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Stochastic control - Wikipedia
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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Stochastic_programming" title="Stochastic programming">Stochastic programming</a></div> <p><b>Stochastic control</b> or <b>stochastic <a href="/wiki/Optimal_control" title="Optimal control">optimal control</a></b> is a sub field of <a href="/wiki/Control_theory" title="Control theory">control theory</a> that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. The system designer assumes, in a <a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a>-driven fashion, that random noise with known <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> affects the evolution and observation of the state variables. Stochastic control aims to design the time path of the controlled variables that performs the desired control task with minimum cost, somehow defined, despite the presence of this noise.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The context may be either <a href="/wiki/Discrete_time" class="mw-redirect" title="Discrete time">discrete time</a> or <a href="/wiki/Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Certainty_equivalence">Certainty equivalence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=1" title="Edit section: Certainty equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An extremely well-studied formulation in stochastic control is that of <a href="/wiki/Linear_quadratic_Gaussian_control" class="mw-redirect" title="Linear quadratic Gaussian control">linear quadratic Gaussian control</a>. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. A basic result for discrete-time centralized systems with only additive uncertainty is the <b>certainty equivalence property</b>:<sup id="cite_ref-Chow_2-0" class="reference"><a href="#cite_note-Chow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> that the optimal control solution in this case is the same as would be obtained in the absence of the additive disturbances. This property is applicable to all centralized systems with linear equations of evolution, quadratic cost function, and noise entering the model only additively; the quadratic assumption allows for the optimal control laws, which follow the certainty-equivalence property, to be linear functions of the observations of the controllers. </p><p>Any deviation from the above assumptions—a nonlinear state equation, a non-quadratic objective function, <a href="/wiki/Multiplier_uncertainty" title="Multiplier uncertainty">noise in the multiplicative parameters</a> of the model, or decentralization of control—causes the certainty equivalence property not to hold. For example, its failure to hold for decentralized control was demonstrated in <a href="/wiki/Witsenhausen%27s_counterexample" title="Witsenhausen's counterexample">Witsenhausen's counterexample</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Discrete_time">Discrete time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=2" title="Edit section: Discrete time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a discrete-time context, the decision-maker observes the state variable, possibly with observational noise, in each time period. The objective may be to optimize the sum of expected values of a nonlinear (possibly quadratic) objective function over all the time periods from the present to the final period of concern, or to optimize the value of the objective function as of the final period only. At each time period new observations are made, and the control variables are to be adjusted optimally. Finding the optimal solution for the present time may involve iterating a <a href="/wiki/Linear-quadratic-Gaussian_control#Discrete_time" class="mw-redirect" title="Linear-quadratic-Gaussian control">matrix Riccati equation</a> backwards in time from the last period to the present period. </p><p>In the discrete-time case with uncertainty about the parameter values in the transition matrix (giving the effect of current values of the state variables on their own evolution) and/or the control response matrix of the state equation, but still with a linear state equation and quadratic objective function, a Riccati equation can still be obtained for iterating backward to each period's solution even though certainty equivalence does not apply.<sup id="cite_ref-Chow_2-1" class="reference"><a href="#cite_note-Chow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup>ch.13</sup><sup id="cite_ref-Turnovsky_3-0" class="reference"><a href="#cite_note-Turnovsky-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The discrete-time case of a non-quadratic loss function but only additive disturbances can also be handled, albeit with more complications.