Discrete Stochastic Calculus

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_1

Mathematical Finance pp 5-96 | Cite as

Discrete Stochastic Calculus

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Ernst Eberlein
Jan Kallsen

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First Online: 04 December 2019

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Part of the Springer Finance book series (FINANCE)

Abstract

The theory of stochastic processes deals with random functions of time such as asset prices, interest rates, and trading strategies. As is also the case for Mathematical Finance, it can be developed in both discrete and continuous time.

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Appendix 1: Problems

The exercises correspond to the section with the same number.

1.1

Denote by Z the density process of Q ∼ P. Show that 1∕Z is the density process of P relative to Q.

1.2

For adapted processes X, Y set $Z:=\mathfrak {E}(X)(Y(0)+\mathfrak {E}(X)^{-1}\bullet Y)$. Show that Z solves the equation Z = Y + Z₋•X.

1.3

For an adapted process X define $\widetilde X:=\mathfrak {L}(e^X)$. Show that $\widetilde X$ is a random walk if and only if X is a random walk. In this case determine the law of $\Delta \widetilde X(1)$ in terms of the law of ΔX(1).

1.4

Suppose that $M\in \mathbb R^{n\times n}$ is a stochastic matrix, i.e. M_ij ≥ 0 and $\sum _{k=1}^nM_{ik}=1$ for i, j = 1, …, n. Moreover, let X be a Markov chain with finite state space E = {x₁, …, x_n} and transition matrix M, i.e. an E-valued process such that

1.
P(X(t + 1) = y_t+1|X(0) = y₀, …, X(t) = y_t) = P(X(t + 1) = y_t+1| X(t) = y_t) for any $t\in \mathbb N$ and any y₀, …, y_t+1 ∈ E such that P(X(0) = y₀, …, X(t) = y_t) > 0,
2.
P(X(t + 1) = x_j|X(t) = x_i) = M_ij for any $t\in \mathbb N$, i, j = 1, …, n.

Show that

1.
X is a Markov process relative to the filtration generated by X,
2.
its transition function p_t and its generator G satisfy p_tf = M^tf and Gf = (M − 1)f if we identity functions $f:E\to \mathbb R$ with vectors $(f(x_1),\dots ,f(x_n))\in \mathbb R^n$ and $1\in \mathbb R^{n\times n}$ denotes the identity matrix,
3.
the adjoint operator A of the generator G satisfies Aμ = G^⊤μ = (M − 1)^⊤μ if we identify measures μ on E with vectors $(\mu (\{x_1\}),\dots ,\mu (\{x_n\})\in \mathbb R^n$ and likewise linear mappings $\mu :B(E)\to \mathbb R$ with $(\mu (1_{\{x_1\}}),\dots ,\mu (1_{\{x_n\}}))\in \mathbb R^n$.

1.5

Elisabeth wants to sell her house within T days. She receives daily offers which are assumed to be independent random variables that are uniformly distributed on [m, M]. Elisabeth has the option of recalling earlier offers and consider them again but she must pay maintenance costs c > 0 for every day the house remains unsold. Determine the optimal time to sell, i.e. the stopping time maximising her expected reward from the sale.

Hint: Try the ansatz that the value function is of the form

$$\displaystyle \begin{aligned}v(t,x)=\left\{ \begin{array}{ll} x-c(t-1)& \mbox{ for } x\geq \underline x,\\ \widetilde v(t,x)-ct &\mbox{ for } x<\underline x, \end{array} \right.\end{aligned}$$

where x represents the maximal offer so far, ${\partial \over \partial x}\widetilde v(t,x)\leq 1$, and the threshold $ \underline x\in [m,M]$ is chosen such that x↦v(t, x) is continuous in $ \underline x$.

1.6

For $x,b_0\in \mathbb R^2$ and $b_1\in \mathbb R^{2\times 2}$ determine the function $X:\mathbb R _+\to \mathbb R^2$ with b^X(t) = b₀ + b₁X(t), where b^X is defined as in (1.132).

Appendix 2: Notes and Comments

The material in this chapter is mostly classical. For an introduction to probability theory including martingales and discrete-time Markov processes see, for example, [153, 275]. Some results in Sects. 1.1–1.3 are discrete-time versions of statements from the general theory in [152, 154, 238]. Many additional references can be found in these texts. The presentation of Sect. 1.4 is based on the parallel more subtle results in Chap. 5. For stochastic optimal control in discrete time see [18, 271] and the references therein. However, we consider a non-Markovian framework similarly as in [96]. Moreover, the exposition here tries to mimic the continuous-time theory of Chap. 7 as much as possible. Section 1.6 presents standard results from calculus in stochastic process notation.

For early solutions to the portfolio problems in Examples 1.48, 1.49, 1.64, 1.65 see [222, 258]. A history on quadratic hedging in the martingale case of Example 1.50 and beyond can be found in [270]. For a nice short introduction to optimal stopping we refer to [203]. The perpetual American put is treated in [277]. For the background of Example 1.58 we refer to [102]. Proposition 1.59 is based on [135, 249]. The dual approach to optimal investment in Examples 1.71, 1.74 is inspired by more general characterisations in [188, 197] but the idea is already present in [27]. The extension of the dual approach underlying Example 1.76 goes back to [65, 161]. Example 1.79 is a special case of the results in [125]. The relationship (1.118) has been stated in [28] in a Brownian motion framework. Problem 1.5 is a slight modification of [271, Example 1.34].

Except for the few examples in Sect. 1.5, we do not discuss Mathematical Finance in discrete time. Classical references include [117, 237, 278].

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Authors and Affiliations

Ernst Eberlein
- 1
Jan Kallsen
- 2

1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
2.Department of MathematicsKiel UniversityKielGermany

Discrete Stochastic Calculus

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