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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.
... Thus, by considering an optimal control of Ito-type processes which satisfy the stochastic differential equation(SDE) w.r.t some Wiener process, our goal is to choose the investment control strategy (i.e. dynamic portfolio strategy) to maximize the expected utility of wealth at some future time τ [18] [19]. ...
... The investor needs to monitor his/her wealth, and therefore, the fraction t θ of the wealth invested in stocks is set to be the control of the system at time t [19]. Thus, here comes ...
... where τ  is the first exit time from the region [19]. ...
... Stochasicity has recently been introduced into the mechanics literature from a geometric variational perspective [1,2]. In these studies, the configuration space of the mechanical system is a Lie group on which the Euler-Poincaré-Stratonovich (EPS) (stochastic) equations of motion are derived; as well as their associated Fokker-Plank equations [16]. These geometric variational approaches have been used to study noisy extensions of a rigid body and the heavy top [1,2], stochasticity in fluid dynamics [12], and noisy energy-Casimir methods have been applied to study steadystate stability [3]. ...
... The authors consider the case studies of a planar rolling ball subject to the noisy nonholonomic constraints: (i) rolling on a randomly moving plate, (ii) random 'skipping' i.e., instantaneous jumps in the point of contact, and (iii) rolling on a rough surface. The noise variables evolve according to a Stratonovich SDE [16] which in some circumstances may be interpreted as noisy coupling forces between the point of contact and the contact surface. In [13], the error in observations of transported velocities is formulated as a set of noisy nonholonomic constraints. ...
... The terms F and Σ represent the deterministic and stochastic coupling between the mechanical system and the contact surface and W is a Wiener process [16]. Take local coordinates q = (r a , ψ α ) such that the constraints are of the form ω(ψ, r, N t ) = dr a +A a α (ψ, r, N t )dψ α = 0, a = 1, . . . ...
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p>In this paper, we investigate the motion of wheeled mobile robots on rough terrains modelled as noisy nonholonomic constraints. Such constraints are the natural extension of ideal nonholonomic constraints when the Stratonovich process is directly introduced in the constraint equations. The resulting stochastic model can capture motion on rough surfaces, random skip/uncertainty in the wheel-ground point of contact, or stochastic motion of the surface. We study a differential robot with ideal noisy and affine noisy constraints, where each case models a certain aspect of motion on rough terrains. We then qualitatively investigate their corresponding stochastic dynamics through Monte-Carlo simulations. The proposed stochastic model for roving rough terrains has the potential to serve as the process model in model-based motion estimators relying on measurements from an interoceptive suite of sensors. The challenge will be dealing with the nonlinear appearance of the noise in the equations of motion. </p
... However, in reality, there are many unpredicted parameters and different types of uncertainties that have not been implemented in system (57). Nonetheless, from [46][47][48], we can consider a stochastic version of system (57) with one-dimensional standard Wiener process w(t) used to model the uncertainties of the form: E 1 dz = M(t)z(t)dt + L(t)u(t)dt + C(t)z(t)dw(t). ...
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Controllability is a basic problem in the study of stochastic generalized systems. Compared with ordinary stochastic systems, the structure of stochastic singular systems is more complex, and it is necessary to study the controllability of stochastic generalized systems in the context of different solutions. In this paper, the controllability of semilinear stochastic generalized systems was investigated by using a GE-evolution operator for integral and impulsive solutions in Hilbert spaces. Some sufficient and necessary conditions were obtained. Firstly, the existence and uniqueness of the integral solution of semilinear stochastic generalized systems were discussed using the GE-evolution operator theory and Banach fixed point theorem. The existence and uniqueness theorem of the integral solution was obtained. Secondly, the approximate controllability of semilinear stochastic generalized systems was studied in the case of the integral solution. Thirdly, the existence and uniqueness of the impulsive solution of semilinear stochastic generalized systems were considered, and some sufficient conditions were provided. Fourthly, the approximate controllability of semilinear stochastic generalized systems was studied for the impulsive solution. At last, the exact controllability of linear stochastic systems was studied in the case of the impulsive solution, with some necessary and sufficient conditions given. The obtained results have important theoretical and practical value for the study of controllability of semilinear stochastic generalized systems.
... To the best of our knowledge, SDE of the third-order with or without time-varying delays naturally appears in multiple applications, where deterministic models are perturbed by the white noise or its generalizations [13,23,27,28]. In most cases, SDEs are understood as a continuous time limit of the corresponding SDEs. ...
... Observe that the coefficients of the SDE (4) are scalar, this simplifies the calculations of the expectation and the second moment. In addition, we have a strong solution for each u k given by (see [15], [20] for instance) ...
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