stochastic odes

notes/math.stats.stochastic-ode.md

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+---
+id: zmvhm0w2tjs2njii7b7nrm5
+title: Stochastic Differential Equations
+desc: ''
+updated: 1685767136408
+created: 1685765664047
+---
+
+# Stochastic processes
+
+Let $(\Omega, B, P)$ be a measure space, with Borel $\sigma$-algebra *B*, and probability measure *P*.
+
+**Definition** A *stochastic process* is a parametrized collection of random variables:
+
+$$\{X_t \mid t \geq 0\},$$
+
+such that each $X_t$ is a random variable in the measure space.
+
+## Brownian movement or Wiener process
+
+Let $\{ w_t \mid t \geq 0 \}$ be a stochastic process such that $w_t$ is continous in the weak sense with respect to $t$. $w_t$ is a *Wiener process* if:
+
+1. $0 \leq t_1 \leq t_2$ implies 
+$$w_{t_2} - w_{t_1} \sim N(0, t_2 - t_1)$$
+2. For any $t_1 < t_2 < t_3$, $$w_{t_3} - w_{t_2}$$ is independent of $$w_{t_2} - w_{t_1}$$.
+3. The probability $w_0 = 0$ satisfies $P(w_0 = 0) = 1$.
+
+*Note:* In general, $w_t$ is not differentiable in any point.
+
+## Ito integral
+
+Let $f(t, x_t) = f(t)$, with $x_t$ an stochastic process, such that 
+
+$$\int_a^b E(f(t)) dt < \infty.$$
+
+We will say that $f(t)$ is a *random function*. Let 
+$$\{a = t_1 < \cdots < t_{n + 1} = b\}$$ 
+be a partition of $[a, b]$, with equally spaced points, and let $\Delta t = (b - a)/n$ and $$\Delta w_i = w_{t_{i + 1}} - w_{t_i}$$. Then, Ito's integral is 
+
+$$\int_a^b f(t) dw_t = \lim\sum_{i = 1}^n f(t_i) \Delta w_i.$$
+
+### Notes
+
+- If $s_n$ represents the nth-partial sum of the integral above, we say that $\lim_{n\to\infty} s_n = I$ in probability if 
+
+$$\lim_{n \to \infty} E((s_n - I)^2) = 0.$$
+
+- Note that there are two stochastic processes involved in the definition: $x_t$, which is implicit in $f(t)$ and $w_t$, which represents *noise* or *decoherence*, depending the problem.
+
+## Ito's stochastic differential equation
+
+**Definition:** $x_t$ is a solution of the *stochastic differential equation*, 
+
+$$dx_t = \alpha(x_t,t) dt + \beta(x_t, t)dw_t,$$
+
+if for any $t > 0$, $x_t$ satisfies
+
+$$x_t = x_0 + \int_0^t \alpha(x_t, t)dt + \int_0^t \beta(x_t, t)dw_t.$$
+
+**Q.** Under which conditions can it be proved that the solution to this equation is unique?.
+
+**Teorem (chain rule)** If $x_t$ is the solution of a stochastic differential equation, and $F(x, t)$ is a real function such that the partial derivatives
+$$\partial_t F, \partial_x F, \partial_{xx} F$$
+are continous functions, then 
+
+$$dF(x_t, t) = f(x_t, t) dt + g(x_t, t) dw_t,$$
+
+where 
+
+$$f(x, t) = \partial_tF + \alpha(x_t, t) \partial_x F + \frac{1}{2} \beta^2(x_t, t) \partial_{xx}F$$
+
+and 
+
+$$g(x_t, t) = \beta(x_t, t) \partial_x F.$$
+
+### Stochastic maltus model
+
+**Theorem** The solution to the stochastic differential equation
+$$dx_t = r x_t dt + c x_t dw_t$$
+is 
+
+$x_t = x_0\exp((r - c^2/2)t + c \cdot w_t)$.
+
+**Proof** Let $F = \ln(x)$, by the chain rule:
+
+$$d\ln(x_t) = \left(rx_t\cdot \frac{1}{x_t} + \frac{1}{2} c^2 x_t^2 \left(- \frac{1}{x_t^2}\right)\right) dt + c x_t \cdot \frac{1}{x_t} dw_t,$$
+
+Simplifying the equation, we get
+
+$$d\ln(x_t) = \left(r - \frac{c^2}{2}\right) dt + c \cdot dw_t.$$
+
+It can be shown that the fundamental theorem of calculus is valid for the stochastic integral of a constant function,  therefore, applying this result to the previous equation, we find,
+
+$$\ln\left(\frac{x_t}{x_0}\right) = \left(r - \frac{c^2}{2}\right) t + c\cdot w_t,$$
+
+and the theorem follows.
+
+### Numerical methods
+
+#### I. Euler-Murayama method
+
+
+$$dx_i = \Delta x_i, \qquad dw_i = \Delta w_i,$$
+
+$$x_{i + 1} = x_i + \alpha(x_i, t_i) \Delta t + \beta(x_i, t_i) \Delta w_i.$$
+
+Note that in order to implement this method we should select $\Delta w_i$ randomly with distribution $N(0, \Delta t)$.
+
+#### II. Milstein method
+
+Modify the last equation into
+
+$$x_{i +1} = x_i + \alpha(x_i, t_i) \Delta t + \beta(x_i, t_i) \Delta w_i + \frac{1}{2} \beta(x_i, t_i) \frac{\partial \beta}{\partial x}(x_i, t_i) \left((\Delta w_i)^2 - \Delta t\right).$$
+
+This last method resembles *predictor-corrector* methods in ordinary differential equations.
+
+---
+
+Originally published by the author in [http://ixxra.github.io/mathannotations/sdes/2014/05/21/stochastic-differential-equations/](http://ixxra.github.io/mathannotations/sdes/2014/05/21/stochastic-differential-equations/)
+
+
+# References
+
+* Seminar notes, René García-Lara, May 2014, available at [http://ixxra.tumblr.com/post/86507241536/today-in-the-mathematics-seminar-stochastic](http://ixxra.tumblr.com/post/86507241536/today-in-the-mathematics-seminar-stochastic)