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Financial Mathematics

创建日期：2024-08-30 15:14:40

所属分类：数学mathematics

标签： Mathematics

Financial Mathematics

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Financial Mathematics 2142

1. Short questions

Complete the sequences!

1.1. Question. The Cauchy problem

is equivalent with the following integral equation:

1.2. Question. Let I denote an open interval such that 0 ∈ I . If the scalar function x : I → R solves the Cauchy problem

then the vector function

solves the following Cauchy problem (involving a system of first order differential equations):

1.3. Question. Let (X, d) denote a metric space.

(a) We call (X, d) complete if

(b) Find an appropriate value of the real number q that makes inequality (5) identically true and admits a valid argument when you fill the blank places with some of the words in brackets [complete, continuous, contraction, root, fixed point, linear]:

Proposition. Let φ : [0, 1] → [0, 1] be given by

Then there exists x0 ∈ [0, 1] fulfilling φ(x0) = x0 .

Proof. One can easily verify the inequality

(with q = ) for every x, y ∈ [0, 1]. Hence φ : [0, 1] → [0,1] is a . . . . . . . . . . . . .

Since [0, 1] is a . . . . . . . . . . . . metric space, we obtain that φ has a . . . . . . . . . . . . .

2. Comprehensive question

2.1. Question. Suppose that n ∈ N , I ⊂ R is an open interval and ai, b: I → R are continuous functions (i = 1, . . . , n). Given τ ∈ I and ξi ∈ R (i = 1, . . . , n), let us consider the equations

Elaborate the theory of the n-th order linear differential equations (without proofs) involving the answers to the following questions:

(a) What can we claim concerning the existence, the uniqueness and the do-main of the maximal solution of the Cauchy problem (6)–(7) ?

(b) Explain the relation between the sets of solutions of the differential equa-tions (6) and (8).

(d) What is a fundamental set of solutions of (8) ? Define its Wronskian as well!

(e) If a fundamental set of solutions of (8) is given, how can we apply the va-riation of constants in order to determine a solution of the differential equation (6) ?

3. Combined question

3.1. Question. Let D ⊂ R 2 be a connected open domain and P, Q : D → R be continuous functions. Let us consider a differential equation of the form.

(a) When do we say that (9) is an exact differential equation? (Write down the proper definition!)

(b) What is the necessary condition fulfilled by exact differential equations with smooth terms?

(d) Verify if the following differential equation is exact and determine its solutions:

4. Practical problems

Details of calculations are required.

4.1. Problem. Solve the Cauchy problem

(involving a separable differential equation).

4.2. Problem. Solve the (Bernoulli type) differential equation

4.3. Problem. Determine the solutions of the following differential equations:

Describe the connections between the solutions of the equations (a) and (b) as well as the connections between the solutions of the equations (a) and (c) !

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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