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B8.2

Ben Tidswell Part B uni English 51 cards

B8.2

Cards in this deck (51)

  1. 1Let {ℙ_𝘁:𝘁∈𝕋} be a consistent family of finite-dimensional distributions. State the Daniell-Kolmogorov Extension TheoremLet {ℙ_𝘁:𝘁∈𝕋} be a consistent family of finite-dimensional distributions. State the Daniell-Kolmogorov Extension Theorem
  2. 2Let (X_t:t≤T) be a stochastic process defined on (Ω,ℱ,ℙ). State the Kolmogorov-Čentsov continuity criterionLet (X_t:t≤T) be a stochastic process defined on (Ω,ℱ,ℙ). State the Kolmogorov-Čentsov continuity criterion
  3. 3Fix a Brownian motion (B_t)_t≥0 on (Ω,ℱ,ℙ). Recall B_0=0. State Blumenthal's 0-1 lawFix a Brownian motion (B_t)_t≥0 on (Ω,ℱ,ℙ). Recall B_0=0. State Blumenthal's 0-1 law
  4. 4Let S_t:=sup_(u≤t)B_u. State and prove The reflection principleLet S_t:=sup_(u≤t)B_u. State and prove The reflection principle
  5. 5State Doob's maximal and L^p inequalitiesState Doob's maximal and L^p inequalities
  6. 6State Doob's Optional Stopping Theorem for discrete supermartingales (bounded case)State Doob's Optional Stopping Theorem for discrete supermartingales (bounded case)
  7. 7Let{Y_k} be a family of random variables and suppose Y_k->Y_∞ in probability. State the Vitali convergence theoremLet{Y_k} be a family of random variables and suppose Y_k->Y_∞ in probability. State the Vitali convergence theorem
  8. 8Let (X_t:t≥0) be a martingale with right-continuous sample paths. State the Martingale Convergence TheoremLet (X_t:t≥0) be a martingale with right-continuous sample paths. State the Martingale Convergence Theorem
  9. 9Let (Y_n)_(n∈ℕ) be a uniformly integrable martingale with respect to the filtration (𝒢_n)_(n∈ℕ), and let Y_∞ be the almost sure limit of Y_n when n -> ∞. State Optional stopping for UI discrete time martingales.Let (Y_n)_(n∈ℕ) be a uniformly integrable martingale with respect to the filtration (𝒢_n)_(n∈ℕ), and let Y_∞ be the almost sure limit of ...
  10. 10Let M,N be continuous local martingales and K,H be two measurable processes. State the Kunita-Watanabe inequalityLet M,N be continuous local martingales and K,H be two measurable processes. State the Kunita-Watanabe inequality
  11. 11Let X^1,...X^d be continuous semimartingales and F:ℝ->ℝ a C^2 function. State Itô's formulaLet X^1,...X^d be continuous semimartingales and F:ℝ->ℝ a C^2 function. State Itô's formula
  12. 12Let M be a continuous local martingale starting at zero. State Lévy's characterisation of Brownian motion.Let M be a continuous local martingale starting at zero. State Lévy's characterisation of Brownian motion.
  13. 13Let M be an ({ℱ_t}_(t≥0),ℙ)-continuous local martingale with M_0=0 and _∞=∞ almost surely. State the Dambis-Dubins-Schwarz Theorem.Let M be an ({ℱ_t}_(t≥0),ℙ)-continuous local martingale with M_0=0 and _∞=∞ almost surely. State the Dambis-Dubins-Schwarz Theorem.
  14. 14Give the covariance function definition for a standard Brownian motionGive the covariance function definition for a standard Brownian motion
  15. 15Define a standard real Brownian motionDefine a standard real Brownian motion
  16. 16Define what it means for a class 𝒞 of random variables to be uniformly integrableDefine what it means for a class 𝒞 of random variables to be uniformly integrable
  17. 17Define the quadratic co-variation between two continuous local martingales M and NDefine the quadratic co-variation between two continuous local martingales M and N
  18. 18Let M be a continuous local martingale and K∈L^2_loc(M). Define K•MLet M be a continuous local martingale and K∈L^2_loc(M). Define K•M
  19. 19Suppose M,N∈ℋ^(2,c) and K∈L^2(M), H∈L^2(N). State the quadratic co-variation of K•M and H•NSuppose M,N∈ℋ^(2,c) and K∈L^2(M), H∈L^2(N). State the quadratic co-variation of K•M and H•N
  20. 20Let M be a continuous local martingale, M_0∈L^2. State and prove two TFAEs regarding the L^2 norm and the long term behaviour of the quadratic co-variation of M. State and prove something about M^2_t-_t in both casesLet M be a continuous local martingale, M_0∈L^2. State and prove two TFAEs regarding the L^2 norm and the long term behaviour of the quad...
  21. 21Define a continuous martingaleDefine a continuous martingale
  22. 22Let X be a random variable on the probability space (Ω,ℱ,ℙ) and 𝒢_1⊆𝒢_2⊆ℱ. State the tower property of conditional expectationLet X be a random variable on the probability space (Ω,ℱ,ℙ) and 𝒢_1⊆𝒢_2⊆ℱ. State the tower property of conditional expectation
  23. 23Define the natural filtration associated with the process XDefine the natural filtration associated with the process X
  24. 24Let (Ω,ℱ,{ℱ}_(t≥0),ℙ) be a filtered probability space. Define a stopping timeLet (Ω,ℱ,{ℱ}_(t≥0),ℙ) be a filtered probability space. Define a stopping time
