).You need to fix your pseudrandom numbers Like this: https://picasaweb.google.com/lh/photo/Z ... linkQuote2. Sharp error results in an appropriate norm are incomplete.2_A. It is missing the concept of a function as in real analysis.This is generally a problem (?) with numerical methods.
Last edited by katastrofa on April 16th, 2015, 10:00 pm, edited 1 time in total.