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=3" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A typical specification of the discrete-time stochastic linear quadratic control problem is to minimize<sup id="cite_ref-Chow_2-2" class="reference"><a href="#cite_note-Chow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: ch. 13,">: ch. 13,  </span></sup><sup id="cite_ref-Turnovsky_3-1" class="reference"><a href="#cite_note-Turnovsky-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} _{1}\sum _{t=1}^{S}\left[y_{t}^{\mathsf {T}}Qy_{t}+u_{t}^{\mathsf {T}}Ru_{t}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mi>Q</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mi>R</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} _{1}\sum _{t=1}^{S}\left[y_{t}^{\mathsf {T}}Qy_{t}+u_{t}^{\mathsf {T}}Ru_{t}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0586342abbb80c5db2be89418c1741e7af4e93ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.462ex; height:7.343ex;" alt="{\displaystyle \mathrm {E} _{1}\sum _{t=1}^{S}\left[y_{t}^{\mathsf {T}}Qy_{t}+u_{t}^{\mathsf {T}}Ru_{t}\right]}"></span></dd></dl> <p>where E<sub>1</sub> is the <a href="/wiki/Expected_value" title="Expected value">expected value</a> operator conditional on <i>y</i><sub>0</sub>, superscript T indicates a <a href="/wiki/Matrix_transpose" class="mw-redirect" title="Matrix transpose">matrix transpose</a>, and <i>S</i> is the time horizon, subject to the state equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{t}=A_{t}y_{t-1}+B_{t}u_{t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{t}=A_{t}y_{t-1}+B_{t}u_{t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60874dfb04f01e84c45266f1906c5a29934e9a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.931ex; height:2.509ex;" alt="{\displaystyle y_{t}=A_{t}y_{t-1}+B_{t}u_{t},}"></span></dd></dl> <p>where <i>y</i> is an <i>n</i> × 1 vector of observable state variables, <i>u</i> is a <i>k</i> × 1 vector of control variables, <i>A</i><sub><i>t</i></sub> is the time <i>t</i> realization of the <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">stochastic <i>n</i> × <i>n</i> state transition matrix</a>, <i>B</i><sub><i>t</i></sub> is the time <i>t</i> realization of the stochastic <i>n</i> × <i>k</i> matrix of control multipliers, and <i>Q</i> (<i>n</i> × <i>n</i>) and <i>R</i> (<i>k</i> × <i>k</i>) are known symmetric positive definite cost matrices. We assume that each element of <i>A</i> and <i>B</i> is jointly <a href="/wiki/Iid" class="mw-redirect" title="Iid">independently and identically</a> distributed through time, so the expected value operations need not be time-conditional. </p><p><a href="/wiki/Bellman_equation" title="Bellman equation">Induction backwards in time</a> can be used to obtain the optimal control solution at each time,<sup id="cite_ref-Chow_2-3" class="reference"><a href="#cite_note-Chow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: ch. 13">: ch. 13 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{t}^{*}=-\left[\mathrm {E} \left(B^{\mathsf {T}}X_{t}B+R\right)\right]^{-1}\mathrm {E} \left(B^{\mathsf {T}}X_{t}A\right)y_{t-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo>=</mo> <mo>−<!-- − --></mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>B</mi> <mo>+</mo> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{t}^{*}=-\left[\mathrm {E} \left(B^{\mathsf {T}}X_{t}B+R\right)\right]^{-1}\mathrm {E} \left(B^{\mathsf {T}}X_{t}A\right)y_{t-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e24e10c2fcf90a05c9e80f309225bd3995ee4e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.705ex; height:3.843ex;" alt="{\displaystyle u_{t}^{*}=-\left[\mathrm {E} \left(B^{\mathsf {T}}X_{t}B+R\right)\right]^{-1}\mathrm {E} \left(B^{\mathsf {T}}X_{t}A\right)y_{t-1},}"></span></dd></dl> <p>with the symmetric positive definite cost-to-go matrix <i>X</i> evolving backwards in time from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{S}=Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{S}=Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b1ab7bc3250dec3463c694c2ec0111ca6d6ed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle X_{S}=Q}"></span> according to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t-1}=Q+\mathrm {E} \left[A^{\mathsf {T}}X_{t}A\right]-\mathrm {E} \left[A^{\mathsf {T}}X_{t}B\right]\left[\mathrm {E} (B^{\mathsf {T}}X_{t}B+R)\right]^{-1}\mathrm {E} \left(B^{\mathsf {T}}X_{t}A\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>Q</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>A</mi> </mrow> <mo>]</mo> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>B</mi> </mrow> <mo>]</mo> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>B</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t-1}=Q+\mathrm {E} \left[A^{\mathsf {T}}X_{t}A\right]-\mathrm {E} \left[A^{\mathsf {T}}X_{t}B\right]\left[\mathrm {E} (B^{\mathsf {T}}X_{t}B+R)\right]^{-1}\mathrm {E} \left(B^{\mathsf {T}}X_{t}A\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc47f7d4ef75d318c8d2e27b9de15fbbed71ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:71.512ex; height:3.843ex;" alt="{\displaystyle X_{t-1}=Q+\mathrm {E} \left[A^{\mathsf {T}}X_{t}A\right]-\mathrm {E} \left[A^{\mathsf {T}}X_{t}B\right]\left[\mathrm {E} (B^{\mathsf {T}}X_{t}B+R)\right]^{-1}\mathrm {E} \left(B^{\mathsf {T}}X_{t}A\right),}"></span></dd></dl> <p>which is known as the discrete-time dynamic Riccati equation of this problem. The only information needed regarding the unknown parameters in the <i>A</i> and <i>B</i> matrices is the expected value and variance of each element of