TypeScript的には型が無くても呼び出しすることは可能です。
しかしそれをやってしまうと何でもアリになってしまうため、静的型付けをしていたメリットが無くなってしまいます。
interface JQuery {
ajaxComplete(handler: (event: JQueryEventObject, XMLHttpRequest: XMLHttpRequest, ajaxOptions: any) => any): JQuery;
ajaxError(handler: (event: JQueryEventObject, jqXHR: JQueryXHR, ajaxSettings: JQueryAjaxSettings, thrownError: any) => any): JQuery;
ajaxSend(handler: (event: JQueryEventObject, jqXHR: JQueryXHR, ajaxOptions: JQueryAjaxSettings) => any): JQuery;
ajaxStart(handler: () => any): JQuery;
ajaxStop(handler: () => any): JQuery;
ajaxSuccess(handler: (event: JQueryEventObject, XMLHttpRequest: XMLHttpRequest, ajaxOptions: JQueryAjaxSettings) => any): JQuery;
load(url: string, data?: string|Object, complete?: (responseText: string, textStatus: string, XMLHttpRequest: XMLHttpRequest) => any): JQuery;
serialize(): string;
serializeArray(): JQuerySerializeArrayElement[];
addClass(className: string): JQuery;
addClass(func: (index: number, className: string) => string): JQuery;
addBack(selector?: string): JQuery;
attr(attributeName: string): string;
attr(attributeName: string, value: string|number): JQuery;
attr(attributeName: string, func: (index: number, attr: string) => string|number): JQuery;
attr(attributes: Object): JQuery;
hasClass(className: string): boolean;
...snip, snip...
}
declare module "jquery" {
export = $;
}
declare var jQuery: JQueryStatic;
declare var $: JQueryStatic;
https://github.com/borisyankov/DefinitelyTyped/blob/master/jquery/jquery.d.ts