Problems and Solutions in Stochastic Calculus with Applications

The following sections are included:

• Dedication
• About the Authors
• Contents
• Introduction

The following sections are included:

• Variation of a Function
• Quadratic Variation of a Function
• Riemann Integral and Riemann–Stieltjes Integral
• Problems

The following sections are included:

• Continuous Probability Model
• Convergence
• Expectation and Lebesgue Integral
• Conditional Expectation
• Stochastic Processes
• Problems

The following sections are included:

• Brownian Motion
• Properties of BM Paths
• Three Martingales of BM
• Markov Property of BM
• Hitting Times, Maximum and Minimum, Joint Distributions and Zeros of BM
• Size of Increments of BM
• BM in Higher Dimensions
• Problems

Arguably, the most important purpose of this book is to study stochastic differential equations (SDEs). Recall that an ordinary differential equation (ODE) with initial value is given as

Consider an ODE with an initial value

The following sections are included:

• Martingales and Dynkin’s Formula
• Calculation of Expectations and PDE
• Time Homogeneous Diffusions
• Exit Times from an Interval
• Explosion
• Recurrence and Transience
• Diffusion on an Interval
• Stationary Distributions
• Multidimensional SDE
• Problems

The concept of martingale, submartingale and supermartingale {M (t)}_t≥0 ({M (t)}_{t∈[0,T ]}) with respect to a filtration {ℱ_t}_t≥0 ({ℱ_t}_t∈[0,T]) has been defined in Chapter 2. The union of these three families of processes can conveniently be called smartingales. So far, we have seen only continuous smartingales, but now we will also discuss more general smartingales with-out this restriction. However, our smartingales are always assumed to be right-continuous with left limits (càdlàg): it can be shown that they can be modified to be càdlàg “without loss” if they were not càdlàg from the beginning…

A semimartingale X_t, 0 ≤ t ≤ T, is a process consisting of a sum of a local martingale and a finite variation process. Semimartingales are the most general processes for which a stochastic integral ${\int }_0^T{H}_{t}d{X}_t$ is defined. The process H_t must be predictable, i.e. measurable with respect to the predictable σ-field, generated by the adapted left-continuous processes. A measurable function of a predictable process gives a predictable process…

A pure jump point process X is a process that changes only by jumps, jumping by size ξ_n at times T_n, n = 1, 2,…. For such a process, X(t) can be expressed as

The following sections are included:

• Change of Measure for Random Variables
• Change of Measure for Processes
• Change of Measure for Point Processes
• Likelihood Functions
• Problems

The following sections are included:

• Stock and FX Options
• Bonds, Rates, and Options
• Problems

Populations developing in time often have uncertain growth factors, be it through random variation in reproduction or environmental factors. Models for developing populations are given by diffusion processes, such as Feller branching diffusion, Wright–Fisher model, birth–death processes and branching processes. Such models generalise classical population models described by differential equations. Often, the classical models are recovered from stochastic ones in the limit, as for example, is the stochastic Lotka–Volterra model for competition of species…

The following section is included:

• Index