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数学代写|概率论代考Probability Theory代写|MAST90081 Itô Integral with Respect to Diffusions

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数学代写|概率论代考Probability Theory代写|Itô Integral with Respect to Diffusions

If
$$H=\sum_{i=1}^n h_{i-1} \mathbb{1}{\left(t{i-1}, t_i\right]} \in \mathcal{E},$$
then the elementary integral
$$I_t^M(H)=\sum_{i=1}^n h_{i-1}\left(M_{t_i \wedge t}-M_{t_{i-1} \wedge t}\right)$$
is a martingale (respectively local martingale) if $M$ is a martingale (respectively local martingale). Furthermore,
\begin{aligned} \mathbf{E}\left[\left(I_{\infty}^M(H)\right)^2\right] & =\sum_{i=1}^n \mathbf{E}\left[h_{i-1}^2\left(M_{t_i}-M_{t_{i-1}}\right)^2\right]=\sum_{i=1}^n \mathbf{E}\left[h_{i-1}^2\left(\langle M\rangle_{t_i}-\langle M\rangle_{t_{i-1}}\right)\right] \ & =\mathbf{E}\left[\int_0^{\infty} H_t^2 d\langle M\rangle_t\right] \end{aligned}

if the expression on the right-hand side is finite. Roughly speaking, the procedure in Sect. $25.1$ by which we defined the Itô integral for Brownian motion and integrands $H \in \overline{\mathcal{E}}$ can be repeated to construct a stochastic integral with respect to $M$ for a large class of integrands $H$. Essentially, in the definition of the norm on $\mathcal{E}$ we have to replace $d t$ (that is, the square variation of Brownian motion) by the square variation $d\langle M\rangle_t$ of $M$ :
$$|H|_M^2:=\mathbf{E}\left[\int_0^{\infty} H_t^2 d\langle M\rangle_t\right] .$$

数学代写|概率论代考Probability Theory代写|The Itô Formula

This and the following two sections are based on lecture notes of Hans Föllmer.
If $t \mapsto X_t$ is a differentiable map with derivative $X^{\prime}$ and $F \in C^1(\mathbb{R})$ with derivative $F^{\prime}$, then we have the classical substitution rule
$$F\left(X_t\right)-F\left(X_0\right)=\int_0^t F^{\prime}\left(X_s\right) d X_s=\int_0^t F^{\prime}\left(X_s\right) X_s^{\prime} d s$$
This remains true even if $X$ is continuous and has locally finite variation (see Sect. 21.10); that is, if $X$ is the distribution function of an absolutely continuous signed measure on $[0, \infty)$. In this case, the derivative $X^{\prime}$ exists as a RadonNikodym derivative almost everywhere, and it is easy to show that (25.10) also holds in this case.

The paths of Brownian motion $W$ are nowhere differentiable (Theorem $21.17$ due to Paley, Wiener and Zygmund) and thus have everywhere locally infinite variation. Hence a substitution rule as simple as (25.10) cannot be expected. Indeed, it is easy to see that such a rule must be false: Choose $F(x)=x^2$. Then the right-hand side in (25.10) (with $X$ replaced by $W$ ) is $\int_0^t 2 W_s d W_s$ and is hence a martingale. The left-hand side, however, equals $W_t^2$, which is a submartingale that only becomes a martingale by subtracting $t$. Indeed, this $t$ is the additional term that shows up in the substitution rule for Itô integrals, the so-called Itô formula. A somewhat bold heuristic puts us on the right track: For small $t, W_t$ is of order $\sqrt{t}$. If we formally write $d W_t=\sqrt{d t}$ and carry out a Taylor expansion of $F \in C^2(\mathbb{R})$ up to second order, then we obtain
$$d F\left(W_t\right)=F^{\prime}\left(W_t\right) d W_t+\frac{1}{2} F^{\prime \prime}\left(W_t\right)\left(d W_t\right)^2=F^{\prime}\left(W_t\right) d W_t+\frac{1}{2} F^{\prime \prime}\left(W_t\right) d t$$

概率论代写

数学代写|概率论代考Probability Theory代写|Itô Integral with Respect to Diffusions

$$H=\sum_{i=1}^n h_{i-1} \mathbb{1}\left(t i-1, t_i\right] \in \mathcal{E},$$

$$I_t^M(H)=\sum_{i=1}^n h_{i-1}\left(M_{t_i \wedge t}-M_{t_{i-1} \wedge t}\right)$$

$$\mathbf{E}\left[\left(I_{\infty}^M(H)\right)^2\right]=\sum_{i=1}^n \mathbf{E}\left[h_{i-1}^2\left(M_{t_i}-M_{t_{i-1}}\right)^2\right]=\sum_{i=1}^n \mathbf{E}\left[h_{i-1}^2\left(\langle M\rangle_{t_i}-\langle M\rangle_{t_{i-1}}\right)\right] \quad=\mathbf{E}\left[\int_0^{\infty} H_t^2 d\langle M\rangle_t\right]$$

$$|H|_M^2:=\mathbf{E}\left[\int_0^{\infty} H_t^2 d\langle M\rangle_t\right]$$

数学代写|概率论代考Probability Theory代写|The Itô Formula

$$F\left(X_t\right)-F\left(X_0\right)=\int_0^t F^{\prime}\left(X_s\right) d X_s=\int_0^t F^{\prime}\left(X_s\right) X_s^{\prime} d s$$

$$d F\left(W_t\right)=F^{\prime}\left(W_t\right) d W_t+\frac{1}{2} F^{\prime \prime}\left(W_t\right)\left(d W_t\right)^2=F^{\prime}\left(W_t\right) d W_t+\frac{1}{2} F^{\prime \prime}\left(W_t\right) d t$$

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