Problem B i can solve but for other problem i do not know how to solve it
Let {Wt : t 2 0} be a Brownian motion on a probability space (ELF, IE”) with afiltration {It : t 2 0}. Consider the Black—Scholes—Merton model with bank accountand stock process dB; = Byrdt, Bo = 1,(£83 = Stadt ‘l’ StUth, 80 = 80, where (1,0 > 0 and r 2 0 are constants. We denote by C(SO,K, T) and byP(So, K, T), the price at time 0 of a Call and respectively Put option on the stock St with strike K and maturity T. (a) Find a probability measure If", equivalent to P, under whichas; = Strdt + 3mm, where W is a Brownian motion under If”(b) Show that e‘T‘St is a P—martingale.(c) Calculate C(SO,K, T) and P090, K, T). ((1) Using Ito’s formula, write the Stochastic Differential Equation verified by theprocess S: (i.e. St raised at the power 7) for a parameter 7 E (0, 1). (e) Give now the price of a call option and put option on the process 8: withmaturity T and strike K.
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Let V Be A Finite Dimensional Inner Product Space And It Be A Subspace 1 Show Or
/in Uncategorized /by developerQuestion is from Advanced Linear Algebra . Question is taken from the topic inner product spaces.
5 . Let V be a finite dimensional inner product space and It be a subspace .( 1 ) Show orthogonal complement WV + of " is " subspace of !`( 2 ) Show that dim ( V ) – dim ( W ) + dim ( It )`
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Let V Be A Nite Dimensional Inner Product Space And Suppose That S And T Are Sel
/in Uncategorized /by developerCompletely lost in this question any help would mean so much, practice midterm question.
Let V be a finite-dimensional inner product space and suppose that Sand T are self-adjoint. Prove that if ST 2 TS then there exists anorthonormal basis (’01, . . . gun) of V which is an eigenbasis for both 8and T. (hint: the A-eigenspace for 8′ is invariant for T and Vice versa)
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Let V Be An Inner Product Space And Let W Be A Subspace Of V The Orthogonal Comp
/in Uncategorized /by developerLet V be an inner product space and let W be a subspace of V . The orthogonal complement of W in V is the set W⊥ = v ∈ V | hv, wi = 0 for all w ∈ W .
Prove the following:
a) W⊥ is a subspace of V .
b) W⊥ ∩ W = {0}. That is, the only vector they have in common is the zero vector.
Hint: For the second statement, assume that there is a vector W not equal to 0 that is common to both. Show that this leads to something that we know to be impossible, and therefore such a W cannot exist. This method is called a proof by contradiction.
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Let V P2 And Define A Pairing Of Poiynomials P And R In 2 By 19 9 1i 0 Q 0 P 1 R
/in Uncategorized /by developerHello, can you help me with the question above? Thanks!
5. Let V = P2 and define a pairing of poiynomials p and r; in [,2 by (19.9) == 1I(0)q(0) +p(1)r1(1) +p(2)q(2). (a) (7 points) Show that this formula defines an inner product on Pg. (1)) (4 points) Show that the polynomials p(t) = 4t2 — 8t + 2 and q(t) = —t2 — t + 4 areorthogonal with respect to this inner product. (0) (5 points) Find the matrix C that represents this inner product with respect to theordered basis S = (t2,t,1). (d) (4 points) Use coordinate vectors and your answer in part (c) to againshow that 4t2 — 8t + 2 and —t2 w t + 4 are orthogonal.
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Let V Span 1 1 0 0 0 1 Let Pr V R 3 R 3 Denote The Linear Transformation Whose V
/in Uncategorized /by developerLet V = span((1, 1, 0), (0, 0, 1)). Let prv : R3 → R3 denote the linear transformation whose value on a vector w ∈ R3 is equal to prv (w), the orthogonal projection of w onto V . Use the Dimension Theorem (Theorem 12.1 from the notes) to prove that the nullity of prv is equal to 1. And then find the standard matrix A of prv : R3 → R3.
Theorem 12.1 (Dimension Theorem). Let F : Rn → Rm be a linear transformation. Then
rank of F + nullity of F = n.
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Let W T 0 T 1 Be A Wiener Process With W 0 0 And The Parameter 2 1 Such A Proces
/in Uncategorized /by developerLet {W(t) : 0 <= t <= 1} be a Wiener process with W(0) = 0 and the parameter α2= 1 (such a process is called a standard Wiener process).
When answering the following questions be sure to justify your answers.
a.) Let X(t) = W(t +α) – W(t) for some α > 0. Is {X(t)} a Gaussian Process?
b.) Is {X(t)} a Wiener process?
c.) Is {X(t)} mean-square continuous?
d.) Let B(t) = W(t) – t*W(1) for 0 <= t <= 1. Is {B(t)} a Gaussian Process?
e.) Derive the covariance function for {B(t)}.
f.) Show that for any t, B(t) is independent of W(1).
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Let W T Be A Brownian Motion For Each Of The Following Processes Verify Whether
/in Uncategorized /by developerLet W(t) be a Brownian motion. For each of the following processes, verify whether it is a martingale process:(1) X(t)=W^2(t)−t(2) X(t)=W^3(t)−3tW(t)(3) X(t)=t^2W(t)−2∫uW(u)du (The limit on the integral is 0 to t)(4) X(t)=W^2(t)
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Let Wt T 2 0 Be A Brownian Motion On A Probability Space Elf Ie With A Ltration
/in Uncategorized /by developerProblem B i can solve but for other problem i do not know how to solve it
Let {Wt : t 2 0} be a Brownian motion on a probability space (ELF, IE”) with afiltration {It : t 2 0}. Consider the Black—Scholes—Merton model with bank accountand stock process dB; = Byrdt, Bo = 1,(£83 = Stadt ‘l’ StUth, 80 = 80, where (1,0 > 0 and r 2 0 are constants. We denote by C(SO,K, T) and byP(So, K, T), the price at time 0 of a Call and respectively Put option on the stock St with strike K and maturity T. (a) Find a probability measure If", equivalent to P, under whichas; = Strdt + 3mm, where W is a Brownian motion under If”(b) Show that e‘T‘St is a P—martingale.(c) Calculate C(SO,K, T) and P090, K, T). ((1) Using Ito’s formula, write the Stochastic Differential Equation verified by theprocess S: (i.e. St raised at the power 7) for a parameter 7 E (0, 1). (e) Give now the price of a call option and put option on the process 8: withmaturity T and strike K.
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Let X 0 Define The Relation On X By X Y Z T Xt Yz For Every X Y Z T X A Show Tha
/in Uncategorized /by developerLet X = ℤ × (ℤ {0}). Define the relation on X by(x, y) (z, t) ↔ xt = yzfor every (x, y), (z, t) ∈ X.(a) Show that this is an equivalence relation on X. 4 marks(b) Find the equivalence classes of (0, 1) and of (3, 3). 4 marks(c) Show that if (x, y) ≡ (x’, y’) and (z, t) ≡ (z’, t’) then (xt + yz, yt) ≡ (x’t’ + y’z’, y’t’).
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Let X 1 X 2 Be An Infinite Sequence Of Independent Identically Distributed Rando
/in Uncategorized /by developerLet X1, X2, · · · be an infinite sequence of independent, identically distributed, random variables with mean µ and variance σ 2 . We define Yn = Xn + Xn+1 + Xn+2, for n = 1, 2, · · ·. For each k ≥ 0, compute Cov(Yn, Yn+k ).
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