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How difficult is stochastic calculus?
Economist
4304euh no, I'm a man of leisure you overworked obese amerifat.
Life is work. Man up and roll up your sleeves.
Economist
a678Two ways to look at it:
PURE: If you look at stochastic calculus from a pure math perspective, then yes, it is quite difficult. One needs to start from measure theoretic probability, stochastic process and then eventually need to pick up decent amounts of PDE theory for any interesting application in optimization problems. But then PDE theory also further depends on functional analysis. And oh, you probably need to get some firm understanding of numerical approximations of PDEs to even draw pictures of your optimization problem.
Having said that... the way finance papers (so not mathematical finance, but bschool finance) has it these days:
APPLIED: Just write down the HJB. Hand wave about "verification theorem" and some guess-and-verify (they always just guess, but never verify...) approach and claim you've solved the problem. No discussion or understanding of existence or uniqueness required. So if you're going at this approach, all you really need is just Ito's lemma and the HJB. Those two just really needs Calculus II.
Economist
c4d2stay in corporate finance kid
Doing corporate actually makes the stochastic calculus stuff worse. You don’t ever actually use any of it, which makes passing the comps harder.
Economist
a7e6This. Stochastic calculus is genuinely hard from a mathematical perspective, but it's routinely applied in finance by people with no serious understanding of the subject.
Two ways to look at it:
PURE: If you look at stochastic calculus from a pure math perspective, then yes, it is quite difficult. One needs to start from measure theoretic probability, stochastic process and then eventually need to pick up decent amounts of PDE theory for any interesting application in optimization problems. But then PDE theory also further depends on functional analysis. And oh, you probably need to get some firm understanding of numerical approximations of PDEs to even draw pictures of your optimization problem.
Having said that... the way finance papers (so not mathematical finance, but bschool finance) has it these days:
APPLIED: Just write down the HJB. Hand wave about "verification theorem" and some guess-and-verify (they always just guess, but never verify...) approach and claim you've solved the problem. No discussion or understanding of existence or uniqueness required. So if you're going at this approach, all you really need is just Ito's lemma and the HJB. Those two just really needs Calculus II.Economist
49e9Two ways to look at it:
PURE: If you look at stochastic calculus from a pure math perspective, then yes, it is quite difficult. One needs to start from measure theoretic probability, stochastic process and then eventually need to pick up decent amounts of PDE theory for any interesting application in optimization problems. But then PDE theory also further depends on functional analysis. And oh, you probably need to get some firm understanding of numerical approximations of PDEs to even draw pictures of your optimization problem.
Having said that... the way finance papers (so not mathematical finance, but bschool finance) has it these days:
APPLIED: Just write down the HJB. Hand wave about "verification theorem" and some guess-and-verify (they always just guess, but never verify...) approach and claim you've solved the problem. No discussion or understanding of existence or uniqueness required. So if you're going at this approach, all you really need is just Ito's lemma and the HJB. Those two just really needs Calculus II.This bro (broette?) speakth the truth.
Economist
bc7cTwo ways to look at it:
PURE: If you look at stochastic calculus from a pure math perspective, then yes, it is quite difficult. One needs to start from measure theoretic probability, stochastic process and then eventually need to pick up decent amounts of PDE theory for any interesting application in optimization problems. But then PDE theory also further depends on functional analysis. And oh, you probably need to get some firm understanding of numerical approximations of PDEs to even draw pictures of your optimization problem.
Having said that... the way finance papers (so not mathematical finance, but bschool finance) has it these days:
APPLIED: Just write down the HJB. Hand wave about "verification theorem" and some guess-and-verify (they always just guess, but never verify...) approach and claim you've solved the problem. No discussion or understanding of existence or uniqueness required. So if you're going at this approach, all you really need is just Ito's lemma and the HJB. Those two just really needs Calculus II.The sticking point is Taylor series typically not included in Calc I, which is why you need Calc II. If you've done that the pplied approach is definitely feasible.
Economist
63dd.
Having said that... the way finance papers (so not mathematical finance, but bschool finance) has it these days:
APPLIED: Just write down the HJB. Hand wave about "verification theorem" and some guess-and-verify (they always just guess, but never verify...) approach and claim you've solved the problem. No discussion or understanding of existence or uniqueness required. So if you're going at this approach, all you really need is just Ito's lemma and the HJB. Those two just really needs Calculus II.Or in finance, not even that: just write down a drift+diffusion SDE for log-returns, invoke a probability triple (which is then never mentioned again), and then wave hands to relate that to some regression. These jokers just use it as a form of intimidation to prevent people from questioning their assumptions. I'd say that describes over 95% of the stochastic calculus I see "used" in finance papers.
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