Posted on Categories:Arbitrage Pricing in Continuous Time, 数学代写, 连续时间的期权定价理论

# 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|DRE4024 The Linear SDE

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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|The Linear SDE

In this section we will study the linear SDE, which in the scalar case has the form
\left{\begin{aligned} d X_t &=a X_t d t+\sigma d W_t, \ X_0 &=x_0 . \end{aligned}\right.
This equation turns up in various physical applications, and we will also meet it below in connection with interest rate theory.

In order to get some feeling for how to solve this equation we recall that the linear ODE
$$\frac{d x_t}{d t}=a x_t+u_t,$$
where $u$ is a deterministic function of time, has the solution
$$x_t=e^{a t} x_0+\int_0^t e^{a(t-s)} u_s d s .$$
If we, for a moment, reason heuristically, then it is tempting to formally divide eqn (5.17) by $d t$. This would (formally) give us
$$\frac{d X_t}{d t}=a X_t+\sigma \frac{d W_t}{d t},$$
and, by analogy with the ODE above, one is led to conjecture the formal solution
$$X_t=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} \frac{d W_s}{d s} d s=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} d W_s .$$
Generally speaking, tricks like this will not work, since the solution of the ODE is based on ordinary calculus, whereas we have to use Itô calculus when dealing with SDEs. In this case, however, we have a linear structure, which means that the second order term in the Itô formula does not come into play. Thus the solution of the linear SDE is indeed given by the heuristically derived formula above. We formulate the result for a slightly more general situation, where we allow $X$ to be vector-valued.

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

Consider, as in Section 5.1, the $n$-dimensional SDE
$$d X_t=\mu\left(t, X_t\right) d t+\sigma\left(t, X_t\right) d W_t .$$
Through the Itô formula, the process above is closely connected to a partial differential operator $\mathcal{A}$, defined below. The next two sections are devoted to investigating the connections between, on the one hand, the analytical properties of the operator $\mathcal{A}$, and on the other hand the probabilistic properties of the process $X$ above.

Definition $5.4$ Given the SDE in (5.21), the partial differential operator $\mathcal{A}$, referred to as the infinitesimal operator of $X$, is defined, for any function $h(x)$ with $h \in C^2\left(R^n\right)$, by
$$\mathcal{A} h(t, x)=\sum_{i=1}^n \mu_i(t, x) \frac{\partial h}{\partial x_i}(x)+\frac{1}{2} \sum_{i, j=1}^n C_{i j}(t, x) \frac{\partial^2 h}{\partial x_i \partial x_j}(x),$$
where as before
$$C(t, x)=\sigma(t, x) \sigma^{\star}(t, x) .$$

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

$\$ \$$left {$$
d X_t=a X_t d t+\sigma d W_t, X_0=x_0 .
$$\正确的。 \ \$$

$$\frac{d x_t}{d t}=a x_t+u_t,$$

$$x_t=e^{a t} x_0+\int_0^t e^{a(t-s)} u_s d s .$$

$$\frac{d X_t}{d t}=a X_t+\sigma \frac{d W_t}{d t},$$

$$X_t=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} \frac{d W_s}{d s} d s=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} d W_s .$$

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

$$d X_t=\mu\left(t, X_t\right) d t+\sigma\left(t, X_t\right) d W_t .$$

$$\mathcal{A} h(t, x)=\sum_{i=1}^n \mu_i(t, x) \frac{\partial h}{\partial x_i}(x)+\frac{1}{2} \sum_{i, j=1}^n C_{i j}(t, x) \frac{\partial^2 h}{\partial x_i \partial x_j}(x),$$

$$C(t, x)=\sigma(t, x) \sigma^{\star}(t, x)$$

## MATLAB代写

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