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1. Let {B}tzo be Brownian motion started at 0 and F³ = o({B₂ : 0 ≤ s ≤ t}) VN. Are the following X₂ (FB)-martingale? (Explain the reason as well.) 1) X₁ := B² 2) Xt=t²B₁ - 2 sb,ds 3) X₁ = B³ - 3t Bt ● {Xt}t≥o is called the Itô process if there exist σ € L² and µ € L¹ such that X₁ (w) = Xo(w) + + [ * o(s,w)dB,(w) + [*µ(s,w)ds, t≥0. Here {B(w)}to is (Ft)-Brownian motion.

1. Let {B}tzo be Brownian motion started at 0 and F³ = o({B₂ : 0 ≤ s ≤ t}) VN. Are the following X₂ (FB)-martingale? (Explain the reason as well.) 1) X₁ := B² 2) Xt=t²B₁ - 2 sb,ds 3) X₁ = B³ - 3t Bt ● {Xt}t≥o is called the Itô process if there exist σ € L² and µ € L¹ such that X₁ (w) = Xo(w) + + [ * o(s,w)dB,(w) + [*µ(s,w)ds, t≥0. Here {B(w)}to is (Ft)-Brownian motion.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Let {B}tzo be Brownian motion started at 0 and FB = o({B₁, : 0 ≤ s≤t}) VN. Are the following X (FB)-martingale? (Explain the reason as well.) 1) Xt := B² 2) Xt t²Bt-2 JsB,ds = 3) Xt := B² - 3tBt • {Xt}tzo is called the Itô process if there exist o € L²2 and μ € L¹ such that X₁(w) = x₁(w) + [* o(s, w)dB₂ (w) + ["*μ(s,w)ds, t≥0. 8 Here {B(w)}tzo is (F)-Brownian motion.
Transcribed Image Text:1. Let {B}tzo be Brownian motion started at 0 and FB = o({B₁, : 0 ≤ s≤t}) VN. Are the following X (FB)-martingale? (Explain the reason as well.) 1) Xt := B² 2) Xt t²Bt-2 JsB,ds = 3) Xt := B² - 3tBt • {Xt}tzo is called the Itô process if there exist o € L²2 and μ € L¹ such that X₁(w) = x₁(w) + [* o(s, w)dB₂ (w) + ["*μ(s,w)ds, t≥0. 8 Here {B(w)}tzo is (F)-Brownian motion.
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