Let X t = b t + W t + Z t X t = b t
Question:
Let Xt=bt+Wt+Zt where W is a Brownian motion and Zt=∑Ntk=1Yk a (λ,F)-compound Poisson process independent of W. The first passage time above the level x is
Tx=inf{t:Xt≥x}
and the overshoot is Ox=XTx−x. Let Φx be the Laplace transform of the pair (Tx,Ox), i.e.,
Φx(θ,μ,x)=E(e−θTx−μOx1{Tx<∞})
Let τ1 be the first jump time of N. We wish to establish an integral equation for Φx using the following computation:
(1) Prove that
E(e−θTx−μOx1{Tx<τ1})=e(b−α)x
where α=√b2+2(θ+λ).
(2) Prove that
E(e−θTx−μOx1{Tx=τ1})=e(b−α)xα(μ−b+α)∫[0,x[(e(α−b)y
Step by Step Answer:
Mathematical Methods For Financial Markets
ISBN: 9781447125242
1st Edition
Authors: Monique Jeanblanc, Marc Yor, Marc Chesney
Students also viewed these Business questions
-
Q1. You have identified a market opportunity for home media players that would cater for older members of the population. Many older people have difficulty in understanding the operating principles...
-
Let A, B be sets. Define: (a) the Cartesian product (A B) (b) the set of relations R between A and B (c) the identity relation A on the set A [3 marks] Suppose S, T are relations between A and B, and...
-
QUIZ... Let D be a poset and let f : D D be a monotone function. (i) Give the definition of the least pre-fixed point, fix (f), of f. Show that fix (f) is a fixed point of f. [5 marks] (ii) Show that...
-
Make use of table 1.4 to express the following quantities in BG units: (a) 14.2 km, (b) 8.14 N/m 3 , (c) 1.61 kg/m 3 , (d) 0.0320 N m/s (e) 5.67 mm/hr.
-
What is the multiples approach to continuation value?
-
In Kerns comparative ratio analysis, the 2018 inventory turnover ratio for Century Chocolate is closest to: A . 5.07. B . 5.42. C . 5.55. Hans Annan, CFA, a food and beverage analyst, is reviewing...
-
Explain why so many fraud complaints are related to online auctions.
-
Ialani Corp. uses a job order costing system for the yachts it constructs. On September 1, 2010, the company had the following account balances: Raw Material Inventory ........ $332,400 Work in...
-
1) Captain Kirk's Cellular begins each month with 740 phones in stock. This stock is depleted each month and reordered. If the carrying cost per phone is $26 per year and the fixed order cost...
-
Let \(\lambda(t, x)=\lambda_{1}+\lambda_{2} x\). Prove that \[d X_{t}=\mu\left(X_{t} ight) d t+\sigma\left(X_{t} ight) d W_{t}+d M_{t}\] where \(M\) is the compensated martingale of an inhomogeneous...
-
Let \(S\) be given by (10.4.2). Give the dynamics of \(S^{k}\). \[\begin{equation*} S_{t}=S_{0} \exp \left[\left(b-\phi \lambda-\frac{\sigma^{2}}{2} ight) t+\sigma W_{t}+[\ln (1+\phi)] N_{t}...
-
Taco Bell is proficient at using social media to drive demand. America's beloved low-price Tex-Mex fast-food chain is a formidable social media marketer because it creates and curates user focused...
-
What can we learn from the ethnic group adaptations in U.S. society?
-
What species might be the earliest hominins, and what does their anatomy and environment tell us about the selective pressures that might have facilitated their evolution?
-
Keynesianism has been subject to much debate over the years. To what extent is the idea that demand management in a period of economic stagnation a recipe for stimulating economic growth once again?...
-
What role did kinship and descent play in the social structure of a chiefdom?
-
What are some methods used to reconstruct past climate, and how did that climate change from the Paleocene to present?
-
Property, plant, and equipment (plant assets) generally represent a material portion of the total assets of most companies. Accounting for the acquisition and usage of such assets is, therefore, an...
-
In Exercises 1-2, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. 1. The logarithm of the difference of two numbers is equal to...
-
Payback Period : Determine the payback period for an investment with an initial cost of $200,000 and annual cash inflows of $50,000. How long will it take for the company to recover its investment?
-
thx. Question 1 (25 marks) Cookware Company manufactures two products, Deluxe and Regular, and uses a traditional TWO-stage cost allocation. The first stage assigns all factory overhead costs to two...
-
Please show your solution tysm! . Presented below is information related to Poulsen Industries for the month of January 3 l, 2020. Ending inventory per perpetual records 'E 2 I .600 Ending...
Study smarter with the SolutionInn App