each matrix and the covariances among elements of the same matrix and among elements across matrices. </p><p>The optimal control solution is unaffected if zero-mean, i.i.d. additive shocks also appear in the state equation, so long as they are uncorrelated with the parameters in the <i>A</i> and <i>B</i> matrices. But if they are so correlated, then the optimal control solution for each period contains an additional additive constant vector. If an additive constant vector appears in the state equation, then again the optimal control solution for each period contains an additional additive constant vector. </p><p>The steady-state characterization of <i>X</i> (if it exists), relevant for the infinite-horizon problem in which <i>S</i> goes to infinity, can be found by iterating the dynamic equation for <i>X</i> repeatedly until it converges; then <i>X</i> is characterized by removing the time subscripts from its dynamic equation. </p> <div class="mw-heading mw-heading2"><h2 id="Continuous_time">Continuous time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=4" title="Edit section: Continuous time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the model is in continuous time, the controller knows the state of the system at each instant of time. The objective is to maximize either an integral of, for example, a concave function of a state variable over a horizon from time zero (the present) to a terminal time <i>T</i>, or a concave function of a state variable at some future date <i>T</i>. As time evolves, new observations are continuously made and the control variables are continuously adjusted in optimal fashion. </p> <div class="mw-heading mw-heading2"><h2 id="Stochastic_model_predictive_control">Stochastic model predictive control</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=5" title="Edit section: Stochastic model predictive control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the literature, there are two types of MPCs for stochastic systems; Robust model predictive control and Stochastic Model Predictive Control (SMPC). Robust model predictive control is a more conservative method which considers the worst scenario in the optimization procedure. However, this method, similar to other robust controls, deteriorates the overall controller's performance and also is applicable only for systems with bounded uncertainties. The alternative method, SMPC, considers soft constraints which limit the risk of violation by a probabilistic inequality.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_finance">In finance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=6" title="Edit section: In finance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a continuous time approach in a <a href="/wiki/Finance" title="Finance">finance</a> context, the state variable in the stochastic differential equation is usually wealth or net worth, and the controls are the shares placed at each time in the various assets. Given the <a href="/wiki/Asset_allocation" title="Asset allocation">asset allocation</a> chosen at any time, the determinants of the change in wealth are usually the stochastic returns to assets and the interest rate on the risk-free asset. The field of stochastic control has developed greatly since the 1970s, particularly in its applications to finance. Robert Merton used stochastic control to study <a href="/wiki/Optimal_portfolio" class="mw-redirect" title="Optimal portfolio">optimal portfolios</a> of safe and risky assets.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Merton%27s_portfolio_problem" title="Merton's portfolio problem">His work</a> and that of <a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a> changed the nature of the <a href="/wiki/Finance" title="Finance">finance</a> literature. Influential mathematical textbook treatments were by <a href="/wiki/Wendell_Fleming" title="Wendell Fleming">Fleming</a> and <a href="/w/index.php?title=Raymond_Rishel&action=edit&redlink=1" class="new" title="Raymond Rishel (page does not exist)">Rishel</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and by Fleming and <a href="/wiki/Halil_Mete_Soner" title="Halil Mete Soner">Soner</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> These techniques were applied by <a href="/w/index.php?title=Jerome_Stein&action=edit&redlink=1" class="new" title="Jerome Stein (page does not exist)">Stein</a> to the <a href="/wiki/Financial_crisis_of_2007%E2%80%9308" class="mw-redirect" title="Financial crisis of 2007–08">financial crisis of 2007–08</a>.<sup id="cite_ref-stein_10-0" class="reference"><a href="#cite_note-stein-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The maximization, say of the expected logarithm of net worth at a terminal date <i>T</i>, is subject to stochastic processes on the components of wealth.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In this case, in continuous time <a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's equation</a> is the main tool of analysis. In the case where the maximization is an integral of a concave function of utility over an horizon (0,<i>T</i>), dynamic programming is used. There is no certainty equivalence as in the older literature, because the coefficients of the control variables—that is, the returns received by the chosen shares of assets—are stochastic. