Applied Stochastic Control of Jump Diffusions by Bernt Øksendal, Agnès Sulem

By Bernt Øksendal, Agnès Sulem

The major function of the booklet is to offer a rigorous, but in general nontechnical, creation to an important and beneficial answer tools of varied kinds of stochastic regulate difficulties for bounce diffusions and its functions. the kinds of keep watch over difficulties lined comprise classical stochastic keep watch over, optimum preventing, impulse keep an eye on and singular keep an eye on. either the dynamic programming strategy and the utmost precept technique are mentioned, in addition to the relation among them. Corresponding verification theorems regarding the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There also are chapters at the viscosity answer formula and numerical tools. The textual content emphasises purposes, in most cases to finance. all of the major effects are illustrated through examples and routines appear at the top of every bankruptcy with whole strategies. it will aid the reader comprehend the idea and spot easy methods to practice it. The e-book assumes a few simple wisdom of stochastic research, degree concept and partial differential equations.

Show description

Read or Download Applied Stochastic Control of Jump Diffusions PDF

Similar probability books

Level crossing methods in stochastic models

On the grounds that its inception in 1974, the extent crossing procedure for interpreting a wide type of stochastic versions has turn into more and more renowned between researchers. This quantity lines the evolution of point crossing idea for acquiring chance distributions of country variables and demonstrates resolution equipment in various stochastic versions together with: queues, inventories, dams, renewal versions, counter types, pharmacokinetics, and the typical sciences.

Structural aspects in the theory of probability

The booklet is conceived as a textual content accompanying the conventional graduate classes on likelihood idea. an incredible function of this enlarged model is the emphasis on algebraic-topological facets resulting in a much wider and deeper knowing of uncomplicated theorems reminiscent of these at the constitution of constant convolution semigroups and the corresponding methods with self sustaining increments.

Steps Towards a Unified Basis for Scientific Models and Methods

Tradition, in reality, additionally performs an immense function in technological know-how that is, in line with se, a large number of alternative cultures. The publication makes an attempt to construct a bridge throughout 3 cultures: mathematical information, quantum conception and chemometrical equipment. after all, those 3 domain names shouldn't be taken as equals in any feel.

Additional resources for Applied Stochastic Control of Jump Diffusions

Example text

3) gets the form H(t, x, u, p, q, r) = {ρt x + (µt − ρt )u}p + σt uq + u γ(t, z)r(t, z)ν(dz) . 4) are ⎧ ⎪ ⎨dp(t) = −ρt p(t)dt + q(t)dB(t) + ⎪ ⎩ r(t− , z)N (dt, dz) ; t

12) holds and hence (ii) is proved. 34 2 Optimal Stopping of Jump Diffusions (iii): In this case ∂D = {(s, x); x = x∗ } and hence ∞ E ∞ P x [X(t) = x∗ ]dt = 0 . X∂D (Y (t))dt = y 0 0 (iv) and (v) are trivial. 9). s. 5) is given by t X(t) = x exp α− 1 2 2β −γ zν(dz) t + ln(1 + γz)N (dt, dz) + βB(t) . s. s. s. 2 Applications and examples e−ρτ X(τ ) 35 is uniformly integrable. τ ∈T For this to hold it suffices that there exists a constant K such that E[e−2ρτ X 2 (τ )] ≤ K for all τ ∈ T . 16) holds and hence (xi) holds also.

If we sell the asset at time s + τ we get the expected discounted net payoff J τ (s, x) := E s,x e−ρ(s+τ ) (X(τ ) − a)X{τ <∞} where ρ > 0 (the discounting exponent) and a > 0 (the transaction cost) are constants. We seek the value function Φ(s, x) and an optimal stopping time τ ∗ ≤ ∞ such that ∗ Φ(s, x) = sup J τ (s, x) = J τ (s, x) . 2 to solve this problem as follows: Put S = R × (0, ∞) and s+t Y (t) = ; t≥0 X(t) Then ⎡ ⎤ ⎡ ⎤ ⎡ 0 ⎥ ⎥ ⎥; z N (dt, dz)⎦ ⎢ ⎦ dt+⎣ ⎦ dB(t)+⎢ dY (t) = ⎣ ⎢ ⎣γX(t− ) αX(t) βX(t) 1 0 ⎤ R ⎡ ⎤ s Y (0) = ⎣ ⎦ x 32 2 Optimal Stopping of Jump Diffusions and the generator A of Y (t) is Aφ(s, x) = ∂φ ∂2φ ∂φ +αx + 21 β 2 x2 2 + ∂s ∂x ∂x φ(s, x+γxz)−φ(s, x)−γxz R ∂φ ν(dz) .

Download PDF sample

Rated 4.28 of 5 – based on 39 votes