  25. 25State Doob's Optional Theorem for martingales with continuous sample pathsState Doob's Optional Theorem for martingales with continuous sample paths
  26. 26Define what it means for a function to be of finite variation and what it means for a stochastic process to be of finite variationDefine what it means for a function to be of finite variation and what it means for a stochastic process to be of finite variation
  27. 27Let X,Y be two stochastic processes defined on a common probability space (Ω,ℱ,ℙ). Define what it means for X to be a modification of Y and what it means to say X and Y are indistinguishableLet X,Y be two stochastic processes defined on a common probability space (Ω,ℱ,ℙ). Define what it means for X to be a modification of Y a...
  28. 28Prove that a continuous local martingale M with M_0=0 a.s., is a process of finite variation iff M is indistinguishable from 0Prove that a continuous local martingale M with M_0=0 a.s., is a process of finite variation iff M is indistinguishable from 0
  29. 29State Fatou's Lemma in the context of expectationState Fatou's Lemma in the context of expectation
  30. 30Let f:I->ℝ be a function defined on a subset I of [0,∞). For aLet f:I->ℝ be a function defined on a subset I of [0,∞). For a
  31. 31State Doob's upcrossing lemma in discrete timeState Doob's upcrossing lemma in discrete time
  32. 32State the moment generating function for a N(μ,σ^2) distribution1State the moment generating function for a N(μ,σ^2) distribution1
  33. 33Prove that exp(λB_t-λ^(2)t/2) is a martingale if B is a standard Brownian motion (just expectation part)Prove that exp(λB_t-λ^(2)t/2) is a martingale if B is a standard Brownian motion (just expectation part)
  34. 34Let B^1 and B^2 be independent Brownian motions. State _tLet B^1 and B^2 be independent Brownian motions. State _t
  35. 35Let X=X_0+M+A be a continuous semimartingale. State an if and only if condition for X to be a submartingaleLet X=X_0+M+A be a continuous semimartingale. State an if and only if condition for X to be a submartingale
  36. 36State what it means for X_n->X a.s., in probability, in distribution, and in L^1State what it means for X_n->X a.s., in probability, in distribution, and in L^1
  37. 37Define what it means to say that a martingale is closedDefine what it means to say that a martingale is closed
  38. 38Let M be a continuous local martingale. State situations where M is a supermartingale and a UI martingale. Give stopping times that reduce M if M_0∈L^1. Comment on what type of process M^ρ is for any stopping time ρLet M be a continuous local martingale. State situations where M is a supermartingale and a UI martingale. Give stopping times that reduc...
  39. 39Define what it means for for a process X to be adapted to a filtration {ℱ_t}_(t≥0)Define what it means for for a process X to be adapted to a filtration {ℱ_t}_(t≥0)
  40. 40Define ℋ^(2,c)Define ℋ^(2,c)
  41. 41For a continuous local martingale M, define L^2_(loc)(M)For a continuous local martingale M, define L^2_(loc)(M)
  42. 42Let B be a Brownian motion, for two times t and s, state 𝔼[B_tB_s]Let B be a Brownian motion, for two times t and s, state 𝔼[B_tB_s]
  43. 43Let M be a continuous martingale. Define the quadratic variation process _t associated with M_tLet M be a continuous martingale. Define the quadratic variation process _t associated with M_t
  44. 44What does it mean for a function to be càdlàg at a point tWhat does it mean for a function to be càdlàg at a point t
  45. 45How can you compute 𝔼[X] if you already know 𝔼[X|Y]How can you compute 𝔼[X] if you already know 𝔼[X|Y]
  46. 46State where M is a continuous local martingale and A is a finite variation processState where M is a continuous local martingale and A is a finite variation process
  47. 47State a theorem regarding the convergence of right-continuous supermartingalesState a theorem regarding the convergence of right-continuous supermartingales
  48. 48Assuming that X_t=exp(θB_t-θ^2t/2) defines a martingale, show that it converges to zero a.s.Assuming that X_t=exp(θB_t-θ^2t/2) defines a martingale, show that it converges to zero a.s.
  49. 49Let M∈ℋ^(2,c) and K∈L^2(M). State the steps to defining K•MLet M∈ℋ^(2,c) and K∈L^2(M). State the steps to defining K•M
  50. 50Define a centred gaussian processDefine a centred gaussian process
  51. 51Name 3 Brownian motions that can be derived from a standard real BM via symmetry, scaling, and time-reversalName 3 Brownian motions that can be derived from a standard real BM via symmetry, scaling, and time-reversal

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uni Part B Flashcards on B8.2, created by Ben Tidswell on 15/04/2023.
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Ben Tidswell
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