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Backward_stochastic_differential_equation" title="Backward stochastic differential equation">Backward stochastic differential equation</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Multiplier_uncertainty" title="Multiplier uncertainty">Multiplier uncertainty</a></li> <li><a href="/wiki/Stochastic_scheduling" title="Stochastic scheduling">Stochastic scheduling</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://answerphone/topic/stochastic-control-theory?cat=technology">Definition from Answers.com</a></span> </li> <li id="cite_note-Chow-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Chow_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Chow_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Chow_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Chow_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFChow1976" class="citation book cs1"><a href="/wiki/Gregory_Chow" title="Gregory Chow">Chow, Gregory P.</a> (1976). <i>Analysis and Control of Dynamic Economic Systems</i>. New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-15616-7" title="Special:BookSources/0-471-15616-7"><bdi>0-471-15616-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+and+Control+of+Dynamic+Economic+Systems&rft.place=New+York&rft.pub=Wiley&rft.date=1976&rft.isbn=0-471-15616-7&rft.aulast=Chow&rft.aufirst=Gregory+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-Turnovsky-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Turnovsky_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Turnovsky_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTurnovsky1976" class="citation journal cs1"><a href="/wiki/Stephen_J._Turnovsky" title="Stephen J. Turnovsky">Turnovsky, Stephen</a> (1976). "Optimal Stabilization Policies for Stochastic Linear Systems: The Case of Correlated Multiplicative and Additive disturbances". <i><a href="/wiki/Review_of_Economic_Studies" class="mw-redirect" title="Review of Economic Studies">Review of Economic Studies</a></i>. <b>43</b> (1): <span class="nowrap">191–</span>94. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2296614">10.2307/2296614</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2296614">2296614</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Review+of+Economic+Studies&rft.atitle=Optimal+Stabilization+Policies+for+Stochastic+Linear+Systems%3A+The+Case+of+Correlated+Multiplicative+and+Additive+disturbances&rft.volume=43&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E191-%3C%2Fspan%3E94&rft.date=1976&rft_id=info%3Adoi%2F10.2307%2F2296614&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2296614%23id-name%3DJSTOR&rft.aulast=Turnovsky&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMitchell1990" class="citation journal cs1">Mitchell, Douglas W. (1990). "Tractable Risk Sensitive Control Based on Approximate Expected Utility". <i>Economic Modelling</i>. <b>7</b> (2): <span class="nowrap">161–</span>164. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0264-9993%2890%2990018-Y">10.1016/0264-9993(90)90018-Y</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Economic+Modelling&rft.atitle=Tractable+Risk+Sensitive+Control+Based+on+Approximate+Expected+Utility&rft.volume=7&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E161-%3C%2Fspan%3E164&rft.date=1990&rft_id=info%3Adoi%2F10.1016%2F0264-9993%2890%2990018-Y&rft.aulast=Mitchell&rft.aufirst=Douglas+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTurnovsky1974" class="citation journal cs1">Turnovsky, Stephen (1974). "The stability properties of optimal economic policies". <i><a href="/wiki/American_Economic_Review" title="American Economic Review">American Economic Review</a></i>. <b>64</b> (1): <span class="nowrap">136–</span>148. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1814888">1814888</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Economic+Review&rft.atitle=The+stability+properties+of+optimal+economic+policies&rft.volume=64&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E136-%3C%2Fspan%3E148&rft.date=1974&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1814888%23id-name%3DJSTOR&rft.aulast=Turnovsky&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHashemianArmaou2017" class="citation journal cs1">Hashemian; Armaou (2017). "Stochastic MPC Design for a Two-Component Granulation Process". <i>IEEE Proceedings</i>: <span class="nowrap">4386–</span>4391. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1704.04710">1704.04710</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017arXiv170404710H">2017arXiv170404710H</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Proceedings&rft.atitle=Stochastic+MPC+Design+for+a+Two-Component+Granulation+Process&rft.pages=%3Cspan+class%3D%22nowrap%22%3E4386-%3C%2Fspan%3E4391&rft.date=2017&rft_id=info%3Aarxiv%2F1704.04710&rft_id=info%3Abibcode%2F2017arXiv170404710H&rft.au=Hashemian&rft.au=Armaou&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMerton1990" class="citation book cs1">Merton, Robert (1990). <i>Continuous Time Finance</i>. Blackwell.