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# 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|FINM2416 STOCHASTIC INTEGRALS

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## 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|STOCHASTIC INTEGRALS

The purpose of this book is to study asset pricing on financial markets in continuous time. We thus want to model asset prices as continuous time stochastic processes, and the most complete and elegant theory is obtained if we use diffusion processes and stochastic differential equations as our building blocks. What, then, is a diffusion?

Loosely speaking we say that a stochastic process $X$ is a diffusion if its local dynamics can be approximated by a stochastic difference equation of the following type.
$$X(t+\Delta t)-X(t)=\mu(t, X(t)) \Delta t+\sigma(t, X(t)) Z(t) .$$
Here $Z(t)$ is a normally distributed disturbance term which is independent of everything which has happened up to time $t$, while $\mu$ and $\sigma$ are given deterministic functions. The intuitive content of (4.1) is that, over the time interval $[t, t+\Delta t]$, the $X$-process is driven by two separate terms.

A locally deterministic velocity $\mu(t, X(t))$.

A Gaussian disturbance term, amplified by the factor $\sigma(t, X(t))$.
The function $\mu$ is called the (local) drift term of the process, whereas $\sigma$ is called the diffusion term. In order to model the Gaussian disturbance terms we need the concept of a Wiener process.

## 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|Information

Let $X$ be any given stochastic process. In the sequel it will be important to define “the information generated by $X$ ” as time goes by. To do this in a rigorous fashion is outside the main scope of this book, but for most practical purposes the following heuristic definitions will do nicely. See the appendices for a precise treatment.

Definition 4.2 The symbol $\mathcal{F}t^X$ denotes “the information generated by $X$ on the interval $[0, t]$ “, or alternatively “what has happened to $X$ over the interval $[0, t]$ “. If, based upon observations of the trajectory ${X(s) ; 0 \leq s \leq t}$, it is possible to decide whether a given event $A$ has occurred or not, then we write this as $$A \in \mathcal{F}_t^X,$$ or say that “A is $\mathcal{F}_t^X$-measurable”. If the value of a given stochastic variable $Z$ can be completely determined given observations of the trajectory ${X(s) ; 0 \leq s \leq t}$, then we also write $$Z \in \mathcal{F}_t^X .$$ If $Y$ is a stochastic process such that we have $$Y(t) \in \mathcal{F}_t^X$$ for all $t \geq 0$ then we say that $Y$ is adapted to the filtration $\left{\mathcal{F}_t^X\right}{t \geq 0}$. For brevity of notation, we will sometimes write the filtration as $\left{\mathcal{F}t^X\right}{t \geq 0}=\mathbf{F}$.

## 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 孝|STOCHASTIC INTEGRALS

$$X(t+\Delta t)-X(t)=\mu(t, X(t)) \Delta t+\sigma(t, X(t)) Z(t) .$$

## 数学代写连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 考|Information

$$A \in \mathcal{F}_t^X,$$

$$Z \in \mathcal{F}_t^X .$$

$$Y(t) \in \mathcal{F}_t^X$$

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