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# Stochastic Differential Equations: An Introduction with Applications

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## Chapters (7)

To convince the reader that stochastic differential equations is an important subject let us mention some situations where such equations appear and can be used:
Having stated the problems we would like to solve, we now proceed to find reasonable mathematical notions corresponding to the quantities mentioned and mathematical models for the problems. In short, here is a first list of the notions that need a mathematical interpretation: (1) A random quantity (2) Independence (3) Parametrized (discrete or continuous) families of random quantities (4) What is meant by a “best” estimate in the filtering problem (Problem 3) (5) What is meant by an estimate “based on” some observations (Problem 3)? (6) What is the mathematical interpretation of the “noise” terms? (7) What is the mathematical interpretation of the stochastic differential equations?
Example 3.8 illustrates that the basic definition of Ito integrals is not very useful when we try to evaluate a given integral. This is similar to the situation for ordinary Riemann integrals, where we do not use the basic definition but rather the fundamental theorem of calculus plus the chain rule in the explicit calculations.
We now return to the possible solutions X t (ω) of the stochastic differential equation$$\frac{{d{X_t}}}{{dt}} = b(t,{\kern 1pt} {X_t}) + \sigma (t,{\kern 1pt} {X_t}){W_t},{\kern 1pt} b(t,{\kern 1pt} x) \in R,\sigma (t,{\kern 1pt} x) \in R$$ (5.1) where W t is 1-dimensional “white noise”. As discussed in Chapter III the Ito interpretation of (5.1) is that X t satisfies the stochastic integral equation$${X_t} = {X_0} + \int\limits_0^t {b(s,{\kern 1pt} {X_s})} ds + \int\limits_0^t {\sigma (s,{\kern 1pt} {X_s})} d{B_s}$$ or in differential form$$d{X_t} = b(t,{\kern 1pt} {X_t})dt + \sigma (t,{\kern 1pt} {X_t})d{B_t}.$$ (5.2)
Problem 3 in the introduction is a special case of the following general filtering problem: Suppose the state X t ∈ R n at time t of a system is given by a stochastic differential equation $$\frac{{d{X_t}}}{{dt}} = b(t,{X_t}) + \sigma \left( {t,{X_t}} \right){W_t},t0,$$ (6.1.1) where b: R n+1 → R n , σ : R n+1 → R n × p satisfy conditions (5.2.1), (5.2.2) and W t is p-dimensional white noise. As discussed earlier the Ito interpretation of this equation is (system) $$d{X_t} = b\left( {t,{X_t}} \right)dt + \sigma \left( {t,{X_t}} \right)d{U_t},$$ (6.1.2) where U t is p-dimensional Brownian motion. We also assume that the distribution of X 0 is known and independent of U t . Similarly to the 1-dimensional situation (3.3.6) there is an explicit several-dimensional formula which expresses the Stratonovich interpretation of (6.1.1): $$d{X_t} = b\left( {t,{X_t}} \right)dt + \sigma \left( {t,{X_t}} \right) \circ d{B_t}$$ in terms of Ito integrals as follows: $$d{X_t} = \tilde b\left( {t,{X_t}} \right)dt + \sigma \left( {t,{X_t}} \right)d{U_t},$$ where $${\tilde b_i}\left( {t,x} \right) = {b_i}\left( {t,x} \right) + \frac{1}{2}\sum\limits_{j = 1}^p {\sum\limits_{k = 1}^n {\frac{{\partial {\sigma _{ij}}}}{{\partial {x_k}}}} } {\sigma _{kj}};1in.$$ (6.1.3)
In this chapter we study some other important topics in diffusion theory and related areas. Some of these topics are not strictly necessary for the remaining chapters, but they are all central in the theory of stochastic analysis and essential for further applications. The following topics will be treated: 8.1 Kolmogorov’s backward equation. The resolvent. 8.2 The Feynman-Kac formula. Killing. 8.3 The martingale problem. 8.4 When is an Ito process a diffusion? 8.5 Random time change. 8.6 The Girsanov formula.
Suppose that the state of a system at time t is described by a stochastic integral Xt of the form $$d{X_t} = dX_t^u = b(t,{X_t},u)dt + \sigma (t,{X_t},u)d{B_t}$$ (11.1) where Xt,b ∈ ℝn×m, σ ∈ ℝn×m and Bt is m-dimensional Brownian motion. Here u ∈ Rk is a parameter whose value we can choose at any instant in order to control the process Xt · Thus u=u(t,ω) is a stochastic process. Since our decision at time t must be based upon what has happened up to time t, the function ω→u(t,ω) must (at least) be measurable wrt. F t, i.e. the process ut must be F t-adapted. Thus the right hand side of (11.1) is well-defined as a stochastic integral, under suitable assumptions on b and σ. At the moment we will not specify the conditions on b and a further, but simply assume that the process Xt satisfying (11.1) exists. See further comments on this in the end of this chapter.