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Continuous+Time+Finance&rft.pub=Blackwell&rft.date=1990&rft.aulast=Merton&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlemingRishel1975" class="citation book cs1">Fleming, W.; Rishel, R. (1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kUrvAAAAMAAJ"><i>Deterministic and Stochastic Optimal Control</i></a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90155-8" title="Special:BookSources/0-387-90155-8"><bdi>0-387-90155-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deterministic+and+Stochastic+Optimal+Control&rft.date=1975&rft.isbn=0-387-90155-8&rft.aulast=Fleming&rft.aufirst=W.&rft.au=Rishel%2C+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkUrvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlemingSoner2006" class="citation book cs1">Fleming, W.; Soner, M. (2006). <i>Controlled Markov Processes and Viscosity Solutions</i>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Controlled+Markov+Processes+and+Viscosity+Solutions&rft.pub=Springer&rft.date=2006&rft.aulast=Fleming&rft.aufirst=W.&rft.au=Soner%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-stein-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-stein_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStein2012" class="citation book cs1">Stein, J. L. (2012). <i>Stochastic Optimal Control and the US Financial Crisis</i>. Springer-Science.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Optimal+Control+and+the+US+Financial+Crisis&rft.pub=Springer-Science&rft.date=2012&rft.aulast=Stein&rft.aufirst=J.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarreiro-GomezTembine2019" class="citation journal cs1">Barreiro-Gomez, J.; Tembine, H. (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FACCESS.2019.2917517">"Blockchain Token Economics: A Mean-Field-Type Game Perspective"</a>. <i>IEEE Access</i>. <b>7</b>: <span class="nowrap">64603–</span>64613. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FACCESS.2019.2917517">10.1109/ACCESS.2019.2917517</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2169-3536">2169-3536</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Access&rft.atitle=Blockchain+Token+Economics%3A+A+Mean-Field-Type+Game+Perspective&rft.volume=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E64603-%3C%2Fspan%3E64613&rft.date=2019&rft_id=info%3Adoi%2F10.1109%2FACCESS.2019.2917517&rft.issn=2169-3536&rft.aulast=Barreiro-Gomez&rft.aufirst=J.&rft.au=Tembine%2C+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1109%252FACCESS.2019.2917517&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stochastic_control&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDixit1991" class="citation journal cs1"><a href="/wiki/Avinash_Dixit" title="Avinash Dixit">Dixit, Avinash</a> (1991). "A Simplified Treatment of the Theory of Optimal Regulation of Brownian Motion". <i>Journal of Economic Dynamics and Control</i>. <b>15</b> (4): <span class="nowrap">657–</span>673. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0165-1889%2891%2990037-2">10.1016/0165-1889(91)90037-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Economic+Dynamics+and+Control&rft.atitle=A+Simplified+Treatment+of+the+Theory+of+Optimal+Regulation+of+Brownian+Motion&rft.volume=15&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E657-%3C%2Fspan%3E673&rft.date=1991&rft_id=info%3Adoi%2F10.1016%2F0165-1889%2891%2990037-2&rft.aulast=Dixit&rft.aufirst=Avinash&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYongZhou1999" class="citation book cs1">Yong, Jiongmin; Zhou, Xun Yu (1999). <i>Stochastic Controls : Hamiltonian Systems and HJB Equations</i>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98723-1" title="Special:BookSources/0-387-98723-1"><bdi>0-387-98723-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Controls+%3A+Hamiltonian+Systems+and+HJB+Equations&rft.place=New+York&rft.pub=Springer&rft.date=1999&rft.isbn=0-387-98723-1&rft.aulast=Yong&rft.aufirst=Jiongmin&rft.au=Zhou%2C+Xun+Yu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+control" class="Z3988"></span></li></ul> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐65b64b4b74‐lwfdk Cached time: 20250219132750 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.307 seconds Real time usage: 0.420 seconds Preprocessor visited node count: 1331/1000000 Post‐expand include size: 22588/2097152 bytes Template argument size: 715/2097152 bytes Highest expansion depth: 16/100 Expensive parser function count: 2/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 41164/5000000 bytes Lua time usage: 0.190/10.000 seconds Lua memory usage: 5656223/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 325.134 1 -total 56.69% 184.333 1 Template:Reflist 39.87% 129.616 6 Template:Cite_book 20.87% 67.857 1 Template:Short_description 14.69% 47.774 6 Template:Cite_journal 12.88% 41.875 2 Template:Pagetype 9.00% 29.252 2 Template:Rp 7.33% 23.845 2 Template:R/superscript 6.25% 20.309 1 Template:See_also 4.42% 14.370 3 Template:Main_other --> <!-- Saved in parser cache with key enwiki:pcache:17058216:|#|:idhash:canonical and timestamp 20250219132750 and revision id 1142603208. 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